These are my live-TeXed notes for Professor Raphael Beuzart-Plessis' topic course at Columbia, Spring 2019.

Any mistakes are the fault of the notetaker. Let me know if you notice any mistakes or have any comments!


TopOverview of the Gan-Gross-Prasad conjectures

TopHecke and Rankin-Selberg integrals

Let us start with the classical work of Hecke on the integral representation of $L$-functions of modular forms. Let $f\in S_k(\Gamma(1))$ be a cusp form of level $k$, which has a Fourier expansion $$f=\sum_{n\ge1} a_nq^n.$$ Let $$L(s, f)=\sum \frac{a_n}{n^s}, \quad \Lambda(s,f)=(2\pi)^{-s}\Gamma(s)L(s,f)$$ be its $L$-function and completed $L$-function. Hecke showed that the completed $L$-function is equal to the Mellin transform of the modular form $$\Lambda(s,f)=\int_0^\infty f(iy)y^s d^\times y,$$ and use it to show the analytic continuation and functional equation of $\Lambda(s,f)$. Evaluating at the center of functional equation $s=k/2$ (which lies outside of the range of convergence of $\Lambda(s,f)$), we obtain a central value formula $$\Lambda(k/2, f)=\int_0^\infty f(iy)y^{k/2}d^\times y.$$ This is an identity which the Gan-Gross-Prasad conjecture aims to generalize.

Let us reformulate Hecke's central value formula in the adelic setting. The modular form $f$ gives rise to a vector $\phi_f\in \pi$ in a cuspidal automorphic representation $\pi$ of $G(\mathbb{A})$, where $G=\PGL_2$. Assume $f$ is normalized (i.e., $a_1=1$) and a Hecke eigenform. Then the central value formula can be rewritten as $$\frac{1}{2}\cdot L(1/2, \pi)=\int_{\mathbb{A}^\times/\mathbb{Q}^\times}\phi_f \left(\begin{smallmatrix}t & \\ & 1\end{smallmatrix}\right) d^\times t.$$ (Here the extra 1/2 comes from normalization of measures). The RHS is an automorphic period $\mathcal{P}_A(\phi_f)$ along the subgroup $A=\mathbb{G}_m\hookrightarrow G, \ t\mapsto \left(\begin{smallmatrix}t & \\ & 1\end{smallmatrix}\right)$ (a real number since $f$ is a normalized eigenform).

Recall also that Rankin-Selberg expressed the Petersson inner product $$\langle f, f\rangle=\int_{\Gamma(1)\backslash \mathcal{H}}|f(z)|^2 y^k\frac{dxdy}{y^2},$$ in terms of the Rankin-Selberg $L$-function $$\langle f, f\rangle=2\pi^{k-1}\res_{s=k}\Lambda(s,f\times f)$$ The RHS is equal to an adjoint $L$-value at the edge of critical strip $$2^{-k}L(1, \pi, \Ad),$$ while the LHS is also equal to $\frac{\pi}{6}$ (again due to normalization of measures) times $$\langle\phi_f,\phi_f\rangle=\int_{[G]}|\phi_f(g)|^2d_\mathrm{tam}g.$$

Squaring Hecke's central value formula and dividing the Rankin-Selberg identity, we obtain $$\frac{\mathcal{P}_A(\phi_f)^2}{\langle\phi_f,\phi_f\rangle}=2^{k-2}\xi(2)\frac{L(1/2,\pi)^2}{L(1,\pi,\Ad)},$$ where $\xi(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$ is the completed Riemann zeta function.

This new identity generalizes to any vector $\phi\in\pi$ of any cuspidal automorphic representation of $G(\mathbb{A})$ over any number field $k$. Let $\phi=\otimes_v \phi_v\in \pi=\otimes_v' \pi_v$. Then $$\frac{|\mathcal{P}_A(\phi)|^2}{\langle\phi,\phi\rangle}=\frac{\zeta_k^S(2)}{2}\frac{L^S(1/2,\pi)^2}{\res_{s=1}\zeta_k^S(s) L^S(1,\pi,\Ad)}\cdot \prod_{v\in S}\alpha_v(\phi_v,\phi_v),$$ for $S$ a sufficiently large finite set of places including the archimedean places, where the local period $$\alpha_v(\phi_v,\phi_v)=\frac{\int_{A(k_v)}\langle\pi_v(a_v)(\phi_v), \phi_v\rangle da_v}{\langle \phi_v,\phi_v\rangle_v}.$$ Here we choose the Tamagawa measures on $A(\mathbb{A})$ and $G(\mathbb{A})$, and the local measures are chosen such that $$\prod_v da_v=d_\mathrm{tam} a$$ and the local period $$\alpha_v(\phi_v,\phi_v)=\frac{\zeta_v(2)}{\zeta_v(1)}\frac{L(1/2, \pi_v)^2}{L(1, \pi_v,\Ad)}$$ for almost all $v$'s. One can show that $\alpha_v$ is not identically zero and $L(s,\pi, \Ad)$ has no pole or zero at $s=1$, hence the new identity implies the equivalence that

there exists $\phi\in \pi$ such that $\mathcal{P}_A(\phi)\ne0$ $\Longleftrightarrow$ $L(1/2, \pi)\ne0$.

TopWaldspurger's formula

Waldspurger proved a remarkable generalization by replacing $A$ with any nontrivial torus $T$ in $G$. Such a torus is isomorphic to $\Res_{k'/k}\mathbb{G}_m/\mathbb{G}_m$ for a quadratic extension $k'/k$ (and the embedding $A\hookrightarrow G$ is unique up to conjugation). Let $\eta: \mathbb{A}^\times/k^\times\rightarrow\{\pm1\}$ be the quadratic character associated to $k'/k$ by global class field theory.

Theorem 1 (Waldspurger) For every $\phi= \otimes_v \phi_v\in \pi$, we have $$\frac{|\mathcal{P}_T(\phi)|^2}{\langle\phi,\phi\rangle}=\frac{\zeta_k^S(2)}{4}\frac{L^S(1/2, \pi) L^S(1/2,\pi \times \eta)}{L^S(1,\eta)L^S(1,\pi,\Ad)}\prod_{v\in S} \alpha_v(\phi_v,\phi_v),$$ where the local periods are defined similarly via integration over $T_v$ instead of $A_v$.

Waldspurger's formula looks exactly the same as the split torus case, but it is much harder to prove: there is no direct relation between the toric period $\mathcal{P}_T(\phi)$ with integral representations of $L(s,\pi)L(s, \pi\times\eta)$ (i.e., only for the central value — no parameter $s$ to vary). Moreover, unlike the split torus case, the local periods can be identically zero: it turns out that

$\alpha_v\ne0$ $\Longleftrightarrow$ $\Hom_{T_v}(\pi_v, \mathbb{C})\ne0$.


$\mathcal{P}_T|_\pi\ne0$ $\Longleftrightarrow$ $L(1/2, \pi)L(1/2, \pi\times\eta)\ne0$ and $\Hom_{T_v}(\pi_v, \mathbb{C})\ne0$ for all $v$.

To obtain a relation only about the central values (i.e., to get rid of the nonvanishing condition of local periods), one considers all inner forms $G'{}=B^\times/k^\times$ of $G$, where $B/k$ is a quaternion algebra with an embedding $k'\hookrightarrow B$. We thus have an embedding $T\hookrightarrow G'$ and we can again consider $T$-period on $G'$.

Assume $\pi$ admits a Jacquet-Langlands lift $\pi'$ to $G'(\mathbb{A})$. In particular, $\pi_v'\cong \pi_v$ for all places $v$ where $B$ splits. Then Waldspurger's formula extends to this more general situation by replacing $(G,\pi)$ with $(G', \pi')$, and the local periods $\alpha_v$ by $\alpha_v'$ (while the $L$-functions stay the same).

The following nice theorem gives a criterion of $\Hom_{T_v}(\pi_v', \mathbb{C})\ne0$, in terms of local root numbers.

Theorem 2 (Saito, Tunnell) $\Hom_{T_v}(\pi_v', \mathbb{C})\ne0$ if and only if $$\varepsilon(\pi_v)\varepsilon(\pi_v\times \eta_v)=
   \eta_v(-1), & B \text{ splits at }v, \\
   -\eta_v(-1), & B \text{ ramified at }v.
$$ Moreover, $\dim \Hom_{T_v}(\pi'_v, \mathbb{C})\le1$.

As a consequence, if $B_v$ is a division algebra, then $$\dim \Hom_{T_v}(\pi_v',\mathbb{C})+\dim \Hom_{T_v}(\pi_v,\mathbb{C})=1.$$ Combining with the classification of quaternion algebra over $k$ and the properties of Jacquet-Langlands correspondence, we know that

there exists $(B, \pi')$ with $\Hom_{T_v}(\pi_v', \mathbb{C})\ne0$ for all $v$ $\Longleftrightarrow$ $\varepsilon(\pi)\varepsilon(\pi\times\eta)=1$.

Therefore Waldspurger's formula implies that

$L(1/2, \pi)L(1/2, \pi\times\eta)\ne0$ $\Longleftrightarrow$ there exists $(B, \pi')$ such that $\mathcal{P}_T|_{\pi'}\ne0$.

Moreover, if such $(B, \pi')$ exists, it is also unique. Since all the local periods are positive definite, it follows from Waldspurger's formula that $$L(1/2,\pi)L(1/2, \pi\times \eta)\ge0,$$ as predicted by the Riemann hypothesis.

TopGan-Gross-Prasad conjectures

One obvious generalization is to replace the pair $(G, A)$ by $(\GL_{n+1}, \GL_n)$ and it leads to the Rankin-Selberg type integral representation of $L$-functions on $\GL_{n+1}\times \GL_n$ due to Jacquet—Piatetski-Shapiro—Shalika. We will discuss this important result later.

The other generalization is by noticing the exceptional isomorphism $$T\cong \SO(W), \quad G'\cong \SO(V),$$ where $W$ is the 2-dimensional quadratic space $(k', N_{k'/k} )$ over $k$, and $V$ is the 3-dimensional quadratic space $(B^\mathrm{tr=0}, \mathrm{Nrd})$ over $k$. Similarly, we also have exceptional isomorphisms $$T\cong \UU(W),\quad G'\cong \mathrm{PU}(V),$$ where $W$ is the 1-dimensional $k'/k$-hermitian space $(k', N_{k'/k})$, and $V$ is the 2-dimensional $k'/k$-hermitian space $(B, \mathrm{Nrd})$. In view of these, Gan-Gross-Prasad proposed the generalization of the results of Waldspurger and Saito-Tunnell to any pair $(W,V)$ of quadratic or hermitian spaces with $W\subseteq V$ a non-degenerate hyperplane.

Let $H=\SO(W)$ (or $\UU(W)$) and $G=\SO(V)$ (or $\UU(V)$). Assume that $\pi_V$, $\pi_W$ are cuspidal automorphic representations of $H(\mathbb{A})$ and $G(\mathbb{A})$ of Ramanujan type.

Conjecture 1 (Gan-Gross-Prasad) TFAE:
  1. There exists $\phi_W \otimes \phi_V\in \pi_V \otimes \pi_W$ such that $\mathcal{P}_H(\phi_W \otimes \phi_V)\ne0$.
  2. $L(1/2, \pi_W\times \pi_V)\ne0$ and $\Hom_{H_v}(\pi_{W,v}\otimes \pi_{V,v},\mathbb{C})\ne0$ for all $v$.

Similarly, to state a version only about central $L$-values, we need the notion of pure inner forms. Let $W'$ be another quadratic or hermitian space with the same dimension (and discriminant in the quadratic case) as $W$. Let $H'{}=\SO(W')$ or $\UU(W')$ and $G'{}=\SO(V')$ or $\UU(V')$, where $V'{}=W' \oplus^{\perp} L$ and $V{}= W \oplus^{\perp} L$ are orthogonal direct sum with the same line $L$. In particular, $W_v\cong W_v'$ for almost all $v$.

Definition 1 Say $\pi_W$ and $\pi_{W}'$ are nearly equivalent if $\pi_{W,v}\cong\pi_{W,v}'$ for almost all places. Unlike the case of $\PGL_2$ this is a nontrivial equivalence relation.
Conjecture 2 (Gan-Gross-Prasad) TFAE:
  1. There exists $W'$ and $\pi_W',\pi_V'$ nearly equivalent to $\pi_W,\pi_V$ respectively such that $\mathcal{P}_{H'}|_{\pi_W' \otimes \pi_V'}$ is nonzero.
  2. $L(1/2, \pi_W\times \pi_V)\ne0$.
Conjecture 3 (Ichino-Ikeda, N. Harris) Assume $H=\UU(W)\hookrightarrow G=\UU(V)$. Assume further that $\pi_{W,v}$, $\pi_{V,v}$ is tempered for all $v$. Let $\phi=\otimes\phi_v\in \pi=\pi_W \otimes \pi_V$. Then we have an identity $$\frac{|\mathcal{P}_H(\phi)|^2}{\langle\phi,\phi\rangle}=2^{-\beta}\prod_{i=1}^{\dim V}L^S(i, \eta_{k'/k}^i)\frac{L^S(1/2, \pi_{W}\times \pi_V)}{L^S(1, \pi_V, \Ad)L^S(1, \pi_W, \Ad)} \prod_{v\in S}\alpha_v(\phi_v,\phi_v).$$ Here $\beta$ is an integer related to the $L$-packet/the size of nearly equivalence classes.
Remark 1
  1. Conjecturally, $\pi_v$ is tempered for all $v$ if and only if $\pi$ is of Ramanujan type.
  2. Sakellaridis-Venkatesh showed $\alpha_v\ne0\Longleftrightarrow\Hom_{H_v}(\pi_v, \mathbb{C})\ne0$.
  3. The local Gan-Gross-Prasad conjecture describes when $\Hom_{H_v}(\pi_v, \mathbb{C})\ne0$ in terms of the local root numbers of the local Langlands parameters for orthogonal or unitary groups.
  4. Known results for the global conjectures: orthogonal case $\dim W=2$ (Waldspurger), $\dim W=3$ (Ichino-Ikeda); unitary case (W. Zhang, H. Xue, Beuzart-Plessis).
  5. Known results for the local conjecture: For non-archimedean places: orthogonal case (Waldspurger); unitary case (Beuzart-Plessis). For archimedean places, partial results in the unitary case (Beuzart-Plessis, H. He).


TopBasics on automorphic forms

(Refs: Borel-Jacquet, Flath in the Corvallis volume. The material is standard and so my notes will be brief.)

Let $G$ be a connected reductive group over a number field $k$. Let $A_G\subseteq Z_G$ be the maximal split torus. Assume $[A_G\backslash G]$ has finite volume. Let $\omega: [A_G]\rightarrow \mathbb{C}^\times$ be a unitary character. We denote by $L^2([G],\omega)$ the space of measurable functions $f:[G]\rightarrow \mathbb{C}$ such that $f(ag)=\omega(a)f(g)$ for any $a\in A_G(\mathbb{A})$ and $$\int_{[A_G\backslash G]}|f(g)|^2 dg<\infty.$$ Let $L^2_\mathrm{cusp}([G],\omega)$ be the subspace of cuspidal functions (i.e. its integral along $[N]$ is zero for any $N $ the nilpotent radical of a proper parabolic subgroup of $G$). Then by a theorem of Gelfand and Piatetski-Shapiro, we have a decomposition $$L^2_\mathrm{cusp}([G], \omega)\cong \widehat\bigoplus_{\pi} m(\pi)\pi, \quad m(\pi)<\infty,$$ where $\pi$ runs over irreducible unitary cuspidal automorphic representations of $G(\mathbb{A})$.

We recall the definition of automorphic forms (smooth, $K_f$-finite, $\mathcal{Z}(\mathfrak{g}_\infty)$-finite, and of uniformly moderate growth functions on $[A_G\backslash G]$). Notice that we don't require $K_\infty$-finite condition (so we obtain a larger space of automorphic forms), which is more suited for the analytic theory. Let $\mathcal{A}(G,\omega)$ be the space of automorphic forms $\phi$ such that $\phi(ag)=\omega(a)\phi(g)$ for any $a\in [A_G]$. In particular, $G(\mathbb{A})$ acts on $\mathcal{A}(G, \omega)$ (not just a $(\mathfrak{g}_\infty, K_\infty)$-module at $\infty$). One can define a nontrivial topology on $\mathcal{A}(G,\omega)$ which makes it a locally Frechet space. This topology gives the global realization of the $(\mathfrak{g}_\infty, K_\infty)$-modules, and agrees with the Casselman-Wallach globalization in this case (see Bernstein-Krotz).

Let $\mathcal{A}_\mathrm{cusp}(G,\omega)\subseteq \mathcal{A}(G,\omega)$ be the subspace of cuspidal automorphic forms. Let $\mathcal{A}_2(G,\omega)=\mathcal{A}(G,\omega)\cap L^2([G],\omega)$. Since every cuspidal form is of rapid decay modulo center (i.e., $|\phi(g)|\ll_N ||g||^{-N}_{[A_G\backslash G]}$), we know that $$\mathcal{A}_\mathrm{cusp}(G,\omega)=\mathcal{A}(G,\omega)\cap L^2_\mathrm{cusp}([G],\omega),$$ which is dense in $L^2_\mathrm{cusp}([G],\omega)$. Moreover, $$\mathcal{A}_\mathrm{cusp}(G,\omega)= \oplus_\pi m(\pi) \pi^\infty,$$ where $\pi^\infty$ is the subspace of smooth vectors. In particular, any automorphic form generates a finite length representation under the right translation of $G(\mathbb{A})$.

Example 1
  1. The constant function $\mathbf{1}\in \mathcal{A}_2(G, \omega)$, but $\mathbf{1}\not\in\mathcal{A}_\mathrm{cusp}(G, \omega)$ unless $G$ has no proper parabolic subgroup (by Borel, Harish-Chandra, this last condition is equivalent to $[A_G\backslash G]$ is compact).
  2. For $k=\mathbb{Q}$, $G=\GL_2$, any $f\in S_k(\Gamma(1))$ gives $\phi_f\in\mathcal{A}_\mathrm{cusp}(G, \mathbf{1})$.

We define an irreducible cuspidal automorphic representation of $G(\mathbb{A})$ to be a topologically irreducible subrepresentation of $\mathcal{A}_\mathrm{cusp}(G,\omega)$, or equivalently an irreducible representation of the form $\pi^\infty$ where $\pi$ is a unitary cuspidal automorphic representation, together with an embedding $\pi^\infty\hookrightarrow \mathcal{A}_\mathrm{cusp}(G,\omega)$. From now on we will work with $\pi^\infty$ and forget about $\pi$, and write $\pi$ for $\pi^\infty$.

We have the factorization theorem: any irreducible cuspidal automorphic representation $\pi$ (or any smooth admissible irreducible representation) of $G(\mathbb{A})$, factorizes as a restricted tensor product $$\pi=\otimes_v' \pi_v.$$ At nonarchimedean places, smooth admissible means the usual notion and irreducible means algebraically irreducible. At archimedean places, apart from usual smooth admissible conditions, one needs the additional topological requirement as a Frechet representation of moderate growth (i.e., for any continuous semi-norm $p$ on $V$, there exists $N>0$ and a continuous semi-norm $q$ such that $p(\pi(g)e)\le ||g||^N q(e)$ for any $g\in G$ and $e\in V$), and irreducible means topologically irreducible. Under these conditions the globalization of Harish-Chandra modules is unique (Casselman-Wallach globalization).

We recalled the notion of unramified representations, the spherical Hecke algebra, and the Satake isomorphism (refs: Cartier and Borel in Corvallis).


We defined the $L$-group, Langlands parameters for unramified representations, and general Langlands $L$-functions. For $\GL_n$, we discussed Whittaker functions and Whittaker models, a sketch of the proof of the Fourier expansion of cusp forms (induction via the mirabolic subgroup), and the uniqueness of local Whittaker models.


For $\GL_n$, we discussed local Kirillov models (restriction of Whittaker models to the mirabolic), and proved that Kirillov model of a generic representation contains a fixed compactly induced module (Gelfand-Kazhdan for nonarchimedean fields, Jacquet-Shalika for $\mathbb{R}$, Kemarsky for $\mathbb{C}$) by induction. We discussed Piatetski-Shapiro's proof of the strong multiplicity one theorem for cusp forms (by the Fourier expansion and the previous property of Kirillov models). We discussed the integral representation of the Rankin-Selberg $L$-function for $\GL_n\times \GL_{n+1}$ (Jacquet—Piatetski-Shapiro—Shalika, generalizing Hecke's integral).

As a consequence of the integral representation, we can express the central $L$-value $L(1/2, \pi\times \sigma)$ for cuspidal automorphic representations on $\GL_n\times \GL_{n+1}$ as an automorphic period over $\GL_n$. Moreover, the local periods are not identically zero (since it has an interpretation of a local zeta integral divided by the local $L$-value, and the local $L$-function is the common gcd of local zeta integrals), and thus the central $L$-value is nonvanishing if and only if an automorphic period is nonvanishing. However, we cannot quite formulate this nonvanishing criterion using the partial $L$-values like we did for $\GL_2$, due to the fact that we do not yet know that the local $L$-function $L(s, \pi_v\times \sigma_v)$ does not have a pole at $s=1/2$. Notice the generalized Ramanujan conjecture implies that $\pi_v,\sigma_v$ are tempered, and in this tempered case the estimates on Whittaker functions imply that $L(s, \pi_v\times \sigma_v)$ is holomorphic for $\Re(s)>0$. However, current bounds on generalized Ramanujan conjecture only implies that $L(s,\pi_v\times \sigma_v)$ is holomorphic for $\Re(s)>1-\delta$ for small $\delta$.


TopThe global Jacquet-Rallis relative trace formula


  • Jacquet-Rallis, On the Gross-Prasad conjecture for unitary groups.
  • W. Zhang, Fourier transform and the global Gan-Gross-Prasad conjecture for unitary groups.
  • Chaudouard, On relative trace formulae: the case of Jacquet-Rallis.

TopBase change for unitary groups

Let $k'/k$ be a quadratic extension of number fields. Let $V/k'$ be an hermitian space of dimension $n$. Recall that $^L\UU(V)=\GL_n(\mathbb{C})\rtimes \Gamma_k$, where $\sigma\in \Gamma_{k}$ acts via the outer automorphism $g\mapsto J{}^tg^{-1}J^{-1}$ through its order two quotient $\Gal(k'/k)$, where $J=\mathrm{antidiag}\{1,\ldots, 1\}$. We define the base change map on $L$-groups $$\mathrm{BC}:{}^L\UU(V)\rightarrow {}^L\mathrm{R}_{k'/k}(G)=(\GL_n(\mathbb{C})\times \GL_n(\mathbb{C}))\rtimes \Gamma_k,\quad  g\mapsto (g, J{}^tg^{-1}J^{-1})$$ where $\Gamma_k$ acts via the order two quotient $\Gal(k'/k)$ by permuting the two copies of $\GL_n(\mathbb{C})$.

Remark 2 More generally, let $G$ be a connected reductive group over $k$. Let $k'/k$ be a finite extension. There is a base change map ${}^LG\rightarrow{}^L\mathrm{R}_{k'/k}G$, and the induced map on for Langlands parameters should correspond to the restriction of the conjectural Langlands group $\mathcal{L}_k$ to $\mathcal{L}_{k'}$.

The Langlands philosophy predicts a base change map $\mathrm{BC}:\mathcal{A}(\UU(V))\rightarrow \mathcal{A}(\GL_{n,k'})$ such that $\mathrm{BC}(\pi)_v=\mathrm{BC}(\pi_v)$ (the local base change map) for almost all $v$. Here:

  1. The local base change for an unramified representation $\pi_v$ on $\UU(V)_v$ is given by the unramified representation of $\GL_n(k_v')$ such that the Langlands parameter of $\mathrm{BC}(\pi_v)$ is given by the base change of the Langlands parameter of $\pi_v$ via the above base change map of $L$-groups.
  2. The local base change map at a split place $v$ is given by $\mathrm{BC}(\pi_v)=\pi_v \boxtimes \pi_v^\vee$ on $\GL_n(k_v')=\GL_n(k_v)\times\GL_n(k_v)$.
Definition 2 A weak base change of $\pi \hookrightarrow\mathcal{A}_\mathrm{cusp}(\UU(V))$ is a $\mathrm{BC}(\pi)\hookrightarrow\mathcal{A}(\GL_{n,k'})$ such that $\mathrm{BC}(\pi)_v\cong\mathrm{BC}(\pi_v)$ for almost all $v$.
Theorem 3 (Mok, Kaletha-Mingeuz-Shin-White) Any $\pi\hookrightarrow \mathcal{A}_\mathrm{cusp}(\UU(V))$ admits a weak base change. Moreover, any such $\pi$ appears with multiplicity one in $\mathcal{A}_\mathrm{cusp}(\UU(V))$.
Remark 3 These works follow the work of Arthur (for split $\SO$ and $\mathrm{Sp}$), and are still conditional on the weighted fundamental lemma (which is proved by Chaudouard-Laumon for any split groups). This weak base change $\mathrm{BC}(\pi)$ is actually isobaric, hence is unique by a version of strong multiplicity one theorem for isobaric representations (due to Jacquet-Shalika).

We will only consider $\pi$ satisfying:

  • there exists a split place $v$ such that $\pi_v$ is supercuspidal.

This implies that $\mathrm{BC}(\pi)_v$ is supercuspidal, hence $\mathrm{BC}(\pi)$ is cuspidal (and thus unique).

Remark 4 Recall the two equivalent characterization of supercuspidal representations $\pi$ for a $p$-adic reductive group $G(F)$:
  1. Any matrix coefficient $c_{v, \tilde v}=\langle \pi(\cdot) v, \tilde v\rangle$ is compactly supported modulo center on $G(F)$.
  2. The Jacquet module $\pi_N=\pi/\langle\pi(u)v-v, u\in N(F), v\in \pi\rangle$ is zero for any proper parabolic subgroup $P=MN\subsetneq G$.

(b) implies (a) is due to Jacquet; (a) implies (b) is more difficult (as Jacquet says) and is due to Casselman and Harish-Chandra.

As an application of (b), we can show that for $\pi\hookrightarrow\mathcal{A}(G)$ such that $\pi_v$ is supercuspidal, then $\pi$ is cuspidal. Indeed, for $P=MN\subsetneq G$ any proper parabolic subgroup, the constant term $\phi_N(g)=\int_{[N]}\phi(ug)du$ factors through $(\pi_v)_N$ for any $v$.

Definition 3 We say a function of the form $f=c_{v,\tilde v}|_{G(F)^1}\in C_c^\infty(G(F))$ essentially a matrix coefficient of $\tilde \pi$. For such $f$, and $\sigma$ an irreducible representation of $G(F)$, we know that if $\sigma(f)=0$, then $\sigma\cong \pi \otimes \chi$, for some unramified character $\chi$ of $G$.

As an application of (a), we see that if $f=\prod_v f_v\in C_c^\infty(G(\mathbb{A}))$ is a test function such that $f_v$ is essentially a matrix coefficient of a supercuspidal representation for some $v$, then its action $R(f)$ on $\mathcal{A}(G)$ has image in $\mathcal{A}_\mathrm{cusp}(G)$.

TopGeneralities on RTFs

Relative trace formulas are analytic tools generalizing the Arthur-Selberg trace formula, introduced by Jacquet to study periods of automorphic forms. In its rough form, it is an equality of the form $$\sum(\text{periods of automorphic forms})^2=\sum \text{relative orbital integrals}.$$ The LHS is called the spectral side and the RHS is called the geometric side. The idea of Jacquet is not to use one relative trace formula in isolation, but rather to compare the geometric side of two relative trace formulas and thus to relate different automorphic periods.

Example 2 Jacquet reproved Waldspurger's formula by comparing $$|\mathcal{P}_T(\phi)|^2\longleftrightarrow \mathcal{P}_A(\phi)\mathcal{P}_{A,\eta}(\phi).$$ The automorphic period on LHS is on a nonsplit torus $T\hookrightarrow \PGL_2$, while the one on RHS is on the split torus $A\hookrightarrow \PGL_2$. Waldspurger's formula the follows since the RHS is related to $L(1/2, \pi)L(1/2, \pi \otimes \eta)$ by Hecke's integral.
Definition 4 Let $f\in C_c^\infty(G(\mathbb{A}))$. Define an operator $R(f)$ on $L^2([G])$ by $$(R(f)\phi)(x)=\int_{[G]}K_f(x,y)\phi(y)dy,\quad K_f(x,y)=\sum_{\gamma\in G(k)}f(x^{-1}\gamma y).$$
Definition 5 For $H_1, H_2\hookrightarrow G$ and $\eta: [H_2]\rightarrow \mathbb{C}^\times$, we define the distribution $$I(f)=\int_{[H_1]\times [H_2]}K_f(h_1, h_2) \eta(h_2) dh_1dh_2$$ (there are convergence issues which we ignore for the moment).

Formally, we have a geometric expansion $$I(f)=\sum_{\gamma\in H_1(k)\backslash G(k)/H_2(k)}\Vol([(H_1\times H_2)_\gamma])\Orb(\gamma,f),$$ where the (relative) orbital integral $\Orb(\gamma,f)$ is defined to be $$\Orb(\gamma, f)=\int_{(H_1\times H_2)_\gamma(\mathbb{A})\backslash (H_1\times H_2)(\mathbb{A})} f(h_1^{-1}\gamma h_2) \eta(h_2)dh_1dh_2.$$

Formally, we have a spectral expansion of the kernel function $$K_f(x,y)=\int_{\hat G^\mathrm{Aut}}\sum_{\phi\in \mathrm{OB}(\pi)} (\pi(f)\phi)(x) \overline{\phi}(y) d\mu_{[G]}\pi,$$ where $$L^2([G])=\int_{\hat G^\mathrm{Aut}} \pi d\mu_{[G]}\pi.$$ So we also have a spectral expansion $$I(f)=\int_{\hat G^\mathrm{Aut}} I_\pi(f) d\mu_{[G]}\pi, $$ where the relative character $I_\pi(f)$ is defined to be $$I_\pi(f)=\sum_{\phi\in \mathrm{OB}(\pi)}\mathcal{P}_{H_1}(\pi(f)\phi)\overline{\mathcal{P}_{H_2,\bar \eta}(\phi)}.$$ (if you replace $\mathcal{P} \otimes \overline{\mathcal{P}}$ by the inner product on $\pi \otimes \tilde\pi$ then $I_\pi(f)$ becomes the usual character $\tr \pi(f)$).

Comparing the geometric expansion and spectral expansion, we arrive at the desired RTF identity.

The problem is that both expansions are not convergent unless $[H_1]$ and $[H_2]$ are compact. We will instead consider simple RTF by choosing the test function $f$ such that both expansions are absolutely convergent.

Definition 6 We say $f=\prod_v f_v$ satisfies the
  1. geometric condition (GC) if there exists $v_0$ such that $f_{v_0}$ is supported on in the (relative) semisimple elliptic locus. Here we say $\gamma\in G_{v_0}$ is semisimple, if the orbit $H_1\gamma H_2$ is closed; and say $\gamma$ is elliptic, if the stabilizer $(H_1\times H_2)_\gamma$ has no split center.
  2. spectral condition (SC) if there exists $v_1\ne v_0$ such that $f_{v_1}$ is essentially a matrix coefficient of a supercuspidal representation (so $\pi(f)=0$ if $\pi$ is not cuspidal).
Proposition 1 Assume $f$ satisfies (GC) and (SC). Assume further that:
  1. $H_1\cap A_G=1$, $A_G\subseteq H_2$, and $\eta|_{[A_G]}=1$,
  2. $A_GH_1\backslash G$, $H_2\backslash G$ are quasi-affine.

Then $I(f)$ is absolutely convergent, and we have the RTF identity $$\sum_{\gamma\in H_1(k)\backslash G(k)_\mathrm{ell}/H_2(k)} \Vol([(H_1\times H_2)_\gamma])\Orb(\gamma,f)=I(f)=\sum_{\pi\hookrightarrow \mathcal{A}_\mathrm{cusp}(G, \mathbf{1})}I_\pi(f).$$

Here $$\mathcal{P}_{H_2, \bar \eta}(\phi)=\int_{[A_G\backslash H_2]} \phi(h_2) \bar \eta(h_2) dh_2,$$ and the orthonormal basis $\mathrm{OB}(\pi)$ is taken with respect to the Petersson inner product on $[A_G\backslash G]$ $$\langle \phi_1, \phi_2\rangle=\int_{[A_G\backslash G]} \phi_1(g)\bar\phi_2(g)dg.$$

TopJacquet-Rallis RTFs

Le $k'/k$ be a quadratic extension of number fields. Let $W\subseteq V$ be hermitian spaces over $k'$ of dimension $n$ and $n+1$ respectively. For simplicity assume $V=W \oplus^\perp \langle e\rangle$, $h(e)=1$. Consider $$H=U(W)\xrightarrow{\mathrm{diag}} G=U(W)\times \UU(V).$$ Let $\pi=\pi_W \boxtimes \pi_V\hookrightarrow \mathcal{A}_\mathrm{cusp}(G)$. The Gan-Gross-Prasad conjecture relates the GGP period $\mathcal{P}_H$ on $\pi$, and its $L$-function $$L(s, \pi_W\times \pi_V):=L(s, \mathrm{BC}(\pi))=L(s, \mathrm{BC}(\pi_W)\times \mathrm{BC}(\pi_V)).$$ Jacquet-Rallis propose to attack the GGP conjecture by comparing two RTFs.

The first RTF (on unitary groups) is associated to $(G, H, H)$. For "good" test functions $f$ on $G$, we consider the distribution $$J(f)=\int_{[H]\times [H]} K_f(h_1, h_2)dh_1 dh_2.$$ Its spectral expansion is related to the GGP period $\mathcal{P}_H|_{\pi}$.

The second RTF (on general linear groups) is associated to $(G', H_1, H_2, \eta)$, where $$H_1=\mathrm{R}_{k'/k}\GL_{n}\hookrightarrow G'{}=\mathrm{R}_{k'/k}(\GL_n\times\GL_{n+1})\hookleftarrow H_2=\GL_{n,k}\times \GL_{n+1,k},$$ and $$\eta(g_n, g_{n+1})=\eta(\det g_n)^{n+1}\eta(\det g_{n+1})^{n}.$$ For "good" test function $f'$ on $G'$, we consider the distribution $$I(f')=\int_{[H_1]\times [H_2]} K_{f'}(h_1, h_2) \eta(h_2) dh_1dh_2.$$ Its spectral expansion is related to $\mathcal{P}_{H_1}\cdot \overline{\mathcal{P}_{H_2, \bar \eta}}$ for $\Pi$ on $G'$. We saw last time that

  • $\mathcal{P}_{H_1}|_\Pi\ne0$ if and only if $L(1/2, \Pi)\ne0$, and we will see that
  • $\mathcal{P}_{H_2}|_\Pi\ne0$ if and only if $\Pi=\mathrm{BC}(\pi)$ for some $\pi\hookrightarrow \mathcal{A}_\mathrm{cusp}(G)$ on some inner form $G$.

To compare the geometric sides of the two RFTs, let us first describe the orbits.

On unitary groups, we can identify $$H\backslash G\cong \UU(V),\quad (g_W, g_V)\mapsto g_W^{-1}g_V,$$ and thus the $H\times H$-orbits in $G$ are given by $\UU(W)$-conjugacy classes in $\UU(V)$.

On general linear groups, we can similarly identify $H_1\backslash G'$ with $\GL_{n+1,k'}$, so the $H_1\times H_2$ orbits are identified with the $\GL_{n,k}\times \GL_{n+1,k}$ orbits in $\GL_{n+1,k'}$. Moreover we can identify $$\GL_{n+1,k'}/\GL_{n+1,k}\cong S_{n+1}:=\{s\in \GL_{n+1,k'}: s \bar s=1\},\quad g\mapsto g\bar g^{-1}.$$ Therefore $H_1\times H_2$ orbits in $G'$ are given by $\GL_{n,k}$-conjugacy classes in $S_{n+1}$.


TopMatching of orbits

Recall $W=V \oplus^\perp \langle e\rangle$, where $h(e)=1$. Fix compatible isomorphisms of $k'$-spaces $W\cong (k')^n$ and $V\cong (k')^n$ such that the vector $e$ gets sent to the last basis vector $e_{n+1}$. We thus obtain an embedding $\UU(W)\hookrightarrow \GL_{n,k'}$ and $\UU(V)\hookrightarrow \GL_{n+1, k'}$. So the orbits on both unitary groups and linear groups can be mapped to the $\GL_{n,k'}$-conjugacy classes in $\GL_{n+1,k'}$.

Definition 7 Say that $g\in \GL_{n+1,k'}$ is regular semisimple (with respect to the conjugation action of $\GL_{n, k'}$) if
  1. $(e_{n+1}, ge_{n+1}, \ldots, g^{n}e_{n+1})$ is a basis of $\mathrm{Mat}_{n+1,1}(k')$, where $e_{n+1}=
  0 \\
  0 \\
$ and
  2. $(e_{n+1}^*, e_{n+1}^*g,\ldots, e_{n+1}^*g^n)$ is a basis of $\mathrm{Mat}_{1,n+1}(k')$, where $e_{n+1}^*=\begin{pmatrix}
0 & \cdots &  0 &  1
Remark 5 One can prove this definition agrees the more general notion. More generally, if $H $ is a reductive group acting on an affine variety $X$. Then $x\in X$ is called semisimple if the orbit $Hx$ is Zariski closed; regular $\mathrm{Stab}_H(x)$ is of minimal dimension. When $X= H\times H$ with $H $ acting diagonally this recovers the usual notion for algebraic groups.

Let $(\GL_{n+1,k'})_\mathrm{rs}\subseteq \GL_{n+1,k'}$ be the subset of regular semisimple elements. Let $\UU(V)_\mathrm{rs}=\UU(V)\cap (\GL_{n+1},k')_\mathrm{rs}$ and $S_{n+1,\mathrm{rs}}=S_{n+1}\cap (\GL_{n+1,k'})_\mathrm{rs}$.

Notice that $g\in (\GL_{n+1,k'})_\mathrm{rs}$ if and only if $$\Delta(g):=\det(\langle e_{n+1}^*g^i, g^j e_{n+1}\rangle)_{0\le i,j\le n}$$ is nonvanishing, so $(\GL_{n+1,k'})_\mathrm{rs}, \UU(V)_\mathrm{rs}, S_{n+1,\mathrm{rs}}$ are all principal Zariski open subsets.

Proposition 2
  1. The centralizer of any $g\in (\GL_{n+1,k'})_\mathrm{rs}$ in $\GL_{n,k'}$ is trivial.
  2. $g_1, g_2\in(\GL_{n+1,k'})_\mathrm{rs}$ are $\GL_{n,k'}$-conjugate if and only if $$\langle e_{n+1}^*, g_1^ie_{n+1}\rangle=\langle e_{n+1}^*, g_2^i e_{n+1}\rangle, \forall i\ge0.$$
  3. $g_1, g_2\in \UU(V)_\mathrm{rs}$ (resp. $s_1, s_2\in S_{n+1,\mathrm{rs}}$) are $\GL_{n,k'}$-conjugate if and only if they are $\UU(W)$-conjugate (resp. $\GL_{n,k}$-conjugate).
  1. If $hgh^{-1}=g$, where $g$ is regular semisimple and $h\in \GL_{n,k'}$. Then $$hg^ie_{n+1}= hg^i h^{-1}e_{n+1}=g^i e_{n+1}$$ for all $i\ge0$. As $(e_{n+1},\ldots, g^ne_{n+1})$ form a basis, we know that $h=1$.
  2. If $g_2=hg_1 h^{-1}$, where $h\in \GL_{n,k'}$. Then $$\langle e_{n+1}^*, g_2^i e_{n+1}\rangle=\langle e_{n+1}^*, hg_1^ih^{-1}e_{n+1}\rangle=\langle e_{n+1}^*, g_1^i e_{n+1}\rangle$$ for all $i\ge0$. Conversely, there exists a unique $h\in\GL_{n+1,k'}$ sending $(e_{n+1}, \ldots, g^n_1e_{n+1})$ to $(e_{n+1},\ldots, g^n_2e_{n+1})$, Moreover $h$ also sends the dual basis $(e_{n+1}^*, \ldots, e_{n+1}^*g_1^n)$ to $(e_{n+1}^*,\ldots, e_{n+1}g_2^*)$ because $$\langle e_{n+1}^* g_1^i, g_1^j e_{n+1}\rangle=\langle e_{n+1}^* g_2^i, g_2^je_{n+1}\rangle$$ by assumption. Since $he_{n+1}=e_{n+1}$ and $e_{n+1}^* h=e_{n+1}^*$, we know that $h\in \GL_{n,k'}$. To check that $g_2=hg_1h^{-1}$, we just need to show that $$(hg_1 h^{-1})^ie_{n+1}=g_2^ie_{n+1}$$ for $0\le i\le n+1$ (because $hg_1h^{-1}$ and $g_2$ then acts in the same way on the basis $(e_{n+1},\ldots, g_2^ne_{n+1})$). For $0\le i\le n$, this is true by the construction of $h$. For $i=n+1$, we check that for $0\le j\le n$, we have $$\langle e_{n+1}^* g_2^j, h g_1^{n+1}e_{n+1}\rangle=\langle e_{n+1}^* g_1^jh^{-1}, h g_1^{n+1}e_{n+1}\rangle,$$ which is equal to $$\langle e_{n+1}^*, g_1^{n+j+1}e_{n+1}\rangle=\langle e_{n+1}^*, g_2^{n+j+1}e_{n+1}\rangle=\langle e_{n+1}^*g_2^j, g_2^{n+1}e_{n+1}\rangle.$$ Thus $(hg_1h^{-1})^{n+1}e_{n+1}=g_2^{n+1}e_{n+1}$ as desired.
  3. Let us show the assertion about unitary groups (the symmetric space is similar). Assume $g_1, g_2\in \UU(V)$ are $\GL_{n,k'}$-conjugate. By (a), there exists a unique $h\in\GL_{n,k'}$ such that $g_1=hg_2h^{-1}$. Let $\gamma\mapsto \gamma^*$ be the adjoint map with respect to the hermitian form on $V$. Then $$g_1^{-1}=g_1^*=(h^*)^{-1}g_2^*h^*=(h^*)^{-1}g_2^{-1}h^*.$$ Therefore $(h^*)^{-1}g_2 h^*=g_1$. By the uniqueness of $h$ we know that $h=(h^*)^{-1}$, and hence $h\in \GL_{n,k'}\cap \UU(V)=\UU(W)$.

The above proposition holds for any $k$-algebra point. For the next proposition we restrict to field value points.

Proposition 3 Let $F/k$ be a field. Let $E=F \otimes_k k'$ so that $\GL_{n,k'}(F)=\GL_{n}(E)$.
  1. Any $g\in \UU(V)_\mathrm{rs}$ is $\GL_n(E)$-conjugate to an element of $S_{n+1,\mathrm{rs}}(F)$. Therefore we have an embedding $$\UU(V)_\mathrm{rs}(F)/\UU(W)(F)\hookrightarrow S_{n+1,\mathrm{rs}}(F)/\GL_{n}(F).$$ Here the actions are by conjugation.
  2. We have a bijection $$\coprod_{W'} \UU(V')_\mathrm{rs}(F)/\UU(W')(F)\cong S_{n+1,\mathrm{rs}}(F)/\GL_n(F).$$ Here $W'$ runs over all $n$-dimensional $E$-hermitian spaces up to isomorphism, and $V'{}=W'\oplus^{\perp}\langle e\rangle$.
Remark 6 In the split case $E=F\times F$, then there is only one isomorphism class of $n$-dimensional $E$-hermitian spaces. In this case $$S_{n+1}(F)\cong \GL_{n+1}(F),\quad (g, g^{-1})\mapsto g,$$ and similarly $$\UU(V)(F)\cong\GL_{n+1}(F),\quad \UU(W)(F)\cong \GL_n(F).$$ So the proposition is clear in this split case (even without regular semisimple condition).
  1. Let $J=\diag(J', 1)$ be the matrix representing the hermitian form on $V$. Let $g\in \UU(V)_\mathrm{rs}(F)$. Then $g^{-1}=J {}^t \bar gJ^{-1}$. So $$\langle e_{n+1}^*, g^{-i}e_{n+1}\rangle=\langle e_{n+1}^*, J{}^t\bar g^i J^{-1} e_{n+1}\rangle=\langle e_{n+1}^*, {}^t\bar g^i e_{n+1}\rangle=\langle e_{n+1}^*, \bar g^i e_{n+1}\rangle.$$ By (a) and (b) of Proposition 2, we know that there exists a unique $\gamma\in\GL_n(E)$ such that $$\bar g{}=\gamma g^{-1}\gamma^{-1}.$$ Then $g=\bar \gamma \bar g^{-1} \bar \gamma^{-1}$, so $\bar g=\bar \gamma^{-1}g^{-1}\bar \gamma$, hence $$\gamma=\bar \gamma^{-1}$$ by the uniqueness of $\gamma$. By Hilbert 90, there exists $h\in \GL_n(E)$ such that $\gamma=\bar h^{-1} h$. We check that $\overline{hgh^{-1}}=hg^{-1}h^{-1}$ so that $hgh^{-1}\in S_{n+1}(F)$.
  2. Let $s\in S_{n+1,\mathrm{rs}}(F)$. We check as in (a) that $s$ and $^t\bar s^{-1}$ are $\GL_n(E)$-conjugate. Thus there exists a unique $\gamma\in\GL_n(E)$ such that $\gamma s\gamma^{-1}={}^t\bar s^{-1}$. Taking adjoint we obtain $s={}^t\bar \gamma^{-1} {}^{t}\bar s^{-1}{}^t\bar \gamma$. By the uniqueness of $\gamma$, we know that $^t \bar\gamma=\gamma$ i.e., $\gamma$ is hermitian. Then $s\in \UU(V_0')_\mathrm{rs}(F)$ for the hermitian space $W_0'{}=(E^n, \gamma)$. This shows the surjectivity of the map. To see the injectivity, assume $hsh^{-1}\in \UU(V')(F)$, $h\in \GL_n(E)$, and $W'{}=(E^n, \gamma')$ for $\gamma'$ a hermitian matrix. Then $$^t \overline{hsh^{-1}}^{-1}=\gamma'hsh^{-1}(\gamma')^{-1},$$ thus $$\Ad({}^t \bar h \gamma' h)s={}^t\bar s^{-1},$$ and so by the uniqueness of $\gamma$ we have $^t \bar h\gamma' h=\gamma$, i.e., $W'\cong W_0'$ as hermitian spaces.
Remark 7 There is another way to look at the bijection in (b). There exists an isomorphism of varieties $$\UU(V)_\mathrm{rs}/\UU(W)\cong S_{n+1, rs}/\GL_{n}.$$ Taking $F $-points, on the RHS we obtain $$(S_{n+1,\mathrm{rs}}/\GL_n)(F)=S_{n+1,\mathrm{rs}}(F)/\GL_n(F).$$ While on the LHS we obtain a disjoint union $$(\UU(V)_\mathrm{rs}/\UU(W))(F)=\coprod_{W'}\UU(V')_\mathrm{rs}(F)/\UU(W')(F).$$

In fact, if $G$ is a reductive group acting freely on an affine variety $X$. Then $(X/G)(F)$ classifies isomorphism classes of pairs $(T,\phi)$, where $T$ is a $G$-torsor over $F $, and $\phi: T\rightarrow X$ is an $G$-equivariant morphism. Since $G$-torsors over $F $ up to isomorphism is in bijection with $H^1(F, G)$, we have $$(X/G)(F)=\coprod_{\alpha\in H^1(F, G)} X_\alpha(F)/G_\alpha(F),$$ where $T_\alpha$ is the $G$-torsor corresponding to $\alpha$, $G_\alpha=\Aut_G(T_\alpha)$, and $X_\alpha=(X\times T_\alpha/G^\mathrm{diag})$ (with $G_\alpha$-action).

We recover the previous result by noticing that $H^1(F, \GL_n)=1$ and $H^1(F, \UU(W))$ is in bijection with isomorphism classes of $n$-dimensional $E$-hermitian spaces, and $G_\alpha=\UU(W_\alpha)$, and $X_\alpha=\UU(V_\alpha)_\mathrm{rs}$.

Definition 8
  1. Say $s\in S_{n+1,\mathrm{rs}}(F)$ and $g\in \UU(V)_\mathrm{rs}(F)$ match if their orbits correspond by the bijection in Proposition 3 (b).
  2. Say $\gamma\in G_\mathrm{rs}'(F)$ and $\delta\in G_\mathrm{rs}(F)$ match if their images in $S_{n+1}(F)$ and $\UU(V)(F)$ match. Here $G_\mathrm{rs}'{}=p^{-1}(S_{n+1,\mathrm{rs}})$, for $p: G'\rightarrow S_{n+1}, g\mapsto g\bar g^{-1}$.

TopComparison of RTFs

Definition 9 Say $f\in C_c^\infty(G(\mathbb{A}))$ is good if $f=\prod_v f_v$ and
  1. (GC'): there exists $v_0$ such that $f_{v_0}$ is supported in $G_\mathrm{rs}(k_{v_0})$ (in particular, in semisimple elliptic locus).
  2. (SC): there exists $v_1\ne v_0$ such that $f_{v_1}$ is essentially a matrix coefficient of a supercuspidal representation.

Define good $f'\in C_c^\infty(G'(\mathbb{A}))$ similarly.

For good functions $f$ and $f'$, we have a simple $\mathrm{RTF}_W$ for $J(f)$, and a simple $\mathrm{RTF}_{G'}$ for $I(f')$. We get further simplification as the orbits are now regular semisimple, and there is no volume terms for the orbital integrals since the centralizer is trivial.

As we have a bijection $$\coprod_W H^W(k)\backslash G^W_\mathrm{rs}(k)/ H^W(k)\cong H_1(k)\backslash G'_\mathrm{rs}(k)/H_2(k),$$ where $W$ runs over all $k'$-hermitian spaces of dimension $n$, it is better to

  • compare $\sum_W \mathrm{RTF}_W$ (rather than a single hermitian space $W$) with $\mathrm{RTF}_{G'}$.

For good test functions $f=(f^W)_W$ and $f'$, we would like to show $$\Orb(\delta, f^W)=\Orb(\gamma, f')$$ for $\delta\in G^W_\mathrm{rs}(k)$ matching with $\gamma\in G'_\mathrm{rs}(k)$. As the orbital integrals are product of local orbital integrals, we would like the global matching to come from a local one $$\Orb(\delta, f_v^W)=\Omega_v(\gamma)\cdot \Orb(\gamma, f_v').$$ Here the transfer factor $\Omega_v(\gamma)$ is necessary as the RHS does not only depends on the orbit of $\gamma$, due to the character $\eta$. Notice the transfer factor should satisfy $$\Omega_{v}(h_1\gamma h_2)=\eta_v(h_2)\Omega_v(\gamma),\quad \gamma\in G_\mathrm{rs}'(k_v), h_i\in H_i(k_v).$$ Moreover, for $\gamma\in G_\mathrm{rs}'(k)$, $\Omega_v(\gamma)=1$ for almost all $v$, and we have a product formula $\prod_v \Omega_v(\gamma)=1$. We will construct such transfer factor explicitly later (it will also satisfy $\Omega_v=1$ for split $v$).

Definition 10 For $v$ a place of $k$. We say that $f'\in C_c^\infty(G_v')$ match with $f=(f^{W_v})_{W_v}$, and write $f' \leftrightarrow (f^{W_v})_{W_v}$ if $$\Orb(\delta, f^{W_v})=\Omega_v(\gamma)\cdot \Orb(\gamma, f')$$ for all matching $\delta\in G_\mathrm{rs}^{W_v}(k_v)$ and $\gamma\in G_\mathrm{rs}'(k_v)$.


(Announcement: the Feb 27 class will be moved to Mar 1 afternoon).

To proceed further, let us recall the classification of hermitian spaces of dimension $n$ over local and global fields.

  • If $v$ is a nonsplit archimedean place, then there are $(n+1)$ isomorphism classes of $k_v'$-hermitian spaces, classified by its signature $(n,0), (n-1,1)\ldots (0,n)$.
  • If $v$ is a nonsplit nonarchimedean place, then there are 2 isomorphism classes $W_+$, $W_-$, with $\eta_{k_v'/k_v}(\disc W_{v,\pm})=\pm1$.
  • If $v$ is a split place, then there is only one isomorphism class.
  • Globally, the isomorphism classes of $k'$-hermitian spaces is determined by its localization $(W_v)_v$ satisfying $W_{v}=W_{v,+}$ for almost all $v$, and $\prod_{v}\eta_{k_v'/k_v}(\disc W_v)=1$.

To construct sufficiently many matching global test functions, we need

Theorem 4 (Fundamental lemma, Yun, Gordon, Beuzart-Plessis) For any global $W$, we have $\mathbf{1}_{G'(O_v)}\leftrightarrow (\mathbf{1}_{G^W(O_v)}, 0)$ for almost all $v$.
Theorem 5 (Smooth transfer, W. Zhang for $p$-adic places, H. Xue for archimedean places) Every $f'\in C_c^\infty(G'(k_v))$ matches some $(f^{W_v})_{W_v}$ and vice versa.
Remark 8 At archimedean places, H. Xue showed the smooth transfer exists for a dense subspace of test functions which suffices for global application. But the smooth transfer may send compactly supported functions to not necessarily compactly supported functions (but still Schwartz).

These two theorems are easy to obtain at split places. Assume $v$ is split in $k'$. Choosing an isomorphism $W\cong (k')^n$, and a place of $k'$ above $v$, we obtain $$G^W_v\cong  \GL_n(k_v)\times \GL_{n+1}(k_v)=:G_v,\quad G'_v\cong G_v\times G_v.$$

Lemma 1 (smooth transfer at a split place) Assume $v$ is split in $k'$. Then any $$f'=f_1\otimes f_2\in C_c^\infty(G_v')=C_c^\infty(G_v)\otimes C_c^\infty(G_v)$$ matches with $$\iota(f'):=f_1 * f_2^\vee\in C_c^\infty(G_v),$$ where $f_2^\vee(g)=f_2(g^{-1})$.
Proof By definition, $g\in G_{v,\mathrm{rs}}$ matches with $(g,1)\in G_{v,\mathrm{rs}}'$. Then by definition $$\Orb(g, f_1*f_2^\vee)=\int_{H_v\times H_v}(f_1*f_2^\vee)(hgh')dhdh',$$ where $H_v\cong \GL_n(k_v)$. Expanding definition, we obtain $$\int_{H_v\times H_v}\int_{G_v}f_1(hgh'\gamma)f_2(\gamma)d\gamma dh dh',$$ making a change of variables, we obtain $$\int_{H_v\times H_v}\int_{G_v}f_1(hg\gamma) f_2(h'\gamma)d\gamma dhdh',$$ which is precisely $$\int_{H_{1,v}}\int_{H_2,v} f'((h,h')(g,1)(\gamma,\gamma) d\gamma dhdh'{}=\Orb((g,1),f'),$$ as desired (as $\eta_v$ is trivial).

Assuming the fundamental lemma and the smooth transfer, let us complete the global comparison of RTFs.

Fix two split places $v_0,v_1$. Choose local test functions $f_v'$ and $f_v=(f^{W_v})_{W_v}$ such that

  1. For $v\ne v_0, v_1$, $f_v'\leftrightarrow f_v$ and $f_v'{}=\mathbf{1}_{G'(O_v)}$, $f_v=(\mathbf{1}_{G^W(O_v)},0)$ for almost all $v$.
  2. For $v=v_0$, $f_{v_0}'$ is essentially a matrix coefficient of a supercuspidal representation, and $f_{v_0}=\iota (f_{v_0}')$.
  3. For $v=v_1$, $f_{v_1}'\in C_c^\infty(G_{v_1,\mathrm{rs}}')$, and $f_{v_1}=\iota(f_{v_1}')$.

Let $f'{}=\prod_vf_v'$, and for a global hermitian space $W$, let $f^W=\prod_v f^{W_v}$. Notice $f^W=0$ for almost all $W$ by (a). Then $f'$ and $(f^W)_W$ are good test functions, and comparing the geometric expansions we obtain $$\sum_W (\mathrm{RTF}_W)(f^W)=\mathrm{RTF}_{G'}(f').$$ It follows from the spectral expansion that $$\sum_W\sum_{\pi\hookrightarrow \mathcal{A}_\mathrm{cusp}(G^W)} J_\pi (f^W)=\sum_{\Pi\hookrightarrow \mathcal{A}_\mathrm{cusp}(G', \mathbf{1})} I_\Pi (f').$$ Fix $\Pi\hookrightarrow \mathcal{A}_\mathrm{cusp}(G', \mathbf{1})$, we would like to isolate $\Pi$ in the RHS. Notice both sides are a prior infinite sum.

Proposition 4 (Global comparison) We have $$\sum_W\sum_{\pi\hookrightarrow \mathcal{A}_\mathrm{cusp}(G^W)\atop \mathrm{BC}(\pi)=\Pi} J_\pi (f^W)= I_\Pi (f').$$
Proof Notice that the fundamental lemma for the full spherical Hecke algebra is not known a priori. But it holds at split places, and this is sufficient for our purpose due to the following automorphic Chebotarev density theorem of Ramakrishnan (for $\GL(2)$ there is a stronger version: a set of places of density $\ge 7/8$ is enough).
Theorem 6 (Ramakrishnan) Let $\sigma,\sigma'\hookrightarrow \mathcal{A}_\mathrm{cusp}(\GL_{m,k'})$ such that $\sigma_w\cong\sigma_w'$ for almost all places $w$ of $k'$ lying above split places of $k$. Then $\sigma=\sigma'$.

Let $\Sigma$ be the set of places of $k$ such that $v$ is split in $k$ and $f_v'{}=\mathbf{1}_{K'_v}$, and $f_v=\mathbf{1}_{K_v}$, where $$K_v=\GL_n(O_v)\times \GL_{n+1}(O_v),\quad K_v'{}=K_v\times K_v.$$ If $\Pi_v$ is not unramified for some $v\in\Sigma$, then both sides of the desired equality are zero. So we may assume $\Pi_\Sigma:= \otimes'_{v\in \Sigma} \Pi_v$ is unramified. Consider the spherical Hecke algebras at $\Sigma$, $$\mathcal{H}(G_\Sigma, K_\Sigma)=\otimes_v'\mathcal{H}(G_v, K_v),\quad \mathcal{H}(G_\Sigma', K_\Sigma')=\mathcal{H}(G_\Sigma, K_\Sigma) \otimes \mathcal{H}(G_\Sigma, K_\Sigma).$$ By Lemma 1, we have a smooth transfer map $$\mathcal{H}(G_\Sigma', K_\Sigma')\rightarrow \mathcal{H}(G_\Sigma, K_\Sigma), \alpha=\alpha_1 \otimes \alpha_2\mapsto \iota(\alpha)=\alpha_1*\alpha_2^\vee=:\alpha^\mathrm{bc}.$$ For $\alpha\in\mathcal{H}(G_\Sigma', K_\Sigma')$, we have $$\Pi'(\alpha)=
  \alpha(\Pi_\Sigma') \Pi'(f_\Sigma'), & \Pi_{\Sigma}' \text{ unramified}\\
  0, & \text{otherwise}.
\end{cases}$$ Here $\alpha(\Pi_\Sigma')$ is a scalar.

Then for $\pi_\Sigma$ unramified, we have $\alpha^\mathrm{bc}(\pi_\Sigma)=\alpha(\pi_\Sigma \boxtimes \tilde\pi_\Sigma)$. Moreover, Lemma 1 implies that $f'* \alpha$ and $f^W*\alpha^\mathrm{bc}$ also match. Therefore comparing the geometric expansion again (but now using the fundamental lemma for the full spherical hecke algebra at split places) we obtain $$\sum_W\sum_{\pi_\Sigma\text{ unramified}} \alpha^\mathrm{bc}(\pi_\Sigma) J_\pi(f^W)=\sum_{\Pi_\Sigma'\text{ unramified}}\alpha(\Pi_\Sigma') I_{\Pi'}(f).$$ Moreover, the contributing $\pi$ and $\Pi$ satisfy $\pi_{v_0}$ is supercuspidal, and $\Pi_{v_0}$ is supercuspidal by our choice of $f_{v_0}$ and $f_{v_0}'$ (hence cuspidal).

The Satake isomorphism identifies $\mathcal{H}(G_\Sigma', K_\Sigma')$ as the ring of regular functions on the space of unramified representations of $\Irr_\mathrm{ur}(G_\Sigma')$. In this way we can view $\alpha$ as a functions on $\Irr_{\mathrm{ur}}(G_\Sigma')$. We now use the fact that the space unitary unramified representations $\Irr_\mathrm{ur}^\mathrm{unit}(G_\Sigma')$ is compact in $\Irr_\mathrm{ur}(G_\Sigma')$, and thus the restriction of $\mathcal{H}(G_\Sigma', K_\Sigma')$ to it is a separating self-adjoint algebra of continuous functions. By the Stone-Weierstrass theorem, $\mathcal{H}(G_\Sigma', K_\Sigma')$ is dense in the space of all continuous functions on $\Irr_\mathrm{ur}^\mathrm{unit}(G_\Sigma')$. This allows us to separate the contribution according to a given $\Pi_\Sigma$. So we obtain on the LHS a sum over $\pi$ such that $\pi_\Sigma \boxtimes \tilde\pi_\Sigma\cong\Pi_\Sigma$, while on the RHS a sum over $\Pi'$ such that $\Pi_\Sigma'\cong\Pi_\Sigma$. By Ramakrishnan's Theorem 6, the RHS is thus equal to $I_\Pi(f')$, and LHS is equal to the sum of $J_\pi(f^W)$ over $\pi$ such that $\mathrm{BC}(\pi)=\Pi$ (as given by local base change at all degree one primes).

Remark 9 One can deduce the fundamental lemma for the full spherical Hecke algebra (after some more local work).

TopApplication to GGP

Theorem 7 (W. Zhang) Assume the fundamental lemma (Theorem 4) and the smooth transfer (Theorem 5). Let $W_0$ be an $n$-dimensional $k'$-hermitian space. Let $\pi\hookrightarrow \mathcal{A}_\mathrm{cusp}(G^{W_0})$ and assume that $\pi$ is supercuspidal at two split places $v_0,v_1$. Then the followings are equivalent:
  1. There exists $W$ of dimension $n$ and $\pi'\hookrightarrow \mathcal{A}_\mathrm{cusp}(G^W)$ such that $\mathrm{BC}(\pi')=\mathrm{BC}(\pi)$ and $\mathcal{P}_{H^W}|_{\pi'}\ne0$.
  2. $L(1/2, \mathrm{BC}(\pi))\ne0$.


Proof of (a) $\Rightarrow$ (b) By replacing $W_0$ with $W$, we may assume that $\pi=\pi'$. Write $G=G^{W_0}$, $H=H^{W_0}$. We use the following important local results.

Theorem 8 (Aizenbud-Gourevitch-Rallis-Schiffman, p-adic case; B. Sun-C. Zhu, archimedean case) For any place $v$, $$\dim \Hom_{H_v}(\pi_v, \mathbb{C})\le1.$$

As a consequence, the global GGP period always factorizes as $$\mathcal{P}_H|_\pi=\prod_v\lambda_v,\quad \lambda_v\in\Hom_{H_v}(\pi_v, \mathbb{C}),$$ and thus the global relative character also factorizes as $$J_\pi=\prod_v J_{\pi_v}, \quad J_{\pi_v}(f_v)=\sum_{\phi\in\mathrm{OB}(\pi_v)}\lambda_v(\pi_v(f_v)\phi_v)\overline{\lambda_v(\phi_v)}.$$

By assumption, $\mathcal{P}_H|_\pi\ne0$, and so $J_\pi\ne0$. We apply the global comparison (Prop. 4) by choosing the local test functions $f_v'$ and $(f^{W_v})_{W_v}$ as follows:

  1. For $v\ne v_0, v_1$, $f^{W_v}=0$ unless $W_{0}=W_{0,v}$, and $f_v:=f^{W_0,v}$ so that $(f_v,0,\ldots 0)\leftrightarrow f_v'$ (which is the unit function at almost all places $v$).
  2. For $v=v_0$, $\pi_{v_0}$ is supercuspidal, we choose $f_0$ to be essentially a matrix coefficient of $\tilde \pi_{v_0}$ such that $J_{\pi_{v_0}}(f_0)\ne0$. (For example, we can take $\phi$ a nonzero matrix coefficient, and take $f_0=\phi*\phi^*$, where $\phi^*(g)=\overline{\phi(g^{-1}})$. Then $$J_{\pi_v}(\phi* \phi^*)=\sum_{e\in\mathrm{OB}(\pi)} |\lambda_v(\pi_v(\phi) e)|^2\ne0.$$ Moreover, there exists $f_0'$ essentially a matrix coefficient of $\tilde \pi_{v_0} \boxtimes \pi_{v_0}$ such that $f_0=\iota(f_0')$. Set $f_{v_0}=f_0$ and $f_{v_0}'=f_0'$.
  3. For $v=v_1$, $\pi_{v_1}$ is supercuspidal, we choose $f_1\in C_c^\infty(G_\mathrm{rs,v_1})$ such that $J_{\pi_{v_1}}(f_1)\ne0$. The existence of $f_1$ follows from the following theorem of Ichino-Zhang (as $G_{v_1}-G_{\mathrm{rs}, v_1}$ is of measure 0). Moreover, there exists $f_1'\in C_c^\infty(G_\mathrm{rs,v_1}')$ such that $f_1=\iota(f_1')$. Set $f_{v_1}=f_1$ and $f_{v_1}{}'{}=f_1'$.
Theorem 9 (Ichino-Zhang) Assume $\pi_{v}$ is tempered. There exists $J_{\pi_{v}}\in L^1_\mathrm{loc}(G_{v})$ (locally integrable functions) such that $$J_{\pi_v}(\phi)=\int_{G_v} J_{\pi_v}(g)\phi(g)dg,\quad \forall \phi\in C_c^\infty(G_{v}).$$

The global comparison (Prop. 4) now gives $$\sum_{\pi'\hookrightarrow \mathcal{A}_\mathrm{cusp}(G), \atop \mathrm{BC}(\pi')=\mathrm{BC}(\pi)} J_{\pi'}(f)=I_{\mathrm{BC}(\pi)}(f').$$ By the factorization $$J_{\pi'}(f)=J_{\pi'}^{0,1}(f^{0,1})\times J_{\pi_{v_0}}(f_0) \times J_{\pi_{v_1}}(f_1),$$ as $\pi_{v_0}\cong \pi_{v_1}'$ and the compatibility of local base change at split places. By the multiplicity one theorem (Theorem 3), we know that the $\pi'$ in the LHS are all distinct, and thus the nonzero $J_{\pi'}^{0,1}$'s are linearly independent (as one can always choose a test function acting as 1 on one $\pi'$ and zero on the other $\pi'$'s). So we can choose $f^{0,1}$ such that $J_{\pi'}^{0,1}(f^{0,1})\ne0$ if and only if $\pi'{}=\pi$. For such a choice, we have the LHS is nonzero, and hence the RHS $I_{\mathrm{BC}(\pi)}(f')\ne0$. Hence $\mathcal{P}_{H_1}|_{\mathrm{BC}(\pi)}\ne0$ and $\mathcal{P}_{H_2,\eta}|_{\mathrm{BC}(\pi)}\ne0$. And $\mathcal{P}_{H_1}|_{\mathrm{BC}(\pi)}\ne0$ is equivalent to $L(1/2, \mathrm{BC}(\pi))\ne0$ by the Rankin-Selberg integral.

Proof of (b) $\Rightarrow$ (a) As a corollary of previous proof,

Corollary 1 If $\pi\hookrightarrow \mathcal{A}_\mathrm{cusp}(G)$ is supercuspidal at two split places and $\mathcal{P}_H|_\pi\ne0$. Then $\mathcal{P}_{H_2,\eta}|_{\mathrm{BC}(\pi)}\ne0$ (i.e., $\mathcal{P}_{\GL_n, \eta_n}|_{\mathrm{BC}(\pi_W)}\ne0$ and $\mathcal{P}_{\GL_{n+1}, \eta_{n+1}}|_{\mathrm{BC}(\pi_V)}\ne0$).

To prove (b) $\Rightarrow$ (a), we need to show that $\mathcal{P}_{\GL_n, \eta_n}|_{\mathrm{BC}(\pi_W)}\ne0$ and $\mathcal{P}_{\GL_{n+1}, \eta_{n+1}}|_{\mathrm{BC}(\pi_V)}\ne0$. This can be proved by combining Flicker-Rallis (relating the period with poles of Asai $L$-functions) and Mok (relating poles of Asai $L$-functions and base change from unitary groups). Here we give a more direct proof assuming the existence of two split supercuspidal places.

Proposition 5 Let $\pi_V\hookrightarrow \mathcal{A}_\mathrm{cusp}(U(V))$ be supercuspidal at two split places $v_0,v_1$. Then there exists $\pi_W\hookrightarrow \mathcal{A}_\mathrm{cusp}(U(W))$ supercuspidal at $v_0, v_1$ such that $\mathcal{P}_H|_{\pi_W \boxtimes \pi_V}\ne0$. In particular, $\mathcal{P}_{\GL_{n+1}, \eta_{n+1}}|_{\mathrm{BC}(\pi_V)}\ne0$ by the previous corollary.
Proof Take any supercuspidal representations $\sigma_0$ and $\sigma_1$ of $U(W)_{v_0}\cong \GL_n(k_{v_0})$ and $U(W)_{v_1}\cong \GL_n(k_{v_1})$ respectively. Since $\sigma_i$ and $\pi_{v_i}$ are supercuspidal, we have by the Rankin-Selberg theory that $$\Hom_{H_{v_i}}(\sigma_i \boxtimes \pi_{v_i}, \mathbb{C})\ne0$$ (in fact this is true for any generic representations $\sigma_v, \pi_v$). Now we apply the following more general globalization of distinguished representations (applied to $H=U(W)$, $G=U(V)$, $S=\{v_0,v_1\}$) to construct the desired $\pi_W$.
Proposition 6 Let $H\hookrightarrow G$ be connected reductive group over $k$. Assume $A_H=1$. Let $\pi\hookrightarrow\mathcal{A}_\mathrm{cusp}(G)$. Let $S$ be a finite set of nonarchimedean places. Let $\sigma_v^0$ be a supercuspidal representations of $H_v$, and $\sigma_S^0=\bigotimes_{v\in S} \sigma_v^0$. Assume $$\Hom_{H_S}(\pi_S \otimes \sigma_S^0, \mathbb{C})\ne0$$ Then there exists $\sigma\hookrightarrow\mathcal{A}_\mathrm{cusp}(H)$ such that
  1. $\mathcal{P}_{H^\mathrm{diag}}|_{\sigma \boxtimes \pi}\ne0$, i.e., there exists $\phi\in \pi,\phi'\in \sigma$ such that $\int_{[H]}\phi\phi'\ne0$.
  2. For all $v\in S$, $\sigma_v$ is an unramified twist of $\sigma_v^0$.
Proof By assumption, $\pi_S|_{H_S}$ surjects onto $\tilde \sigma_S^0$. As $\tilde \sigma_S^0$ is supercuspidal and hence a projective module when restricted to $H^1_S$, we obtain an embedding $$\tilde \sigma_S^0|_{H^1_S}\hookrightarrow \pi_S|_{H^1_S}.$$ Take $\phi_S\in \pi_S$ be a nonzero vector in the image of this embedding, and take a nonzero vector of the form $\phi=\phi_S \otimes \phi^S\in \pi$. Then $\langle R(H^1_S)\phi\rangle$ is a quotient of $\tilde \sigma_S^0$. The same holds for $\phi$ replaced by $R(g)\phi$ for any $g\in  G(\mathbb{A}^S)$. But $G(k) G(\mathbb{A}^S)$ is dense in $G(\mathbb{A})$ by weak approximation, we may assume that $\phi(1)\ne0$. Since the restriction of $\phi$ to $H_v$ is supercuspidal, we know that $\phi|_{[H]}\in \mathcal{A}_\mathrm{cusp}(H)$ is a nonzero cusp form. So there exists $\sigma\hookrightarrow \mathcal{A}_\mathrm{cusp}(H)$ not perpendicular to $\phi|_{[H]}$, i.e, $\int_{[H]}\phi\phi'\ne0$ for some $\phi'\in \sigma$. For such $\sigma$, (a) is satisfied. Moreover, the inner product gives an $H^1_S$-invariant paring between $\sigma$ and $\langle R(H_S^1)\phi\rangle$, and thus $$\Hom_{H^1_S}(\sigma \otimes \tilde \sigma_S^0, \mathbb{C})\ne0.$$ Therefore, $$\Hom_{H^1_S}(\sigma, \sigma_S^0)\ne0,$$ and hence $\sigma_S$ is an unramified twist of $\sigma_S^0$.

Now we can finish the proof of (b) $\Rightarrow$ (a). By the Rankin-Selberg integral, we have $\mathcal{P}_{H_1}|_{\mathrm{BC}(\pi)}\ne0$. By Proposition 5 applied to $\pi_W$ and $\pi_V$, we obtain $\mathcal{P}_{H_2,\eta}|_{\mathrm{BC}(\pi)}\ne0$. Thus $I_{\mathrm{BC}(\pi)}(f')\ne0$ for some $f'$. We can again modify $f'$ at $v_0$ and $v_1$ such that $f_{v_i}'$ are as before, and use the global comparison to conclude that $J_{\pi}(f^W)\ne0$ for some $W$ and some $\pi\hookrightarrow \mathcal{A}_\mathrm{cusp}(G^W)$.


TopThe Ichino-Ikeda conjecture


  • N. Harris, The Refined Gross-Prasad Conjecture for Unitary Groups
  • W. Zhang, Automorphic period and the central value of Rankin-Selberg L-function
  • Ichino-Ikeda, On the Periods of Automorphic Forms on Special Orthogonal Groups and the Gross-Prasad Conjecture

TopTamagawa measures

Let $G$ be a connected reductive group over a number field $k$. Let $\omega_G$ be a nonzero $G$-invariant volume form on $G$. For example for $G=\GL_n$, we may take $\omega_G=\frac{\wedge dg_{i,j}}{(\det g)^n}$. Let $\psi: \mathbb{A}/k\rightarrow \mathbb{C}^\times$ be a nontrivial character (these all conjugate under $k^\times$).

Weil associated to $(\omega_{G,v},\psi_v)$ a Haar measure $|\omega_G|_{\psi,v}$ on $G(k_v)$ as follows. Locally in the analytic topology, write $$\omega_G=f(x_1,\ldots, x_d) dx_1 \cdots dx_d,$$ then we define $$|\omega_{G}|_{\psi,v}=|f(x_1,\ldots,x_d)|_v d_{\psi_v} x_1 \cdots d_{\psi_v} x_d,$$ where $d_{\psi_v}x$ is the $\psi_v$-self-dual measure on $k_v$, i.e., the Fourier transform $$\phi\mapsto\hat\phi(x)=\int_{k_v}\phi(y) \psi_v(xy)d_{\psi_v}y$$ preserves the $L^2$-norm. The definition of $|\omega_G|_{\psi,v}$ does not depend on the choice of coordinates and hence give rise to a measure on $G(k_v)$.

For $\lambda\in k^\times$, we have $$|\lambda \omega_G|_{\psi,v}=|\lambda|_v |\omega_G|_{\psi,v},\quad |\omega_G|_{\psi^\lambda,v}=|\lambda|_v^{\dim G/2}|\omega_G|_{\psi,v}.$$ Therefore formally the product $\prod_v |\omega_G|_{\psi,v}$ does not depend on the choice of $(\omega_G, \psi)$. However, the product $\prod_{v\not\in S}|\omega_G|_{\psi,v}(G(O_v))$ may diverge. In fact, by Weil-Steinberg, we have at unramified places $v$ (where $\omega_G$ comes from $G/O_v$ and $\psi_v$ of conductor 1), $$|\omega_G|_{\psi_v}(G(O_v))=L_v(0, M_G)^{-1}=:\Delta_{G,v}^{-1},$$ where $M_G$ is the Artin-Tate motive over $k$ associated to $G$.

Example 3 When $G=\GL_n$, $M_G=\mathbb{Q}(1)+ \cdots+ \mathbb{Q}(n)$, and $$L(s, M_G)=\zeta_k(s+1)\cdots \zeta_k(s+n).$$ (complete zeta functions).
Example 4 When $G=\UU(V)$, $M_G=\eta_{k'/k}(1)+\eta^2_{k'/k}(2)+\cdots +\eta^{n}_{k'/k}(n)$, and $$L(s, M_G)=\prod_{i=1}^nL(s+i, \eta^i_{k'/k}).$$

Since 0 may be outside the range of convergence of $\prod_v L_v(s, M_G)$, we modify the local measures and define

Definition 11 The Tamagawa measure is $$d_\mathrm{tam}g:=(\Delta_G^*)^{-1}\prod_v \Delta_{G,v}|\omega_G|_{\psi,v},$$ where $\Delta_G^*$ is the leading coefficient of $L(s, M_G)$ at $s=0$.

From now on, we will always take global measure to be the Tamagawa measure, and the local measure to be $dg_v:=|\omega_G|_{\psi,v}$. So by definition, $$d_\mathrm{tam}g=(\Delta_G^*)^{-1}\prod_v \Delta_{G,v}dg_v=``\prod_v dg_v

TopAsai L-functions and factorization of Flicker-Rallis period

Let $k'/k$ be a quadratic extension of number fields.

Definition 12 Recall that ${}^L \GL_{n,k'}=(\GL_n(\mathbb{C})\times \GL_n(\mathbb{C}))\rtimes \Gamma_k$. We define its Asai representation to be $$\mathrm{As}: {}^L\GL_{n,k'}\rightarrow \GL_n(\mathbb{C}^ n \otimes \mathbb{C}^n)$$ given by $$(g_1,g_2)\mapsto g_1 \otimes g_2,\ \sigma\mapsto
  1, & \sigma\in\Gamma_{k'},\\
  \iota, & \text{otherwise},
\end{cases} \quad \iota(u \otimes v)=v \otimes u.
Definition 13 Let $\Pi\hookrightarrow \mathcal{A}_\mathrm{cusp}(\GL_{n,k'})$ such that $\omega_\Pi|_{\mathbb{A}^\times_k}=1$. The Flicker-Rallis period is defined to be $$\mathcal{P}_{\GL_n}: \phi\in \Pi\mapsto \int_{[Z_n\backslash \GL_n]} \phi(h)dh.$$
Theorem 10 (Rallis, Flicker) Let $S$ be a sufficiently large finite set of places. The partial Asai $L$-function $L^S(s, \Pi, \mathrm{As})$ has meromorphic continuation to $\mathbb{C}$ with at most a simple pole at $s=1$. For every $\phi=\otimes_v \phi_v\in \Pi$, we have $$\mathcal{P}_{\GL_n}(\phi)=n\cdot \frac{\res_{s=1}L^S(s, \Pi, \mathrm{As})}{\Delta_{\GL_n}^{S,*}}\prod_{v\in S}\int_{N_{n-1}(k_v)\backslash \GL_{n-1}(k_v)}W_v(h_v)dh_v,$$ where the Whittaker function factors as $$W_\phi(g)=\int_{[N_{n,k'}]}\phi(ug)\psi_n(u)^{-1}du=\prod_vW_v, $$ and $$\Delta_{\GL_n}^{S,*}=\res_{s=0}L^S(s, M_{\GL_n}).$$

This theorem is an analogue to (by letting $k'{}=k\times k$):

Theorem 11 (Jacquet-Shalika) Let $\Pi, \Pi'\hookrightarrow\mathcal{A}_\mathrm{cusp}(\GL_n)$ such that $\omega_\Pi\overline{\omega_{\Pi'}}=1$. Then $L^S(s, \Pi\times \tilde \Pi')$ has meromorphic continuation to $\mathbb{C}$ with at most a simple pole at $s=1$. For every $\phi=\otimes_v \phi_v\in\Pi$ and $\phi'=\otimes_v\phi_v'\in \Pi'$, we have $$\langle \phi,\phi'\rangle_\mathrm{Pet}=n\cdot\frac{\res_{s=1} L^S(s,\Pi\times \tilde \Pi')}{\Delta_{\GL_n}^{S,*}}\prod_{v\in S}\int_{N_{n-1}(k_v)\backslash \GL_{n-1}(k_v)}W_v(h_v)\overline{W_v'(h_v)}dh_v,$$ here we normalize the Petersson inner product $$\langle\phi,\phi'\rangle_\mathrm{Pet}=\int_{[Z_n\backslash \GL_n]}\phi(g)\overline{\phi'(g)}dg.$$
Remark 10 Notice the local functionals at $v\in S$ appearing in the period formula are always convergent and nonzero (as the Kirillov model $K(\pi_v,\psi_v)\supseteq C_c^\infty(N_{n-1}(k_v)\backslash \GL_{n-1}(k_v),\psi_v)$). Therefore
  • $L^S(s, \Pi, \mathrm{As})$ has a pole at $s=1$ if and only if $\mathcal{P}_{\GL_n}|_{\Pi}\ne0$,
  • $L^S(s, \Pi\times\tilde \Pi')$ has a pole at $s=1$ if and only if $\Pi\cong\Pi'$.

When there is a pole at $s=1$, these local functionals are $\GL_n(k_v)$-invariant a priori.

Proof (Sketch of Theorem 11) The proof uses the Rankin-Selberg method for $\GL_n\times \GL_n$.
Definition 14 Let $\Phi=\prod_v \Phi_v\in \mathcal{S}(\mathbb{A}^n)$ be a Schwartz function, i.e.,
  • $\Phi_v$ is a smooth rapidly decay function for $v$ archimedean,
  • $\Phi_v\in C_c^\infty(k_v^n)$ for $v$ nonarchimedean,
  • $\Phi_v=\mathbf{1}_{O_v^n}$ for almost all $v$.
Definition 15 Define $$f(g,s)=\int_{\mathbb{A}^\times}\Phi(e_nzg)|\det zg|^s dz,\quad e_n=(0,\ldots,0,1),\ g\in \GL_n(\mathbb{A}).$$ It converges when $\Re(s)>1/n$. This function can be viewed as a section of a parabolic induction. In fact, let $P_n\subseteq \GL_n$ be the mirabolic subgroup , and $Q=P_nZ_n$ be a parabolic subgroup of type $(n-1,1)$. Then $$f(-,s)\in \Ind_{Q(\mathbb{A})}^{\GL_n(\mathbb{A})}(|\det|^s \otimes |\cdot|^{-(n-1)s}).$$
Definition 16 Define the Eisenstein series $$E(g,\Phi, s)=\sum_{\gamma\in Q(k)\backslash\GL_n(k)}f(\gamma g,s).$$ It converges when $\Re(s)>1$.
Lemma 2 $s\mapsto E(g,\Phi,s)$ has meromorphic continuation to $\mathbb{C}$ and has a simple pole at $s=1$ with residue $\frac{\hat\Phi(0)}{n}$, where $\hat \Phi(x)=\int_{\mathbb{A}^n}\Phi(y)\psi(x{}^ty)dy$.
Proof We use Poisson summation as in Tate's thesis (but now for $k^n$). Using $$P_n(k)\backslash \GL_n(k)\cong k^n-\{0\}, \quad g\mapsto e_ng,$$ we can rewrite $$E(g, \Phi,s)=\int_{\mathbb{A}^\times/k^\times}\sum_{\xi\in k^n-\{0\}}\Phi(\xi zg)|\det zg|^s dz.$$ This is equal to $$E_{\ge1}(g, \Phi,s)+E_{\le1}(g,\Phi,s)$$ depending on $|z|\ge1$ or $|z|\le1$. The first term is homomorphic in $s$. We then apply the Poisson summation to obtain $$E_{\le 1}(g, \Phi,s)=E_{\ge1}({}^tg^{-1}, \hat \Phi, 1-s)+\int_{|z|\le1}\hat\Phi(0)|\det zg|^{s-1}dz-\int_{|z|\le1}\Phi(0)|\det zg|^s,$$ the last two terms give $$|\det g|^{s-1}\frac{\hat\Phi(0)}{n(s-1)},\quad |\det g|^s\frac{\Phi(0)}{ns}$$ respectively.
Definition 17 Define the global zeta integral $$Z(s,\phi,\phi',\Phi)=\int_{[Z_n\backslash \GL_n]}\phi(g)\overline{\phi'(g)} E(g,\Phi,s)dg.$$

By Lemma 2, the global zeta integral has meromorphic continuation to $\mathbb{C}$ with at most a simple pole at $s=1$ (the residue is $\frac{\hat \Phi(0)}{n}\langle\phi,\phi'\rangle_\mathrm{Pet}$). Now unfolding the global zeta integral we obtain $$Z(s, \phi,\phi',\Phi)=\int_{N_n(\mathbb{A})\backslash \GL_n(\mathbb{A})}W_\phi(g)\overline{W_{\phi'}(g)}\Phi(e_ng)|\det g|^sdg,\quad \Re(s)\gg 0.$$ It now factorizes as product of local zeta integrals $$Z(s,\phi,\phi',\Phi)=(\Delta_{\GL_n}^*)^{-1}\prod_v\Delta_{\GL_n,v} Z_v(s),$$ where $$Z_v(s)=\int_{N_n(k_v)\backslash \GL_n(k_v)}W_v(h_v)\overline{W_v'(h_v)}\Phi_v(e_nh_v)|\det h_v|^s dh_v.$$ The local zeta integrals are convergent for $\Re(s)>1-\varepsilon$, and computation at unramified places $v$ shows that $$\Delta_{\GL_n,v} Z_v(s)=L(s,\Pi_v\times \tilde \Pi_v').$$ Thus $$Z(s,\phi,\phi',\Phi)=\frac{L^S(s,\Pi\times\tilde \Pi')}{\Delta_{\GL_n}^{S,*}}\prod_{v\in S}Z_v(s).$$ It follows that $L^S(s, \Pi\times\tilde \Pi')$ has meromorphic continuation to $\mathbb{C}$ with at most simple pole at $s=1$. Moreover, comparing the residue at $s=1$ of $Z(s,\phi,\phi',\Phi)$ we obtain $$\frac{\res_{s=1}L^S(s,\Pi\times\tilde \Pi')}{\Delta_{\GL_n}^{S,*}}\prod_v Z_v(1)=\frac{\hat\Phi(0)}{n}\langle\phi,\phi'\rangle_\mathrm{Pet}.$$ Now we compute $$Z_v(1)=\int_{N_n(k_v)\backslash \GL_n(k_v)}W_v(h_v)\overline{W_v'(h_v)}\Phi_v(e_nh_v)|\det h_v|dh_v,$$ which is equal to $$\int_{P_n(k_v)\backslash \GL_n(k_v)}\int_{N_{n-1}(k_v)\backslash \GL_{n-1}(k_v)}W_v(hg)\overline{W_v'(hg)}dh\Phi_v(e_ng)|\det g|dg.$$ Notice the outer integral is over $k^n-\{0\}$ and $|\det g|dg$ is the Haar measure on $k_v^n$. Now we done by choosing $\Phi_v\rightarrow \delta_{e_n}$ and so $\hat \Phi_v(0)\rightarrow 1$.

Remark 11 Theorem 10 is proved similarly using a generalization of the Rankin-Selberg method to $k'\ne k\times k$.
Remark 12 Let $\mathrm{As}^-{}=\mathrm{As} \otimes \eta_{k'/k}$ and $\mathrm{As}^+=\mathrm{As}$. Twisting everything by $\eta_n=(\eta_{k'/k}\circ \det)^{n+1}$ and apply Theorem 10, we obtain $$ \mathcal{P}_{\GL_n,\eta_n}(\phi)=n\cdot \frac{\res_{s=1}L^S(s, \Pi, \mathrm{As}^{\scriptscriptstyle (-1)^{n+1}})}{\Delta_{\GL_n^{S,*}}}\prod_{v\in S}\int\limits_{\scriptscriptstyle N_{n-1}(k_v)\backslash \GL_{n-1}(k_v)}W_v(h_v)\eta_n(h_v)dh_v.$$ Therefore
  • $L^S(s, \Pi, \mathrm{As}^{(-1)^{n+1}})$ has a pole at $s=1$ if and only if $\mathcal{P}_{\GL_n,\eta_n}|_\Pi\ne0$.

Let $\Ad_n$ be the adjoint representation of ${}^L\GL_{n,k'}$ on $\Lie(^{L}\GL_{n,k'})=\mathfrak{gl}_n(\mathbb{C}) \oplus  \mathfrak{gl}_n(\mathbb{C})$. Let $\Ad_V$ be the adjoint representation on ${}^L\UU(V)$ on $\Lie {}^L\UU(V)=\mathfrak{gl}_n(\mathbb{C})$. Let $\mathrm{BC}: {}^L\UU(V)\hookrightarrow {}^L\GL_{n,k'}$ be the base change map. Then one can check $$\Ad_n\circ \mathrm{BC}=\mathrm{As}\circ \mathrm{BC} \oplus \mathrm{As}^-\circ \mathrm{BC},\quad  \Ad_V=\mathrm{As}^{(-1)^n}\circ \mathrm{BC}.$$ Therefore for $\pi\hookrightarrow\mathcal{A}_\mathrm{cusp}(U(V))$, we have $$L^S(s, \mathrm{BC}(\pi), \Ad)=L^S(s, \mathrm{BC}(\pi), \mathrm{As})L^S(s, \mathrm{BC}(\pi), \mathrm{As}^-),$$ and $$L^S(s, \pi, \Ad)=L^S(s, \mathrm{BC}(\pi), \mathrm{As}^{(-1)^n}).$$ When $\mathrm{BC}(\pi)$ is cuspidal, by Theorem 11 (as $L^S(s, \mathrm{BC}(\pi), \Ad)=L^S(s, \mathrm{BC}(\pi)\times\widetilde {\mathrm{BC}(\pi)})$) we know that exactly one of $L^S(s, \mathrm{BC}(\pi), \mathrm{As}^{\pm})$ has a pole at $s=1$ and the other is nonzero at $s=1$. Moreover, we have the following theorem.

Theorem 12 (Mok, Kaletha-Mingeuz-Shin-White) Let $\Pi\hookrightarrow \mathcal{A}_\mathrm{cusp}(\GL_{n,k'})$. Then $\Pi$ is in the image of base change for some $\UU(V)$ if and only if $L^S(s, \Pi, \mathrm{As}^{(-1)^{n+1}})$ has a pole at $s=1$.

Thus we obtain the following corollary.

Corollary 2 Let $\pi\hookrightarrow \mathcal{A}_\mathrm{cusp}(\UU(V))$ such that $\mathrm{BC}(\pi)$ is cuspidal. Then
  • $L^S(s, \pi, \Ad)$ has no pole and is nonzero at $s=1$.
  • $\mathcal{P}_{\GL_n,\eta_n}|_{\mathrm{BC}(\pi)}\ne0$.


TopLocal and global packets for unitary groups

Let $F $ be a local field. Let $W_F$ be the Weil group and $\mathcal{L}_F$ be the Weil-Deligne group ($\mathcal{L}_F=W_F\times \mathrm{SU}_2(\mathbb{R})$ or $W_F$ depending on $F $ is nonarchimedean or archimedean).

Definition 18 Let $G/F$ be a connected reductive group. A $L$-parameter is a homomorphism $\phi: \mathcal{L}_F\rightarrow {}^{L}G$ such that
  1. $\phi$ is continuous and semisimple (i.e., $\phi(\tau)$ is semisimple for all $\tau\in \mathcal{L}_F$).
  2. The composition $W_F\rightarrow {}^{L}G\rightarrow \Gamma_F$ is the natural inclusion.

Two $L$-parameters are equivalent if they are $\hat G(\mathbb{C})$-conjugate.

Let $E/F$ be a quadratic extension of local fields. Fix $\sigma\in \mathcal{L}_F-\mathcal{L}_E$.

Definition 19 A complex representation $\phi: \mathcal{L}_E\rightarrow\GL(M)$ is called $\varepsilon$-conjugate-dual for $\varepsilon\in\{\pm1\}$, if there exists an isomorphism $T: M\cong \tilde M^\sigma$ (where $\tilde\phi(g)={}^t\phi(g)^{-1}$, and $\phi^\sigma(g)=\phi(\sigma g\sigma^{-1})$) such that $^tT^\sigma: M^{\sigma^2}\cong \tilde M^\sigma$ is equal to $\varepsilon T$ (notice $M\cong M^{\sigma^2}$). Equivalently, there exists a nondegerarte pairing $B(\cdot, \cdot): M\times M \rightarrow \mathbb{C}$ such that $$B(\phi(\tau)u, \phi(\sigma\tau\sigma^{-1})v)=B(u, v), \quad\forall\tau\in \mathcal{L}_E,$$ and $$B(u, \phi(\sigma^2)v)=\varepsilon B(v,u).$$ Such $B$ is called an $\varepsilon$-conjugate-dual form for $\phi$.
Remark 13 This definition does not depent on the choice of $\sigma$.
Proposition 7
  1. We have a commutative diagram $$\xymatrix{\left\{
  L\text{-parameters }\\
  \mathcal{L}_F\rightarrow {}^L \GL_{n,E}
\right\}/\sim \ar[r]^{\phi\mapsto\mathrm{pr}_1\circ\phi|_{\mathcal{L}_E}}_{\cong} & \left\{
n\text{-dim adm reps } \\
\mathcal{L}_E\rightarrow \GL(M)
\right\}/\sim \\ \left\{{
  L\text{-parameters }\\
  \mathcal{L}_F\rightarrow {}^L\UU(V)
\right\}/\sim \ar[u]^{\mathrm{BC}} \ar[r]^-{\phi\mapsto \phi|_{\mathcal{L}_E}}_-{\cong} & \left\{{
  n\text{-dim }(-1)^{n+1}\text{-conjugate-dual}\\
 \text{ adm reps } \mathcal{L}_E\rightarrow\GL(M)
\right\}/\sim. \ar@{^(->}[u]}$$
  1. Let $\phi_E\mapsto \bar \phi_E$ be the inverse of the top arrow. Then an irreducible admissible $\phi_E: \mathcal{L}_E\rightarrow \GL(M)$ is $\varepsilon$-conjugate-dual if and only if $\mathrm{As}^{\varepsilon}\circ \bar\phi_E$ has nonzero fixed vector. If $\phi_E$ is moreover unitary, then this is further equivalent to $L(s, \mathrm{As}^\varepsilon\circ \bar\phi_E)$ has a pole at $s=0$.
  1. The commutativity is clear. The inverse of the top horizontal arrow is given by $\phi_E\mapsto \bar\phi_E$, where $$\bar \phi_E(\tau)=(\phi_E(\tau), \phi_E(\sigma\tau\sigma^{-1}),\tau),\quad\forall \tau\in \mathcal{L}_E,$$ and $$\bar\phi_E(\sigma)=(1, \phi_E(\sigma^2), \sigma).$$ For the bottom horizontal arrow, if $\phi: \mathcal{L}_F\rightarrow {}^L\UU(V)$ is an $L$-parameter with $\phi(\sigma)=(A,\sigma)$. Then $(u,v)\mapsto {}^tu (AJ)^{-1}v$, where $J=\mathrm{antidiag}((-1)^{n-1}, \ldots, 1)$, is a nondegerarte $(-1)^{n+1}$-conjugate-dual form for $\phi|_{\mathcal{L}_E}$. Conversely, if $\phi_E$ has a nondegerarte $(-1)^{n+1}$-conjugate-dual form $(u,v)\mapsto {}^tu Bv$, then $\phi_E$ extends to an $L$-parameter $L_F\rightarrow {}^L\UU(V)$ with $\sigma\mapsto ((JB)^{-1},\sigma)$.
  2. $\mathrm{As}^{\varepsilon}\circ \bar\phi_E$ is the representation of $\mathcal{L}_F$ on $M \otimes M$ given by $$\tau\mapsto \phi_E(\tau) \otimes \phi_E(\sigma\tau\sigma^{-1}),\quad \forall\tau\in \mathcal{L}_E,$$ and $$\sigma\mapsto\varepsilon(1 \otimes \phi_E(\sigma^2))\circ \iota.$$ A fixed vector of its dual representation is nothing but a $\varepsilon$-conjugate-dual form $B$ on $M$. Moreover, as $\phi_E$ is irreducible, then $B$ is nonzero implies that $B$ is nondegerarte by Shur's lemma. The second claim follows from a general property of local $L$-functions: if $r: \mathcal{L}_F\rightarrow \GL(N)$ is unitary, then $N^{\mathcal{L}_F}\ne0$ if and only if $L(s, r)$ has a pole at $s=0$.
Remark 14 Let $\Phi_n(E/F)$ be the set of isomorphism classes of $L$-parameters $\mathcal{L}_F\rightarrow {}^L\UU(V)$. For $\phi\in \Phi_n(E/F)$, we define its component group $S_\phi:=\pi_0(\mathrm{Cent}_{\hat \UU(V)}(\im \phi))$.

If $B$ is a nondegerarte $(-1)^{n+1}$-conjugate-dual form on $\phi_E$, then $$S_\phi\cong\pi_0(\mathrm{Cent}_{\GL_n(\mathbb{C})}(\phi_E, B)).$$ Moreover, if we decompose $$\phi_E= \bigoplus_{i\in I} \phi_i \oplus  \bigoplus_{j\in J} 2 m_j \phi_j \oplus \bigoplus_{k\in K} l_k( \phi_k \oplus \tilde \phi_k^\sigma),$$ where $\phi_i$ is irreducible and $(-1)^{n+1}$-conjugate-dual, $\phi_j$ is irreducible $(-1)^n$-conjugate-dual, and $\phi_k$ is irreducible but not conjugate-dual. Then $$\mathrm{Cent}_{\GL_n(\mathbb{C})}(\phi_E, B)\cong\prod_{i\in I} \OO(n_i, \mathbb{C})\times \prod_{j\in J}\Sp(2m_j, \mathbb{C})\times \prod_{k\in K}\GL(\ell_k, \mathbb{C}),$$ and thus $$S_\phi\cong (\mathbb{Z}/2 \mathbb{Z})^I.$$

Now we can state the local Langlands correspondence for unitary groups.

Theorem 13 (Mok, Kaletha-Mingeuz-Shin-White, Langlands) There is a natural finite-to-one map (local Langlands parametrization) $$\Irr(\UU(V))\rightarrow \Phi_n(E/F), \quad \pi\mapsto \phi_\pi.$$ Setting the $L$-packet $$\Pi^V(\phi)=\{\pi\in\Irr(\UU(V)): \phi_\pi=\phi\}.$$ Then there are bijections (depending on some auxiliary choice of the Whittaker datum) $$\coprod_{V}\Pi^V(\phi)\cong\hat S_\phi=\{\chi: S_\phi\mapsto\{\pm1\}\},\quad \pi\mapsto\chi_\pi.$$ Here $V$ runs over all isomorphism classes of $n$-dimensioanl hermitian spaces.

Moreover, these are compatible with unramified local Langlands correspondence: if $\UU(V)$ and $\pi$ are unramified, then $\phi_\pi$ is trivial on $I_F$ and $\SU(2)$ and $\phi_\pi(\Frob)$ given by the unramified Langlands parameter $L(\pi)$ and $\chi_\pi=1$. It also satisfies local-global compatibility and endosocpic relations which characterize this correspondence.

Definition 20 Let $\pi \in\Irr(\UU(V))$, by the local Langlands correspondence for $\GL_n(E)$, the representation $\phi_{\pi, E}$ gives $\Pi\in\Irr(\GL_n(E))$. We denote by $\mathrm{BC}(\pi):=\Pi$.

By Proposition 7, $\mathrm{BC}(\pi)$ determines $\phi_\pi$ and thus can serve as a substitute for it. For example, if $\mathrm{BC}(\pi)$ is generic, we can recover $S_{\phi_\pi}$ as follows. By Bernstein-Zelevinsky, we may write $\mathrm{BC}(\pi)$ uniquely as a parabolic induction (an isobraic sum) $$\mathrm{BC}(\pi)=I_{P(E)}^{\GL_n(E)}(\Pi_1\otimes \cdots \otimes \Pi_k)=:\Pi_1 \boxplus \cdots \boxplus \Pi_k,$$ wehre $\Pi_i$ are irreducible essentially square-integrable representations. Rewriting it according to the shape of the $\phi_E$, $$\boxplus_{i\in I} n_i\Pi_i\boxplus \boxplus_{j\in J}  2m_j\Pi_j\boxplus \boxplus_{k\in K}l_k(\Pi_k \boxplus \tilde \Pi_k^\sigma),$$ where $\Pi_i$ are square-integrable and $(-1)^{n+1}$-conjugate-dual (equivalently, $L(s,\Pi_i, \mathrm{As}^{(-1)^{n+1}})$ has a pole at $s=0$), and similarly for $\Pi_j$ and $\Pi_k$. Then $$S_{\mathrm{BC}(\pi)}:=(\mathbb{Z}/ 2 \mathbb{Z})^I\cong S_{\phi_\pi}.$$ Let $\Pi\in\Irr(\GL_n(E))$ be $(-1)^{(n+1)}$-conjugate-dual, correpsonding to a Langlands parameter $\phi$ of $\UU(V)$. Then $$\Pi^V(\phi)=\mathrm{BC}^{-1}(\Pi),$$ which is compatible with the previous identification $\hat S_\phi\cong\hat S_{\Pi}$, and we may recover an element of $\Pi^V(\phi)$ by $\pi$ and $\chi_\pi\in \hat S_\Pi$.

This substitute is purley cosmetic in the local case, but will be more essential in the global case (as the global Langlands parametrization is not yet available).

Let $k'/k$ be a quadratic extension of global fields.

Definition 21 Define the space of cuspidal automorphic forms of Ramanujan type $$\mathcal{A}_\mathrm{cusp,  Ram}(\UU(V))=\bigoplus_{\pi\hookrightarrow \mathcal{A}_\mathrm{cusp}(\UU(V))\atop \mathrm{BC}(\pi)\text{ generic}}\pi.$$

The hypothetical global Langlands correpsondence gives a map from $\mathcal{A}_{\mathrm{cusp,Ram}}(\UU(V))$ to global discrete (i.e., with finite centralizer) $L$-parameters $$\phi:\mathcal{L}_k\rightarrow {}^L\UU(V).$$ These discrete $L$-parameters $\phi$ (in view of Proposition 7) should be the same as $n$-dimensional $(-1)^{n+1}$-conjugate-dual admissible representations $$\phi_{k'}: \mathcal{L}_{k'}\rightarrow \GL(M)$$ such that $\phi_{k'}$ is a direct sum of distinct irreducible $(-1)^{n+1}$-conjugate-daul representations (notice $n_i=1$, $m_j=0$, $l_k=0$ is the only way to get finite centralizer). These $\phi_{k'}$, by the hypothetical global Langlands correpsondence, should in turn be in bijection with automorphic representations $$\Pi\hookrightarrow \mathcal{A}_{\GL_{n,k'}},\quad \Pi=\Pi_1\boxplus \cdots\boxplus\Pi_k,$$ where each $\Pi_i$ is unitary and $(-1)^{n+1}$-conjugate-dual, i.e., $L^S(s,\Pi_i,\mathrm{As}^{(-1)^{n+1}})$ has a pole at $s=1$ (taken as the definition in the global case). Forgetting about the hypothetical global Langlands parameters, we have the following theorem.

Theorem 14 (Mok, Kaletha-Mingeuz-Shin-White, endoscopic classification for unitary groups)
  1. For each $\pi\hookrightarrow\mathcal{A}_\mathrm{cusp, Ram}(\UU(V))$, $\mathrm{BC}(\pi)$ is of the form $$\mathrm{BC}(\pi)=\Pi_1\boxplus \cdots \boxplus \Pi_k,$$ where $\Pi_i$'s are distict unitary cuspidal automorphic representations of $\GL_{n_i, k'}$ such that each $L^S(s, \Pi_i, \mathrm{As}^{(-1)^{n+1}})$ has a pole at $s=1$. Moreover, $\mathrm{BC}(\pi)_v=\mathrm{BC}(\pi_v)$ for all $v$.
  2. Conversely, let $\pi\hookrightarrow\mathcal{A}(\GL_{n,k'})$ be of the form $$\Pi=\Pi_1\boxplus\cdots \boxplus  \Pi_k,$$ satisfying the conditions as in (a). Set $S_\Pi=(\mathbb{Z}/2 \mathbb{Z})^{k}$. Then for all $v$, $\Pi_v$ is $(-1)^{n+1}$-conjugate-dual and we have a natural morphism $$S_\Pi\rightarrow S_{\Pi_v}$$ (the identity if $v$ splits). If $\pi= \otimes_v'\pi_v$ is an admissible irreducible representations of $\UU(V)(\mathbb{A})$ with $\mathrm{BC}(\pi_v)=\Pi_v$, then $\pi\hookrightarrow\mathcal{A}_\mathrm{cusp}(\UU(V))$ if and only if $$\prod_v \chi_{\pi_v}|_{S_\Pi}=1.$$
Remark 15
  1. This theorem can be restated as $$\mathcal{A}_\mathrm{cusp, Ram}(\UU(V))=\bigoplus_{\Pi} \bigoplus_{\pi= \otimes_v'\mathrm{BC}^{-1}(\Pi_v)}\langle \chi_{\pi}, \mathbf{1}_{S_\Pi}\rangle\pi,$$ where $\Pi$ is as in (b), and $\chi_\pi=\prod_v\chi_{\pi_v}|_{S_\Pi}$.
  2. Generalized Ramanujan conjecture for $\UU(V)$ predicts that any $\pi\hookrightarrow \mathcal{A}_\mathrm{cusp,Ram}(\UU(V))$ is tempered everywhere, i.e, $\pi_v$ is tempered for all $v$. As local base change sends tempered representations to tempered representations and nontempered representations to nontempered representations, we know that this theorem reduces the Generalized Ramanujan conjecture for $\UU(V)$ to the Ramanujan conjecture for $\GL_{n,k'}$ (which is still open).


TopThe Ichino-Ikeda conjecture

Let $k'/k$ a quadratic extension of number fields. Let $W$ be a hermitian space over $k'$ of dimension $n$. Let $V =W\oplus^{\perp} \langle e\rangle$ with $h(e)=1$. Let $H=U(W)\hookrightarrow G=U(W)\times U(V)$. Let $\pi \hookrightarrow\mathcal{A}_\mathrm{cusp}(G)$. Assume $\pi_v$ is tempered for all $v$ (in particular, $\pi\hookrightarrow \mathcal{A}_\mathrm{cusp, Ram}(G))$ is of Ramanujan type: $\mathrm{BC}(\pi)$ is tempered at one place, which implies it is cuspidal and hence generic). Fix $\langle\cdot,\cdot \rangle_v$ a $G_v$-invariant inner product on $\pi_v$, recall the local periods $$\mathcal{P}_{H,v}: (\phi_v,\phi_v')\mapsto \int_{H_v}\langle \pi_v(h_v)\phi_v, \phi_v'\rangle_v dh_v.$$ The integral is convergent since $\pi_v$ is tempered. Then $\mathcal{P}_{H,v}$ is an $H_v\times H_v$-invariant sesquilinear form on $\pi_v$.

Conjecture 4 (Ichino-Ikeda, N. Harris) For $\phi=\otimes_v\phi_v\in \pi$ nonzero, we have $$\frac{|\mathcal{P}_H(\phi)|^2}{\langle\phi,\phi\rangle_\mathrm{Pet}}=|S_{\mathrm{BC}(\pi)}|^{-1}\cdot\frac{\Delta_G^S}{(\Delta_H^S)^2}\cdot \frac{L^S(1/2, \mathrm{BC}(\pi))}{L^S(1,\pi,\mathrm{Ad})}\prod_{v\in S}\frac{\mathcal{P}_{H,v}(\phi_v,\phi_v)}{\langle\phi_v,\phi_v\rangle_v},$$ where
  1. $S$ is a sufficiently large finite set of places,
  2. the component group $S_{\mathrm{BC}(\pi)}=S_{\mathrm{BC}(\pi_W)}\times S_{\mathrm{BC}(\pi_V)}$ if $\pi= \pi_W \boxtimes \pi_V$.
  3. $\Delta_G^S/(\Delta_H^S)^2=L^S(n+1, \eta_{k'/k}^{n+1})$.
Remark 16
  1. $L^S(1, \pi, \Ad)$ is well defined and nonzero (Corollary 2).
  2. Unramified computation (N. Harris) shows that $$\frac{\mathcal{P}_{H,v}(\phi_v,\phi_v)}{\langle \phi_v,\phi_v\rangle_v}=\frac{\Delta_{G,v}}{\Delta_{H,v}^2}\cdot\frac{L(1/2, \mathrm{BC}(\pi_v))}{L(1,\pi_v,\Ad)}$$ for almost all $v$.
  3. If $\mathrm{BC}(\pi)$ is cuspidal, then $S_\mathrm{BC}(\pi)=(\mathbb{Z}/2 \mathbb{Z})^2$.
  4. This conjecture can be reformulated in terms of the relative character $$J_\pi(f)=\sum_{\mathrm{OB}(\pi)}\mathcal{P}_H(\pi(f)\phi)\overline{\mathcal{P}_H(\phi)}.$$ Namely, for $f=\prod_v f_v$, we have an identity $$J_\pi(f)=|S_{\mathrm{BC}(\pi)}|^{-1}(\Delta_H^S)^{-2}\frac{L^S(1/2, \mathrm{BC}(\pi))}{L^S(1,\pi,\Ad)}\prod_{v\in S}J_{\pi_v}(f_v),$$ where $$J_{\pi_v}(f_v)=\sum_{\mathrm{OB}(\pi_v)}\mathcal{P}_{H,v}(\pi_v(f_v)\phi_v,\phi_v)$$ is the local relative character. This identity can be thought of as a regularized version of the factorization "$|S_\mathrm{BC}(\pi)|^{-1}\prod_v' J_{\pi_v}(f_v)$".
  5. Both sides of the conjectural Ichino-Ikeda formula can be seen as $H(\mathbb{A})\times H(\mathbb{A})$-invariant sesquilinear forms on $\pi$, hence are necessarily proportional by local multiplicity one.
Theorem 15 (W. Zhang, H. Xue, Beuzart-Plessis) Let $\pi\hookrightarrow\mathcal{A}_\mathrm{cusp}(G)$ be tempered at all places, and supercuspidal at two split places. Then the Ichino-Ikeda conjecture holds for $\pi$.

TopThe proof

We may assume that $\mathcal{P}_H|_\pi\ne0$, otherwise $L^S(1/2,\mathrm{BC}(\pi))=0$ by the global Gan-Gross-Prasad conjecture and thus both sides vanish. By multiplicity one, it suffices to show the relative character identity for $J_\pi(f)$ for one test function $f$ such that $J_\pi(f)\ne0$. Choose $f=\prod f_v\in C_c^\infty(G(\mathbb{A}))$, and $f'{}=\prod_v f_v'\in C_c^\infty(G'(\mathbb{A}))$ such that

  1. $f$ and $f'$ are good.
  2. $f_v'\leftrightarrow (f_v',0,\ldots,0)$ for all $v$.
  3. $J_\pi(f)\ne0$ (we have seen this is possible in the proof the global GGP).

By the global comparison we have $$\sum_{\pi'\hookrightarrow\mathcal{A}_\mathrm{cusp}(G) \atop \mathrm{BC}(\pi')=\mathrm{BC}(\pi)} J_{\pi'}(f)=I_{\mathrm{BC}(\pi)}(f).$$ For $\pi'$ in the LHS, we have $\mathrm{BC}(\pi_v')=\mathrm{BC}(\pi_v)$ by Theorem 14. Now we need the following theorem.

Theorem 16 (Weak local GGP, Beuzart-Plessis) For any $v$ and any $\Pi_v\in\Irr(G_v')$ tempered, there is at most one $\pi_v'\in \mathrm{BC}^{-1}(\Pi_v)$ such that $$\Hom_{H_v}(\pi_v', \mathbb{C})\ne0.$$

Consequently, there is only one nonzero term on the LHS of the global comparison, and hence $$J_\pi(f)=I_{\mathrm{BC}(\pi)}(f').$$ Now we can use the factorization of the RHS to obtain the desire factorization for the LHS.

Write $\Pi=\mathrm{BC}(\pi)$, then $$I_{\mathrm{BC}(\pi)}(f')=\sum_{\phi\in\mathrm{OB}(\Pi)}\mathcal{P}_{H_1}(\Pi(f')\phi)\overline{\mathcal{P}_{H_2,\eta}(\phi)},$$ where the orthogonormal basis is taken with respect to $$\langle\phi,\phi\rangle_\mathrm{Pet}=\int_{[A_{G'}\backslash G']}|\phi(g)|^2 dg=\Vol([A_{G'}\backslash Z_{G'}])\int_{[Z_{G'}\backslash G']}|\phi(g)|^2dg.$$ where the volume factor is equal to 4 under the Tamagawa measure. Using the explicit factorization of $\mathcal{P}_{H_1}$ (Jacquet—Piatetski-Shapiro—Shalika), $\mathcal{P}_{H_2,\eta}$ (Theorem 10) and $\langle\cdot,\cdot\rangle_\mathrm{Pet}$ (Theorem 11) in terms of the Whittaker model of $\Pi$, we have $$I_\Pi(f')=\frac{1}{4}\prod_v{}'I_{\Pi_v}(f_v'),$$ where $$I_{\Pi_v}(f_v')=\sum_{W\in \mathrm{OB}(\mathcal{W}(\Pi_v,\psi_v))} \mathcal{P}_{H_1,v}(\Pi_v(f_v')W)\mathcal{P}_{H_2,\eta,v}(W).$$ Here $\mathcal{W}(\Pi_v, \psi_v)$ is the local Whittaker model with respect to $N'{}=N_{n,k'}\times N_{n+1,k'}$, and $\psi_{N'}=\psi_n \otimes \psi_{n+1}^{-1}$. Then $$\mathcal{P}_{H_1,v}(W)=\int_{N_{1,v}\backslash H_{1,v}} W(h_1)dh_1.$$ $$\mathcal{P}_{H_2,v}(W)=\int_{N_{2,v}\backslash H_{2,v}^P} W(h_2) dh_2,$$ where $P=P_{n,k'}\times P_{n+1,k'}$ $N_2=N'\cap H_2$, and $H_2^P=H_2\cap P$. The orthogonormal basis is take with respect to the scalar product $$(W,W)_v=\int_{N'_v\backslash P_v}|W(p)|^2 dp.$$ By this factorization, it suffices check the following local relative character relations.

Theorem 17 (Local relative character relations, Beuzart-Plessis) There exists explicit constants $(\kappa_v)_v$ such that $\prod_v \kappa_v=1$, and for every tempered $\pi_v\in\Irr(G_v)$ with $\Hom_{H_v}(\pi_v, \mathbb{C})\ne0$, and $f_v\leftrightarrow f_v'$, we have $$J_{\pi_v}(f_v)=\kappa_v \cdot I_{\mathrm{BC}(\pi_v)}(f_v').$$
Remark 17
  1. This was proved by W. Zhang for unramified or supercuspidal representations (and easy at split places).
  2. We can show that this gives a spectral characterization of transfers. Namely, $f_v\leftrightarrow f_v'$ if and only if the local relative character relation holds for every tempered $\pi_v\in\Irr(G_v)$ such that $\Hom(\pi_v, \mathbb{C})\ne0$. Using this spectral characterization, for example, one can check the fundamental lemma for the full spherical Hecke algebra as both sides of the local relative character relation can be explicitly computed.

Next we will discuss previously mentioned key ingredients in the proof of the global GGP and Ichino-Ikeda conjectures:

  1. the existence of smooth matching,
  2. the fundamental lemma,
  3. the local GGP conjectures,
  4. the local relative character relation.

TopSmooth transfer for p-adic fields

Now we come back to the local situation. Let $E/F$ be a quadratic extension of $p$-adic fields. Let $W\subseteq V$ hermitian spaces over $E$ of dimension $n$ and $n+1$, with $V=W \oplus^{\perp} \langle e\rangle$, $h(e)=1$. Let $H^W=\UU(W)$, $G^W=\UU(W)\times \UU(V)$.

TopDefinition of transfer factors

Recall that $$H_1(F)\backslash G'(F)\cong \GL_{n+1}(F),\quad (g_n,g_{n+1})\mapsto g_n^{-1}g_{n+1},$$ and $$\GL_{n+1}(E)/\GL_{n+1}(F)\cong S(F),\quad g\mapsto\nu(g):=g \bar g^{-1}.$$ Recall that $G_\mathrm{rs}'(F)$ to be the inverse image of $S_\mathrm{rs}(F)$ defined by the conditions in Definition 7. Fix $\eta': E^\times \rightarrow \mathbb{C}^\times$ extending $\eta_{E/F}$ (when $E/F$ is unramified, we may choose $\eta'$ to be the unramified quadratic character).

Definition 22 For $\gamma=(g_n,g_{n+1})\in G_\mathrm{rs}'(F)$, we define the transfer factor to be $$\Omega(\gamma):=\eta'(\det g_{n}^{-1}g_{n+1})^n\eta'(\det(e_{n+1}, se_{n+1},\ldots, s^ne_{n+1})),$$ where $s=\nu(g_n^{-1}g_{n+1})$. Different choices of $\eta'$ define transfer factors differing from each other by a smooth function.

It is easy to check that $$\Omega(h_1\gamma h_2)=\eta(h_2)\Omega(\gamma),\quad (h_1, h_2)= H_1(F)\times H_2(F).$$ In the global setting (relative to $k'/k$), if we pick $\eta': \mathbb{A}_{k'}^\times/(k')^\times\rightarrow \mathbb{C}$ extending $\eta_{k'/k}$ and construct $\Omega_v$ using $\eta_v'$ for all $v$, then $$\prod_v\Omega_v(\gamma)=1, \quad \gamma\in G_\mathrm{rs}'(k).$$


TopReduction to Lie algebras

Definition 23 Define Lie algebras $$\mathfrak{u}^V(F):=\Lie(\UU(V)(F))=\{X\in\mathfrak{gl}_E(V): X+X^*=0\},$$ $$\mathfrak{s}(F):=\Lie (S(F))=\{X\in\mathfrak{gl}_{n+1}(E): X+\bar X=0\}.$$

Fix an identification $W\cong E^n$, which in turn gives an identification $V\cong E^{n+1}$ by mapping $e$ to $e_{n+1}$, and an inclusion $\mathfrak{u}^V(F)\hookrightarrow\mathfrak{gl}_{n+1}(E).$

Definition 24 We say $X\in\mathfrak{gl}_{n+1}(E)$ is regular semisimple if $(e_{n+1}, X e_{n+1}, \ldots, X^ne_{n+1})$ and $(e_{n+1}^*, e_{n+1}^*X,\ldots, e_{n+1}^*X^n)$ are basis of $\mathrm{Mat}_{n+1,1}(E)$ and $\mathrm{Mat}_{1,n+1}$ respectively. We denote the set of regular semisimple elements by $\mathfrak{gl}_{n+1,\mathrm{rs}}(E)$. We define $\mathfrak{u}^V_\mathrm{rs}(F):=\mathfrak{u}^V(F)\cap\mathfrak{gl}_{n+1,\mathrm{rs}}(E)$, and similarly $\mathfrak{s}_\mathrm{rs}(F)$.

For $X\in \mathfrak{u}_\mathrm{rs}^V(F)$ and $Y\in \mathfrak{s}_\mathrm{rs}(F)$, they have trivial stabilizers and closed orbits. Analogous to the group case, we introduce the following definitions.

Definition 25 We define orbital integrals $$\Orb(X, \phi^V)=\int_{\UU(W)(F)} \phi^V(hXh^{-1})dh,\quad \phi^V\in C_c^\infty(\mathfrak{u}^V(F)),$$ $$\Orb(Y, \phi')=\int_{\GL_n(F)} \phi'(hYh^{-1})\eta(\det h)dh,\quad \phi'\in C_c^\infty(\mathfrak{s}_\mathrm{rs}(F)).$$
Definition 26 Say $X\in \mathfrak{u}_\mathrm{rs}^V(F)$ and $Y\in \mathfrak{s}_\mathrm{rs}(F)$ match if they are $\GL_n(E)$-conjugate in $\mathfrak{gl}_{n+1}(E)$, or equivalently if $\langle e_{n+1}^*, X^ie_{n+1}\rangle=\langle e_{n+1}^*, Y^i e_{n+1}\rangle$ for all $i\ge0$. Like the group case, this matching induces a bijection $$\coprod_W \mathfrak{u}_\mathrm{rs}^V(F)/\UU(W)(F)\cong \mathfrak{s}_\mathrm{rs}(F)/\GL_n(F).$$
Definition 27 For $Y\in \mathfrak{s}_\mathrm{rs}(F)$., we define its transfer factor to be $$\omega(Y)=\eta'(\det(e_{n+1}, Ye_{n+1},\ldots, Y^ne_{n+1})).$$ It satisfies $$\omega(hYh^{-1})=\eta(h)\omega(y).$$
Definition 28 We say $(\phi^W)_W\in C_c^\infty(\mathfrak{u}^V(F))$ and $\phi'\in C_c^\infty(\mathfrak{s}(F))$ match if $$\Orb(X, \phi^W)=\omega(Y)\Orb(Y, \phi')$$ for all $X\in \mathfrak{u}_\mathrm{rs}^V(F)\leftrightarrow Y\in \mathfrak{s}_\mathrm{rs}(F)$.
Theorem 18 (Smooth transfer, Lie algebra version) Every $(\phi^W)_W\in  C_c^\infty(\mathfrak{u}^V(F))$ matches some $\phi'\in C_c^\infty((\mathfrak{s}(F)))$, and vice versa.
Proposition 8 Theorem 18 implies Theorem 5.

We first reduce Theorem 5 to a transfer statement between $\UU(V)$ and $S(F)$ (inhomogeneous version). For $f^W\in C_c^\infty(G^W(F))$, we define its projection $$\tilde f^W(g_v)=\int_{H^W(F)}f^W(h(1, g_v))dh\in C_c^\infty(\UU(V)(F)).$$ Similarly for $f'\in C_c^\infty(G'(F))$, we define its projection $$\tilde f'(s)=\int_{H_1(F)\times \GL_{n+1}(F)} f'(h_1(1,gh_2))\eta'(\det gh_2)^ndh_1dh_2\in C_c^\infty(S(F)),$$ where $g\in \GL_{n+1}(E)$ is chosen such that $\nu(g)=s$. We have the surjective projections $p: G^W(F)\rightarrow \UU(V)(F)$ and $p': G'(F)\rightarrow S(F)$ with $$\Orb(\delta, f^W)=\Orb(p(\delta), \tilde f^W),$$ and $$\Orb(\gamma,f')=\eta'(\det \gamma_{n}^{-1}\gamma_{n+1})\Orb(p'(\gamma),\tilde f').$$

So Theorem 5 reduces to

Theorem 19 (Smooth transfer, inhomogeneous version) For every $(\Phi^W)_W\in C_c^\infty(\UU(V)(F))$, there exists $\Phi'\in C_c^\infty(S(F))$ such that $$\Orb(g, \Phi^W)=\tilde \Omega(s)\Orb(s,\Phi')$$ for all $g\in \UU(V)_\mathrm{rs}(F)\leftrightarrow s\in S_\mathrm{rs}(F)$. Here $\tilde \Omega(s)=\eta'(\det(e_{n+1},\ldots, s^ne_{n+1}))$.

To pass from the group to the Lie algebra, one usually use the exponential map (Harish-Chandra's descent to Lie algebra). It turns out that in our setting it is simpler to use some approximation of the exponential map, namely the Cayley map. For example, the group is not covered by the image of the exponential map from the Lie algebra, but is covered by the image of Cayley maps.

Definition 29 Take $\xi\in \ker N_{E/F}$, we define the Cayley map $c_\xi: \mathfrak{u}^V(F)$ (resp. $\mathfrak{s}(F)$) to $\UU(V)(F)$ (resp. $S(F)$) by $$c_\xi(X):=\xi\frac{X+1}{X-1}.$$ This induces $\UU(W)(F)$ (resp. $\GL_n(F)$) equivariant isomorphisms $$\mathcal{U}^V:=\{ X\in \mathfrak{u}^V(F): \det(X-1)\ne0\}\cong \mathcal{V}_\xi^V:=\{g\in \UU(V)(F): \det (g+\xi)\ne0\}.$$ and $$\mathcal{U}'{}:=\{Y \in \mathfrak{s}(F): \det(Y-1)\ne0\}\cong \mathcal{V}_\xi':=\{s\in S(F): \det (s+\xi)\ne0\}.$$

Since varying $\xi $ we obtain coverings $$\UU(V)(F)=\cup_\xi \mathcal{V}_\xi^V,\quad S(F)=\cup_\xi \mathcal{V}_\xi',$$ we know that any $\Phi^W$ or $\Phi'$ can be written as a finite sum (as there are only only finitely many eigenvalues) $$\Phi^W=\sum_ic_{\xi_i,*}\phi_i^W, \quad\Phi'{}=\sum_ic_{\xi_i,*}\phi_i',$$ where $\phi_i^W\in C_c^\infty(\mathcal{U}^V)$, $\phi_i'\in C_c^\infty(\mathcal{U}')$ and the pushforward function is defined as $$(c_{\xi,*}\phi)(g)=
  \phi(x), & g=c_\xi(x), \\
  0, & \text{otherwise}.

We can easily check the following lemma.

Lemma 3
  1. $X\in \mathcal{U}_\mathrm{rs}^V\leftrightarrow Y\in \mathcal{U}_\mathrm{rs}'$ if and only if $c_\xi(X)\in \UU(V)_\mathrm{rs}\leftrightarrow c_\xi(Y)\in S_\mathrm{rs}(F)$.
  2. For $X\in \mathcal{U}^V_\mathrm{rs}$, $Y\in \mathcal{U}_\mathrm{rs}'$, we have $$\Orb(c_\xi(X), c_{\xi,*}\phi^W)=\Orb(X, \phi^W),$$ $$\Orb(c_\xi(Y), c_{\xi,*}\phi')=\Orb(Y, \phi').$$
  3. For $Y\in \mathcal{U}_\mathrm{rs}'$, $$\tilde\Omega(c_\xi(Y))=\alpha_\xi\cdot \eta'(\det(1-Y))^{-n}\omega(Y),$$ for some $\alpha_\xi\in \mathbb{C}^\times$.
  4. If $X\in \mathcal{U}_\mathrm{rs}^V(F)\leftrightarrow Y\in \mathfrak{s}_\mathrm{rs} (F)$, then $\det(1-X)=\det(1-Y)$.

Now we can finish the proof.

Proof (Proposition 8) It remains to show that Theorem 18 implies Theorem 19. By the linearity of the transfer, we may assume $\Phi^W=c_{\xi,*}\phi^W$ for some $\xi\in \ker N_{E/F}$ and $(\phi^W)_W\in C_c^\infty(\mathcal{U}^V)$. By Theorem 18, there exists $\phi'\in C_c^\infty(\mathfrak{s}(F))$ matching $(\phi^W)_W$. Since $\supp(\phi^W)\subseteq \mathcal{U}^V$, there exists $\beta\in C_c^\infty(E^\times)$ such that $$\phi^W(X)=\beta(\det(1-X))\phi^W(X).$$ By Lemma 3 (d), we have $\beta(\det(1-Y))\phi'(Y)$ also matches $(\phi^W)_W$. So we may assume that $\supp(\phi')\subseteq \mathcal{U}'$.

We set $\phi''(Y)=\alpha_\xi^{-1}\eta'(\det(1-Y))^n \phi'(Y)$, then Lemma 3 (b) and (c) imply that $$\tilde\Omega(c_\xi(Y))\Orb(c_\xi(Y), c_{\xi,*}\phi'')=\omega(Y)\Orb(Y, \phi')$$ which is equal to $$\Orb(X,\phi^W)=\Orb(c_\xi(X), \Phi^W).$$ Therefore by Lemma 3 (a), $\Phi^W$ matches $\Phi'{}=c_{\xi,*}\phi''$.

TopReduction to transfer around zero

Let $X$ be an affine variety over a field $F $ with an action by a reductive group $G$ over $F $. We define the GIT quotient $X\sslash G:=\Spec F[X]^G$. The GIT quotient is a categorical quotient, namely any $G$-invariant morphism $X\mapsto Y$ factors uniquely through $\pi: X\rightarrow X\sslash G$. The geometric points of the GIT quotient in general are not in bijection with the $G$-orbits in $X$. However, for $x\in (X\sslash G)(\bar F)$, $\pi_{\bar F}^{-1}(x)$ contains a unique closed $G(\bar F)$-orbit (called a semisimple orbit), giving a bijection $$(X\sslash G)(F)\cong\{\text{semisimple } G(\bar F)\text{-orbits in }X(\bar F)\}.$$ This is not true over $F $: for $x\in (X\sslash G)(F)$, $\pi_F^{-1}(x)$ might be empty, and even if it is not empty $\mathcal{O}_x\cap X(F)$ can split into more than one $G(F)$-orbit (e.g., the difference between stable conjugacy and usual conjugacy in the group case).

To bypass this difficulty, assume there is a $G$-invariant nonempty open subset $X_\mathrm{rs}\subseteq X$ such that every $x\in X_\mathrm{rs}(F)$ has closed orbit and trivial stabilizer. Then we define $$(X\sslash G)_\mathrm{rs}:=\pi(X_\mathrm{rs})\cong X_\mathrm{rs}/G,$$ which is a geometric quotient. For $x\in (X\sslash G)_\mathrm{rs}(F)$, $\pi_F^{-1}(x)$ is either empty or exactly one $G(F)$-orbit in $X_\mathrm{rs}(F)$. Therefore we obtain an open embedding in the analytic topology that $$X_\mathrm{rs}(F)/G(F)\hookrightarrow (X\sslash G)_\mathrm{rs}(F).$$

Proposition 9 There is a (necessarily unique) isomorphism $$\mathfrak{u}^V\sslash \UU(W)\cong \mathfrak{s}\sslash \GL_n$$ such that its restriction to the regular semisimple loci gives back the matching of orbits as $$\mathfrak{u}_\mathrm{rs}^V(F)/\UU(W)(F)\hookrightarrow (\mathfrak{u}^V\sslash \UU(W))_\mathrm{rs}(F)\cong (\mathfrak{s}\sslash \GL_n)_\mathrm{rs}(F)=\mathfrak{s}_\mathrm{rs}(F)/\GL_n(F).$$
Proof Notice that the action of $\UU(W)$ on $\mathfrak{u}^V$ and $\GL_n$ on $\mathfrak{s}$ both become the action of $\GL_{n,E}$ on $\mathfrak{gl}_{n+1,E}$ after base change to $E$. Therefore $$\mathfrak{u}^V\sslash \UU(V)\cong (\mathfrak{gl}_{n+1,E}\sslash \GL_{n,E})^\sigma, \quad \mathfrak{s}\sslash \GL_n\cong (\mathfrak{gl}_{n+1,E}\sslash \GL_{n,E})^\tau,$$ where $\sigma$, $\tau$ are involutions of $\mathfrak{gl}_{n+1,E}$ with fixed points $\mathfrak{u}^V, \mathfrak{s}$ respectively. Explicitly, $$\sigma(X)=-J {}^t\bar X J^{-1},\quad \tau(X)=-\bar X$$ where $J\in \GL_n(E)$ represents the hermitian form on $W$.

It remains to check for that the actions of $\sigma$ and $\tau$ are the same on the GIT quotients. It is enough to check this for an open dense subset of the GIT quotients, namely for $X\in\mathfrak{gl}_{n+1}(E)$ in general position, $\sigma(X)\sim \tau(X)$ under $\GL_n(E)$-action. Equivalently, we need to check that $X\sim\sigma\tau(X)=J{}^tX J^{-1}$, i.e., $X\sim {}^tX$. But as in the group case (Proposition 2 (b)), for $X, Y\in\mathfrak{gl}_{n+1,\mathrm{rs}}(E)$, then $X\sim Y$ under $\GL_n(E)$ if and only if $\langle e_{n+1}^*, X^i e_{n+1}\rangle=\langle e_{n+1}^*, Y^i e_{n+1}\rangle$ for all $i\ge0$. Applying this criterion to $Y={}^tX$ it follows that $X\sim {}^tX$ as desired.


Definition 30 Let $\mathcal{A}=\mathfrak{s}\sslash \GL_n\cong \mathfrak{u}^V\sslash \UU(W)$. Let $\mathcal{A}_\mathrm{rs}=\mathfrak{s}_\mathrm{rs}\sslash \GL_n=\mathfrak{u}^\vee_\mathrm{rs} /\UU(W)\subseteq \mathcal{A}$. Then $$\mathcal{A}_\mathrm{rs}(F)=\mathfrak{s}_\mathrm{rs}(F)/\GL_n(F)=\coprod_W \mathfrak{u}^V _\mathrm{rs}(F)/\UU(W)(F).$$ Let $\pi: \mathfrak{u}\rightarrow \mathcal{A}$, $\pi' : \mathfrak{s}\rightarrow\mathcal{A}$ be the projections. Let $C_c^\infty(\mathfrak{u}(F))=\bigoplus_W C_c^\infty(\mathfrak{u}^V(F))$. Then for any $a\in \mathcal{A}_\mathrm{rs}(F)$, we have $\pi^{-1}(a)$ is one regular semisimple $\UU(W)(F)$-orbit in some $\mathfrak{u}^V(F)$, and $\pi'^{-1}(a)$ is one regular semisimple $\GL_n(F)$-orbit.
Definition 31 For $a\in \mathcal{A}_\mathrm{rs}(F)$ and $\phi\in C_c^\infty(\mathfrak{u}(F))$, we define $$\Orb_a(\phi):=\Orb(X, \phi^W)$$ for any $X\in \pi^{-1}(a)$, where $W$ is such that $X\in \mathfrak{u}^V_\mathrm{rs}(F)$. Similarly define $$\Orb_a(\phi'):=\omega(Y) \Orb(Y,\phi')$$ for any $Y\in \pi'^{-1}(a)$. We define spaces of functions on $\mathcal{A}_\mathrm{rs}(F)$, $$\Orb(\mathfrak{u}):=\{\Orb(\phi), \phi\in C_c^\infty(\mathfrak{u}(F))\},$$ $$\Orb(\mathfrak{s}):=\{\Orb(\phi'), \phi'\in C_c^\infty(\mathfrak{s}(F)).$$

Then Theorem 18 can be reformulated as the identity $$\Orb(\mathfrak{u})=\Orb(\mathfrak{s}).$$

Proposition 10 A function $\mathcal{F}: \mathcal{A}_\mathrm{rs}(F)\rightarrow \mathbb{C}$ belongs to $\Orb(?)$ ($?=\mathfrak{u}$ or $\mathfrak{s}$) if and only if it is compactly supported in $\mathcal{A}(F)$ and it coincides near every $a\in\mathcal{A}(F)$ with a function in $\Orb(?)$.
Proof Necessity is clear. Conversely, assume that $\mathcal{F} $ is compactly supported and locally an orbital integral. Then there exists a finite covering of $\supp(\mathcal{F})$ by $\cup_i U_i$ such that $\phi_i\in C_c^\infty(?)$ and $\mathcal{F}|_{U_i}=\Orb(\phi_i)|_{U_i}$. Up to refining the cover, we may assume that $U_i$'s are disjoint. Then $$\mathcal{F}=\sum_i\mathbf{1}_{U_i}\Orb(\phi_i)=\sum_i\Orb(\mathbf{1}_{\pi^{-1}(U_i)}\phi_i)\in\Orb(?).$$ Hence $\mathcal{F}$ is an orbital integral.
Definition 32 We say that the transfer exists near $a\in \mathcal{A}(F)$ if there exists $U\subseteq \mathcal{A}(F)$ an open neighborhood of $a$ such that $\Orb(\mathfrak{u})|_U=\Orb(\mathfrak{s})|_U$. By Proposition 10, to prove that the transfer exists if suffices to show that the transfer exists near every $a\in \mathcal{A}(F)$.

TopSlices and Harish-Chandra's semisimple descent

Ref: J.-M. Drezet, Luna's slice theorem and applications

Let $X$ be a connected smooth affine variety over $F $ with an action of a reductive group $G$ over $F $.

Definition 33 We say $x\in X(F)$ is semisimple if $G.x$ is Zariski closed in $X$ (equivalently, $G(F).x$ is closed in the analytic topology in $X(F)$). Then by Matsushima's criterion, the stabilizer $G_x$ is also reductive. It acts on $N_x:=T_xX/T_x(G.x)$, the normal space to $G.x$ at $x$, called the slice representation.
Theorem 20 (etale Luna's slice) There exists a locally closed $G_x$-invariant smooth affine subvariety $Z\subseteq X$ containing $x$ and a $G_x$-morphism $Z\rightarrow N_x$ sending $x$ to 0, such that we have the following Cartesian diagram with etale horizontal maps $$\xymatrix{G\times^{G_x}Z \ar[r]^{\text{etale}} \ar[d] & X \ar[d] ^{\pi_X}  \\ Z\sslash G_x \ar[r]^{\text{etale}} & X\sslash G,}\quad \xymatrix{ Z \ar[r]^{\text{etale}}  \ar[d]^{\pi_Z} & N_x \ar[d]^{\pi_N} \\ Z\sslash G_x \ar[r]^{\text{etale}} & N_x\sslash G_x,}$$ where $G\times^{G_x} Z=G\times Z/G_x$ with diagonal $G_x$-action given by $g_x(g, y)=(gg_x^{-1},g_xy)$. In particular, we have $G\times^{G_x}Z\sslash G\cong Z/G_x$.
Definition 34 Such $Z$ is called an etale slice (think: $Z$ cut every $G$-orbit in a single $G_x$-orbit; the $G$-action around $x$ can be described in terms of $G_x$-action on $Z$, which can be further linearized to the slice representation $N_x$). We may thus localize and linearize the orbital integrals to the etale slice $Z$.
Definition 35 We define $$\mathcal{A}_X:=X\sslash G,\quad \mathcal{A}_Z:=Z\sslash G_x, \quad\mathcal{A}_N:=N_x\sslash G_x.$$

Taking $F $-points, we obtain Cartesian diagrams with etale (i.e. local isomorphism) horizontal maps $$\xymatrix{(G\times^{G_x} Z)(F) \ar[r] \ar[d]  & X(F) \ar[d] \\ \mathcal{A}_Z(F) \ar[r] & \mathcal{A}_X(F),}\quad \xymatrix{Z(F) \ar[r] \ar[d] & N_x(F) \ar[d] \\ \mathcal{A}_Z(F) \ar[r] & \mathcal{A}_N(F).}$$

Definition 36 Let $\bar{\mathcal{U}}\subseteq \mathcal{A}_Z(F)$ be an open neighborhood of $\pi_Z(x)$ such that $\bar{\mathcal{U}}\rightarrow \mathcal{A}_X(F)$ and $\bar{\mathcal{U}}\rightarrow\mathcal{A}_N(F)$ are both open embeddings. Then $\mathcal{U}:=\pi_Z^{-1}(\bar{\mathcal{U}})$ is a $G_x(F)$-invariant open neighborhood of $x$, and $$\mathcal{U}\rightarrow N_x(F),\quad \mathcal{V}_x:=G(F)\times^{G_x(F)}\mathcal{U}\rightarrow X(F)$$ are both open embeddings. In this way $\mathcal{U}$ can be viewed as a $G_x(F)$-invariant open neighborhood of $0\in N_x(F)$ and $\mathcal{V}_x$ is a $G(F)$-invariant open neighborhood of $x\in X(F)$. Such $\mathcal{U}$ is called an analytic slice.

We have a commutative diagram $$\xymatrix{ & G(F) \times \mathcal{U} \ar[rd] \ar[ld]   & \\ N_x(F)\supseteq \mathcal{U} \ar[rd]  && \mathcal{V}_x\subseteq X(F) \ar[ld] \\ &\mathcal{A}_N(F)\supseteq\bar{\mathcal{U}}\subseteq \mathcal{A}_X(F)&}$$ It will help us compare $G_x(F)$-orbital integrals on $\mathcal{U}$ with $G(F)$-orbital integral on $\mathcal{V}_x$.

Assume that the regular semisimple locus $X_\mathrm{rs}\subseteq X$ is nonempty (hence open and dense), and assume that for any $y\in X_\mathrm{rs}$, we have $G_y=1$ (hence the same holds for $Z_\mathrm{rs}\subseteq X$ and $N_{x,\mathrm{rs}}\subseteq N_x$, and moreover $Z_\mathrm{rs}=Z\cap X_\mathrm{rs}$ and is equal to the preimage of $N_{x,\mathrm{rs}}$).

Definition 37 Given a character $\eta: G(F)\rightarrow\{\pm1\}$, we choose transfer factors $\omega: X_\mathrm{rs}(F)\rightarrow \mathbb{C}^\times$ such that $$\omega(g.y)=\eta(g)\omega(y),\quad g\in G(F), y\in X_\mathrm{rs}(F),$$ and $\omega_x: N_{x,\mathrm{rs}}(F)\rightarrow \mathbb{C}^\times$ such that $$\omega_x(g_x. y)=\eta(g_x)\omega_x(y),\quad g_x\in G_x(F), y\in N_{x,\mathrm{rs} }(F) .$$ We assume that $\omega$ and $\omega_x$ coincide on $\mathcal{U}_\mathrm{rs}:=\mathcal{U}\cap X_\mathrm{rs}(F)=\mathcal{U}\cap N_{x,\mathrm{rs}}(F)$.
Definition 38 For $\phi\in C_c^\infty(X(F))$, $a\in \mathcal{A}_{X,\mathrm{rs}}(F)$, we define $$\Orb_a(\phi)=\omega(y)\int_{G(F)}\phi(g.y)\eta(g)dg$$ for any $y\in \pi_X^{-1}(a)$. For $\phi_x\in C_c^\infty(N_x(F))$ and $a\in \mathcal{A}_{N, \mathrm{rs}}(F)$, we define $$\Orb_a(\phi_x)=\omega_x(y)\int_{G_x(F)}\phi_x(g.y)\eta(g)dg,$$ for any $y\in \pi_N^{-1}(a)$.

Then we have $$\Orb(\mathcal{V}_x)\subseteq \Orb(X),\quad \Orb(\mathcal{U})\subseteq \Orb(N).$$ Both subspaces can be considered as spaces of functions on $\bar{\mathcal{U}}_\mathrm{rs}$.

Proposition 11 (Harish-Chandra's semisimple descent) We have $\Orb(\mathcal{V}_x)=\Orb(\mathcal{U})$. In other words, the orbital integral locally around $x\in X(F)$ can be identified as orbital integral locally around $0\in N_x(F)$.
Proof The result follows from the commutative diagram of spaces of functions, $$\xymatrix{ & C_c^\infty(G(F) \times \mathcal{U}) \ar[rd]^{p_2} \ar[ld]_{p_1}   & \\ C_c^\infty(\mathcal{U}) \ar[rd]^{\Orb}  && C_c^\infty(\mathcal{V}_x) \ar[ld]_{\Orb} \\ & C_c^\infty(\bar{\mathcal{U}}) &}$$

where $p_1, p_2$ are given by $$p_1(\phi)(y)=\int_{G(F)}\phi(g,y)\eta(g)dg,$$ $$p_2(\phi)(g,y)=\int_{G_x(F)}\phi(gg_x^{-1}, g_xy)dg_x,$$ which are both surjective.

Remark 18 Let $\mathcal{V}=\pi_X^{-1}(\bar{\mathcal{U}})$. Then $$\Orb(X)|_{\bar{\mathcal{U}}}=\Orb(\mathcal{V}).$$ We have $$\mathcal{V}=\coprod_i G(F)\times^{G_{x_i}(F)} \mathcal{U}_i,$$ where $x_i$'s are representatives of the semisimple $G(F)$-orbits in $\pi_X^{-1}(x)$, and the $\mathcal{U}_i$ is the analytic slices for $x_i$. Moreover, we have $$N_{x_i}\sslash G_{x_i}\cong N_x\sslash G_x$$ and $U_i\subseteq N_{x_i}(F)$ is the preimage of $\bar{\mathcal{U}}$. By Proposition 11, we have $$\Orb(\mathcal{V})=\sum_i\Orb(\mathcal{U}_i).$$ The latter is also equal to $\Orb(\coprod_i \mathcal{U}_i)=\Orb(\coprod_i N_{x_i})|_{\bar{\mathcal{U}}}.$


TopProof of the local transfer away from the center

Definition 39 Consider the space $$\mathfrak{z}':=\{Y\in \mathfrak{s}(F): hYh^{-1}=Y, \forall h\in\GL_n\}.$$ It is the most singular locus in $\mathfrak{s}$ (maximal stabilizers), and consists of elements of the form $\diag(\lambda I_n, \mu)$, where $(\lambda, \mu)\in (\ker N_{E/F})^2$. Similarly, the most singular locus in $\mathfrak{u}^V$ is given by $$\mathfrak{z}^V:=\{X\in \mathfrak{u}^V(F): hXh^{-1}=X, \forall h\in \UU(W)\}$$ consists of elements of the form $\diag(\lambda\Id_W, \mu\Id_{\langle e\rangle})$, where $(\lambda, \mu)\in (\ker N_{E/F})^2$. Moreover, both spaces $\mathfrak{z}'$ and $\mathfrak{z}^V$ are mapped isomorphically to $\mathfrak{z}\hookrightarrow \mathcal{A}$, called the center of $\mathcal{A}$.

Using Harish-Chandra's semisimple descent and induction, we will prove the following.

Proposition 12 The transfer exists locally near every $a\in \mathcal{A}(F)-\mathfrak{z}(F)$, i.e., there exits an open neighborhood $\mathcal{U}\subseteq \mathcal{A}(F)$ of $a$ such that $\Orb(\mathfrak{u})|_\mathcal{U}=\Orb(\mathfrak{s})|_\mathcal{U}$.

By Harish-Chandra's semisimple descent, we have a decomposition of both orbital integral spaces in terms of slice representations. Let $\pi: \mathfrak{u}\rightarrow \mathcal{A}$, and $\pi':\mathfrak{s}\rightarrow \mathcal{A}$ be the projections. Let $X_1,\ldots,X_k\in \pi_F^{-1}(a)$ be a set of representatives of semisimple orbits. We denote $Y\in \pi_F'^{-1}(a)$ be a representative of the unique semisimple orbit. Then we have the slice representations $$N_{X_i}=T_{X_i}\mathfrak{u}/T_{X_i} \UU(W).X_i=\mathfrak{u}^V/[\mathfrak{u}(W), X_i] \curvearrowleft \UU(W)_{X_i},$$ $$N_Y=T_Y \mathfrak{s}/T_Y \GL_n.Y=\mathfrak{s}/[\mathfrak{gl}_n,Y] \curvearrowleft \GL_{n,Y}.$$ Let $$N_X:=\coprod_{i=1}^k N_{X_i}\curvearrowleft \UU_X:=\coprod \UU(W)_{X_i},$$ and $$\pi_X: N_X\rightarrow \mathcal{A}_X\cong N_{X_i}\sslash \UU(W)_{X_i},\quad \pi_Y: N_Y\rightarrow \mathcal{A}_Y\cong N_Y\sslash \GL_{n,Y}.$$ By Harish-Chandra's semisimple descent, if $\mathcal{U}$ is small enough, then $\mathcal{U}$ be comes an open neighborhood of 0 in $\mathcal{A}_X(F)$ and $\mathcal{A}_Y(F)$, and $$\Orb(\mathfrak{u})|_\mathcal{U}=\Orb(N_X)|_\mathcal{U},\quad \Orb(\mathfrak{s})|_\mathcal{U}=\Orb(N_Y)_\mathcal{U}.$$ So it remains to compare the orbital integrals on the slice representations $N_X$ and $N_Y$.

Remark 19 Two warnings are in order (which we will ignore):
  1. the identification of $\mathcal{U}$ as open neighborhoods of $\mathcal{A}_X(F)$ and $\mathcal{A}_Y(F)$ are not canonical and one needs to check the choice made are compatible.
  2. the orbital integrals on $\mathfrak{s}(F)$ and $N_Y(F)$ are twisted by $\eta_{E/F}\circ\det$ and we need to check the transfer factors are also compatible.

We will see that essentially (i.e. up to the center) there exists $(F_j)_{j\in I}$ finite extensions of $F $ and $1\le n_j<n$ such that we have a "decomposition" $$N_X\cong \prod_{j\in I} R_{F_j/F} \mathfrak{u}_{n_j+1, E_j/F_j},\quad \UU_X\cong \prod_{j\in I} R_{F_j/F}\UU_{n_j+1, E_j/F_j}.$$ where $E_j=E \otimes_F F_j$. And similarly $$N_Y\cong\prod_{j\in I}R_{F_j/F} \mathfrak{s}_{n_j+1, E_j/F_j},\quad \GL_{n,Y}\cong\prod_{j\in I}R_{F_j/F}\GL_{n_j,F_j}.$$ So by induction on $n$ (assuming the transfer exists near all $a\in \mathcal{A}(F)$ for smaller $n$), we obtain the desired comparison between the orbital integral spaces on $N_X$ and $N_Y$. Thus our remaining goal is to prove such decompositions.

Let us first describe the semisimple orbits. Let $E=F[\sqrt{\tau}]$. Then we have $$\mathfrak{gl}_{n+1,F}\cong \mathfrak{s},\quad Y\mapsto \sqrt{\tau}Y,$$ $$\mathfrak{h}(V):=\{ X\in \mathfrak{gl}_E(V): X=X^*\}\cong \mathfrak{u}^V,\quad X\mapsto \sqrt{\tau}X.$$ Define $$\widetilde{\mathfrak{gl}_n}=\mathfrak{gl}_n \oplus F^n \oplus F_n\curvearrowleft \GL_n,\quad h(A, b,c)=(hAh^{-1}, hb, ch^{-1})$$ Similarly define $$\widetilde{\mathfrak{h}(W)}:=\mathfrak{h}(W)\oplus W, \curvearrowleft \UU(W),\quad h(A, w)=(hAh^{-1}, hw).$$ Then we have identifications $$\mathfrak{gl}_{n+1}\cong\widetilde{\mathfrak{gl}_n}\oplus \mathbb{G}_a,\quad \left(\begin{smallmatrix} A & b\\ c & \lambda\end{smallmatrix}\right) \mapsto (A, b, c,\lambda)$$ and $$\mathfrak{h}(V)\cong\widetilde{\mathfrak{h}(W)} \oplus \mathbb{G}_a,\quad \left(\begin{smallmatrix}A & w \\ h(\cdot,w) &\lambda \end{smallmatrix}\right) \mapsto (A, w,\lambda)$$ compatible with the action of $\GL_n$ and $\UU(W)$ respectively. Moreover, $$\mathcal{A}=\tilde{\mathcal{A}}\times \mathbb{G}_a,\quad \tilde{\mathcal{A}}=\widetilde{\mathfrak{gl}_n}\sslash \GL_n\cong\widetilde{\mathfrak{h}(W)}\sslash \UU(W).$$ Define $$\widetilde{\mathfrak{gl}_{n,\mathrm{rs}}}=\left\{(A, b,c):
  (b, \ldots, A^{n-1}b) \text{ a basis of }F^n\\
  (c,\cdots,cA^{n-1}) \text{ a basis of }F_{n}
\right\}$$ and $$\widetilde{\mathfrak{h}(W)_\mathrm{rs}}=\{(A,w): (w,\ldots, A^{n-1}w) \text{ a basis of } W\}$$ Then $$\mathcal{A}_\mathrm{rs}=\tilde{\mathcal{A}_\mathrm{rs}}\times \mathbb{G}_a.$$

Define $$\pi: \tilde{\mathfrak{h}}:=\coprod_W \widetilde{\mathfrak{h}(W)}\rightarrow \tilde{\mathcal{A}},\quad \pi': \widetilde{\mathfrak{gl}_n}\rightarrow \tilde{\mathcal{A}}.$$ The local transfer now becomes a comparison between orbital integral spaces on $\tilde{\mathfrak{h}}$ and $\widetilde{\mathfrak{gl}_n}$. ; $$\Orb(\tilde{\mathfrak{h}})=\Orb(\widetilde{\mathfrak{gl}_n}).$$

Proposition 13 (Rallis-Schiffmann)
  1. $X=(A,w)\in \widetilde{\mathfrak{h}(W)}$ is semisimple if and only if $W_X^+:=\mathrm{span}\{A^kw\}_{k\ge0}$ is nondegenerate and $X^-:= A|_{W_X^-}$ is semisimple (in the usual sense), where $W_X^-:=(W_X^+)^\perp$.
  2. $Y=(A, b, c)\in \widetilde{\mathfrak{gl}_n}$ is semisimple if and only if $V_Y^+:=\mathrm{span}\{A^kb\}_{k\ge0}$ and $V^{+*}:=\mathrm{span}\{cA^k\}_{k\ge0}$ are in perfect duality, and $Y^-:=A|_{V_Y^-}$ is semisimple (in the usual sense), where $V_Y^-:=(V_Y^{+*})^\perp$.
  3. $X\in \widetilde{\mathfrak{h}(W)}_\mathrm{ss}$, $Y\in \widetilde{\mathfrak{gl}_n}_{,\mathrm{ss}}$ have the same image in $\tilde{\mathcal{A}}$ if and only if
    1. $\dim (W_X^+)=\dim (V_Y^+)$ and $X^+=X|_{W_X^+}\in \widetilde{\mathfrak{h}(W_X+)}_\mathrm{rs}$, $Y^+=Y|_{V_Y^+}\in \widetilde{\mathfrak{gl}_n(V_Y^+)}_\mathrm{rs}$ have matching orbits.
    2. $X^-$, $Y^-$ have the same characteristic polynomial.

We will not prove this proposition. Instead we draw the following consequence.

Corollary 3 $Y, Y'\in \widetilde{\mathfrak{gl}_n}_{,\mathrm{ss}}(F)$ have the same image in $\tilde{\mathcal{A}}(F)$ if and only if they are $\GL_n(F)$-conjugate.
Proof By Proposition 13 (c), and that there exists $X\in \tilde{\mathfrak{h}}_\mathrm{ss}(F)$ with the same image in $\tilde{\mathcal{A}}(F)$, we know that $Y^+$ and $Y'^+$ match the same regular semisimple element, hence they are conjugate. Similarly $Y^-$ and $Y'^-$ are semisimple elements with the same characteristic polynomial, hence they are also conjugate.

Let $Y\in \widetilde{\mathfrak{gl}_n}_{,\mathrm{ss}}(F)$, and $a=\pi'(Y)\in\tilde{\mathcal{A}}(F)$. Let $P_{Y^-}$ be the characteristic polynomial of $Y^-$. Let $W^+$ be the unique hermitian space over $E$ such that $Y^+$ matches $X^+\in \widetilde{\mathfrak{h}(W^+)}_\mathrm{rs}$. Then by Proposition 13 (c), the orbits in $\pi_{F}^{-1}(a)\cap \tilde{\mathfrak{h}}_\mathrm{ss}(F)$ is in bijection with conjugacy classes of pairs $(W^-, X^-)$ where

  1. $W^-$ is an hermitian space over $E$, and
  2. $X^-\in \mathfrak{h}(W^-)$ such that $P_{X^-}=P_{Y^-}$.

In this way these semisimple orbits are further in bijection with $I $-tuple of hermitian spaces $(W_j)_{j\in I}$ over $E_j$ of dimension $n_j$, where $P_{Y^-}=\prod_{j\in I} P_j^{n_j}$, $F_j=F[T]/P_j$ and $E_j=E \otimes_F F_j$, given by $$W^-{}=\bigoplus R_{E_j/E}W_j,\quad X^-{}=\text{multiplication by } T \text{ on each }W_j.$$ This finishes the description of the semisimple orbits.

Now we can also compute the centralizers $$\GL_{n,Y}=\GL(V^-)_{Y^-}=\prod_{j\in I}R_{F_j/F}\GL_{n_j,F_j}$$ and $$\UU(W)_X=\UU(W_X^-)_{X^-}=\prod_{j\in I}R_{F_j/F}\UU(W_j).$$

Finally we can describe the slice representations. Let $N_Y^+$ be the slice representation at $Y^+$ inside $\widetilde{\mathfrak{gl}(V^+)}$, which is a representation of the trivial group (as $Y^+$ is regular semisimple). Then $$N_Y=\mathfrak{gl}(V^-)_{Y^-} \oplus V^- \oplus V^{-,*} \oplus N_Y^+\cong \prod_{j\in I}R_{F_j/F}\widetilde{\mathfrak{gl}_{n_j, F_j}} \oplus N_Y^+,$$ as a representation of $\GL_{n,Y}=\GL(V^-)_{Y^-}$. Similarly, let $N_X^+$ be the slice representation at $X^+$ in $\widetilde{\mathfrak{h}(W^+)}$, which is a representation of the trivial group. Then $$N_X=\mathfrak{h}(W_X^-)_{X^-} \oplus W_X^- \oplus N_X^+\cong\prod_{j\in I}R_{F_j/F}\widetilde{\mathfrak{h}(W_j)} \oplus N_X^+,$$ as a representation of $\UU(W)_X=\UU(W_X^-)_{X^-}$.

This proves the desired decomposition (up to the central part $N_Y^+$ and $N_X^+$). Next time we will study what happens at the center.


Definition 40 We say $X=(A,w)\in \widetilde{\mathfrak{h}(W)}$ is regular semisimple if $(w, \ldots, A^{n-1}w)$ is a basis of $W$. We say $Y=(A,b,c)\in \widetilde{\mathfrak{gl}_n}$ is regular semisimple if $(b,\ldots, A^{n-1}b)$ is a basis of $F^n$ and $(c,\cdots, cA^{n-1})$ is a basis of $F_n$.

Then we have a matching of orbits $$\coprod_W\widetilde{\mathfrak{h}(W)}_\mathrm{rs}/\UU(W)(F)\cong \widetilde{\mathfrak{gl}_{n,\mathrm{rs}}}(F)/\GL_n(F),\quad X=(A,w)\leftrightarrow Y=(A',b,c)$$ given by the condition $h(A^iw, w)=c(A')^ib$ for all $i\ge0$, or equivalently, $X$ and $Y$ are in the same $\GL_n(E)$-orbit in $\widetilde{\mathfrak{gl}_n}(E)$.

Definition 41 For $f^W\in C_c^\infty(\widetilde{\mathfrak{h}(W)})$ and $X\in \widetilde{\mathfrak{h}(W)}_\mathrm{rs}(F)$, we define $$\Orb(X, f^W)=\int_{\UU(W)(F)}f^W(h.x)dh.$$ For $f'\in C_c^\infty(\widetilde{\mathfrak{gl}_n})$ and $Y\in \widetilde{\mathfrak{gl}_{n,\mathrm{rs}}(F)}$, we define $$\Orb(Y,f')=\int_{\GL_n(F)}f'(h.Y)\eta_{E/f}(h)dh.$$ We say $(f^W)_W\in C_c^\infty(\widetilde{\mathfrak{h}})\leftrightarrow f'\in C_c^\infty(\widetilde{\mathfrak{gl}_n})$ match if for all matching $X\in \widetilde{\mathfrak{h}(W)}_\mathrm{rs}\leftrightarrow Y\in\widetilde{\mathfrak{gl}_{n}}_{,\mathrm{rs}}$, we have $$\Orb(X, f^W)=\omega(Y)\Orb(Y,f'),\quad \omega(A,b,c):=\eta(\det(b,\ldots, A^{n-1}b)).$$
Definition 42 Define $\mathcal{N}^W\subseteq \widetilde{\mathfrak{h}(W)}(F)$, $\mathcal{N}_n\subseteq \widetilde{\mathfrak{gl}_n}(F)$ be the preimage of the center $\widetilde{\mathfrak{z}}\hookrightarrow \widetilde{\mathcal{A}}$, known as the nilpotent cone (up to the center). Then $X\in \mathcal{N}^W$ (resp. $Y\in \mathcal{N}$) if and only if the closure of its orbit orbit meets the center $\mathfrak{z}^W$ (resp. $\mathfrak{z}_n $) of $\widetilde{\mathfrak{h}}(W)$ (resp. $\widetilde{\mathfrak{gl}_n}$). Define $\mathcal{N}=\coprod_W \mathcal{N}^W$.

We can now reformulate Proposition 12 in a more down-to-earth way as follows.

Proposition 14
  1. Let $f=(f^W)_W\in C_c^\infty(\widetilde{\mathfrak{h}(F)}-\mathcal{N})$. Then there exists $f'\in C_c^\infty(\widetilde{\mathfrak{gl}_n}(F))$ matching $f$.
  2. Let $f'\in C_c^\infty(\widetilde{\mathfrak{gl}_n}(F)-\mathcal{N}_n)$. Then there exists $f\in C_c^\infty(\widetilde{\mathfrak{h}}(F))$ matching $f'$.

TopTransfer and Fourier transforms

To deal with the missing functions supported on the nilpotent cone, we shall use (partial) Fourier transform following W. Zhang, to generate new transfers from old ones. This idea of studying the compatibility of transfer and Fourier transform goes back to Waldspurger (for proving the endoscopic transfer using the fundamental lemma). However unlike Waldspurger, the proof of W. Zhang will be entirely local.

Definition 43 Let $\mathcal{F}_1$ be the partial Fourier transform with respect to $W$ or $F^n \oplus F_n$. Namely, for $f\in C_c^\infty(\widetilde{\mathfrak{h}(W)})$, we define $$(\mathcal{F}_1f)(A,w)=\int_W f(A,w')\psi(\Tr_{E/F}h(w,w'))dw'.$$ For $f'\in C_c^\infty(\widetilde{\mathfrak{gl}_n})$, we define $$(\mathcal{F}_1f')(A,b,c)=\int_{F^n \oplus F_n}f'(A,b',c')\psi(cb'+c'b) db'dc'.$$ Here we fix a nontrivial character $\psi:F\rightarrow \mathbb{C}^\times$ and the Haar measures are chosen to be the self-dual ones (i.e. $\mathcal{F}_1$ preserves the $L^2$-norms).
Definition 44 Let $\mathcal{F}_2$ be the partial Fourier transform with respect to $\mathfrak{h}(W)$ or $\mathfrak{gl}_n$. Namely, we define $$(\mathcal{F}_2f)(A,w)=\int_{\mathfrak{h}(W)}f(A',w)\psi(\Tr AA')dA',$$ and $$(\mathcal{F}_2f')(A,b,c)=\int_{\mathfrak{gl}_n(F)}f'(A',b,c)\psi(\Tr AA')dA'.$$
Definition 45 Let $\mathcal{F}_3$ be the total Fourier transform on $\widetilde{\mathfrak{h}}$ or $\widetilde{\mathfrak{gl}_n}$. Namely, we define $$(\mathcal{F}_3f)(X)=\int_{\widetilde{\mathfrak{h}(W)}} f(X')\psi\langle X,X'\rangle dX', $$ where $\langle(A,w), (A',w')\rangle:=\Tr AA'+\Tr_{E/F} h(w,w')$, and $$(\mathcal{F}_3f')(Y)=\int_{\widetilde{\mathfrak{gl}_n}(F)}f'(Y')\psi(\langle Y,Y'\rangle)dY',$$ where $\langle (A,b,c),(A', b', c')\rangle:=\Tr AA'+c'b+cb'$.

The following basic properties are easy to check.

Proposition 15
  1. (equivariance of Fourier transforms) Let $(^hf)(X):=f(h^{-1}X)$. Then $\mathcal{F}_i(^hf)=^h(\mathcal{F}_i(F))$ for $i=1,2,3$ for any $f\in C_c^\infty(\widetilde{\mathfrak{h}(W)})$ and $h\in \UU(W)(F)$. Similarly statement holds for $\widetilde{\mathfrak{gl}_n}$.
  2. $\mathcal{F}_3=\mathcal{F}_2\circ \mathcal{F}_1=\mathcal{F}_1\circ \mathcal{F}_2$.
Theorem 21 (W. Zhang, Fourier transforms preserve smooth transfer) There exists explicit constants $c_1^W, c_2^W, c_3^W\in \mathbb{C}^\times$ such that if $f=(f^W)_W\leftrightarrow f'$, then $(c_i^W \mathcal{F}_if^W)\leftrightarrow \mathcal{F}_if'$ for $i=1,2,3$.

Consequently, if $(f^W)_W$ is transferable, then $(c_i^W\mathcal{F}_if^W)_W$ is also transferable; and if $f'$ is transferable, then $\mathcal{F}_if'$ is also transferable. As a sum of transferable functions are transferable, the existence of smooth transfer follows from Theorem 21 combined with the following result.

Theorem 22 (Aizenbud, uncertainty principle) Let $\mathcal{F}_0=\Id$. We have $$C_c^\infty(\widetilde{\mathfrak{h}(W)})=\sum_{i=0}^3 \mathcal{F}_i C_c^\infty(\widetilde{\mathfrak{h}(W)}-\mathcal{N}^W)+C_c^\infty(\widetilde{\mathfrak{h}(W)})_0,$$ and $$C_c^\infty(\widetilde{\mathfrak{gl}_n})=\sum_{i=0}^3\mathcal{F}_i C_c^\infty(\widetilde{\mathfrak{gl}_n}-\mathcal{N}_n)+C_c^\infty(\widetilde{\mathfrak{gl}_n})_0.$$
Remark 20
  1. Here $$C_c^\infty(\widetilde{\mathfrak{h}(W)})_0:=\langle f- {}^hf: f\in C_c^\infty(\widetilde{\mathfrak{h}(W)}), h\in \UU(W)(F)\rangle,$$ and $$C_c^\infty(\widetilde{\mathfrak{gl}_n})_0:=\langle f'-\eta(h)\cdot{}^hf': f'\in C_c^\infty(\widetilde{\mathfrak{gl}_n}), h\in \GL_n(F)\rangle$$ are kernels of the coinvariant maps. On these spaces the orbital integrals are zero. One expects that the converse is also true (the density principle), which is known when $n=1,2$.
  2. Aizenbud proved the dual statement: there is no $\UU(W)(F)$-invariant distribution $T$ on $\widetilde{\mathfrak{h}(W)}$ such that $\mathcal{F}_iT$ is supported on $\mathcal{N}^W$ for $i=0,\ldots,3$ (distribution = linear functional on test functions in the nonarchimedean case). This dual statement also holds in the archimedean case. But since the distribution in the archimedean case needs to take the topology into account, it only gives (a) and (b) after taking the closure of the RHS.
Remark 21 We will also consider a slight variant given by $$\mathfrak{gl}_{n+1}=\widetilde{\mathfrak{gl}_n}\oplus F\curvearrowleft \GL_n,\quad \mathfrak{h}(V)=\widetilde{\mathfrak{h}(W)} \oplus F\curvearrowleft \UU(W).$$ Let $\mathcal{F}_4$ be the total Fourier transform on $C_c^\infty(\mathfrak{gl}_{n+1})$ or $C_c^\infty(\mathfrak{h}(V))$. The following is an easy exercise.
Lemma 4 Theorem 21 for $i=3$ is equivalent to the following: if $f\in C_c^\infty(\mathfrak{h}(V))\leftrightarrow f'\in C_c^\infty(\mathfrak{gl}_{n+1})$, then $c_3^W\mathcal{F}_4 f\leftrightarrow \mathcal{F}_4 f'$.

TopThe base case $n=1$

Today we will explain the proof of Theorem 21 when $n=1$. In this case $$\widetilde{\mathfrak{gl}_1}(F)=F \oplus F \oplus F \curvearrowleft F^\times,\quad t(\lambda,x,y)=(\lambda, tx, t^{-1}y).$$ There are two isomorphism classes of hermitian lines $$W_+=(E, N_{E/F}), \quad W_-=(E, \varepsilon_- N_{E/F}),$$ where $ \varepsilon_-\in F^\times-N_{E/F}(E^\times)$, and $$\widetilde{\mathfrak{h}(W_{\pm}})=F \oplus E \curvearrowleft \UU(W_{\pm})=E^1:=\{u\in E^\times: N(u)=1\},\quad u(\lambda, z)=(\lambda, uz).$$ Then

  1. $(\lambda, x,y)$ is regular semisimple if and only if $xy\ne0$,
  2. $(\lambda, z)$ is regular semisimple if and only if $z\ne0$.
  3. they match if and only if $xy=\varepsilon_{\pm} N(z)$ (by convention $\varepsilon_{+}=1$).

Notice the center $F\subseteq \widetilde{\mathfrak{gl}_1}(F)$ and $F\subseteq \widetilde{\mathfrak{h}(W_\pm)}$ does not play a role, and $\mathcal{F}_2$ is the Fourier transform with respect to the center, we know easily that $\mathcal{F}_2$ commutes with transfer. Hence it is enough to consider $\mathcal{F}_1$.

For $f=(f_+, f_-)\in C_c^\infty(E)^2$, it has orbital integrals $$\Orb(z, f_{\pm})=\int_{E^1}f_{\pm}(uz)du,\quad z\ne0.$$ For $f'\in C_c^\infty(F^2)$, it has orbital integral $$\Orb((x,y), f')=\int_{F^\times} f'(tx,t^{-1}y)\eta(t)d^\times t,\quad xy=0.$$ Then $f\leftrightarrow f'$ if and only if $$\Orb(z, f_{\pm})=\eta(x)\Orb((x,y),f'), \quad 0\ne xy=\varepsilon_{\pm}N(z).$$

Definition 46 Define the Fourier transform $$\hat f_{\pm}(z)=(\mathcal{F}_1f_\pm)(z)=\int_{E}f_{\pm}(z')\psi_{E}(\varepsilon_{\pm}zz')dz',\quad \psi_E:=\psi\circ \Tr_{E/F}$$ and $$\hat f'(x,y)=(\mathcal{F}_1f')(x,y)=\int_{F^2}f'(x',y')\psi(xy'+x'y)dxdy'.$$
Definition 47 For $a\in F^\times$, we denote $$\Orb_a(f):=\Orb(z, f_{\pm}),\quad a=\varepsilon_{\pm}N(z),$$ and $$\Orb_a(f'):=\eta(x)\Orb((x,y),f'),\quad xy=a.$$ Then $f\leftrightarrow f'$ if and only if $\Orb_a(f)=\Orb_a(f')$ for all $a\in F^\times$.

Theorem 21 in this case $n=1$ can be made more precise as

Proposition 16 If $(f_+,f_-)\leftrightarrow f'$, then $(\varepsilon(\eta,\psi)\hat f_+, -\varepsilon(\eta,\psi)\hat f_-)\leftrightarrow \hat f'$.
Remark 22 We will use Tate's thesis to prove this result. For any character $\chi: F^\times\rightarrow \mathbb{C}^\times$, we write $\Re(\chi)\in \mathbb{R}$ such that $$|\chi(t)|=|t|^{\Re(\chi)}.$$ For $f\in C_c^\infty(F)$, recall the Tate integral is defined to be the Mellin transform $$\int_{F^\times} f(t)\chi(t) d^\times t$$ which converges absolutely for $\Re(\chi)>0$. Recall that Tate's theorem says that for $0<\Re(\chi)<1$, we have $$\int_{F^\times}\hat f(t)\widetilde \chi(t) d^\times t=\gamma(\chi,\psi)\int_{F^\times} f(t)\chi(t)d^\times t,$$ where $\gamma(\chi,\psi)$ is Tate's $\gamma$-factor, $\hat f(y)=\int_F f(y)\psi(xy)dy$ is the usual Fourier transform, and $\widetilde \chi(t)=\chi(t)^{-1}|t|$. Let $\gamma(\chi_E, \psi_E)$ be the $\gamma$-factor for $\chi_E=\chi\circ N$. Recall that the inductivity of $\gamma$-factors in degree 0 says that $$\gamma(\chi_E, \psi_E)=\frac{\gamma(\chi,\psi)\gamma(\chi\eta,\psi)}{\varepsilon(\eta,\psi)}.$$
Proof (Proposition 16) Assume $f\leftrightarrow f'$. Let $\hat f=(\hat f_+, -\hat f_-)$. We would like to show that $$\varepsilon(\eta,\psi)\Orb_a(\hat f)=\Orb_a(\hat f').$$ We show this by computing the Mellin transform of both sides. For $\Re(\chi)>0$, the Mellin transform of the RHS is given by $$\int_{F^\times}\int_{F^\times} \hat f'(t,t^{-1}a)\eta(t)d^\times t\ \chi(a)d^\times a=\int_{(F^\times)^2}\hat f'(t,a)\eta(t)\chi(ta)d^\times t d^\times a,$$ which is a Tate integral in two variables, and hence converges. If $f'{}=f'_1 \otimes f_2'$, where $f_i'\in C_c^\infty(F)$, then it is easy to see that $\hat f'=\hat f_2' \otimes \hat f_1'$. Therefore by Tate's theorem we have when $0<\Re(\chi)<1$, the Mellin transform of the RHS is equal to $$\gamma(\widetilde\chi,\psi)\gamma(\widetilde \chi \eta,\psi)\int_{(F^\times)^2}f'(a,t)\eta(t)\widetilde \chi(ta)d^\times td^\times a,$$ which is the same as $$\gamma(\widetilde\chi,\psi)\gamma(\widetilde \chi \eta,\psi)\int_{F^\times}\Orb_a(f')\eta(a)\widetilde\chi(a)d^\times a.$$

Using $F^\times=N(E^\times)\cup \varepsilon_-N(E^\times)$, we know the Mellin transform of the LHS is equal to $$\varepsilon(\eta,\psi)\left(\int_{E^\times/E^1}\Orb(z,\hat f_+)\chi_E(z)d^\times z-\chi(\varepsilon_-)\int_{E^\times/E^1}\Orb(z,\hat f_-)\chi_E(z)d^\times z\right).$$ Notice $$\int_{E^\times/E^1} \Orb(z,\hat f_{\pm})\chi_E(z)d^\times z=\int_{E^\times}\hat f_{\pm}(z)\chi_E(z)d^\times z,$$ which by Tate's theorem (when $0<\Re(\chi)<1$) is equal to $$\gamma(\widetilde \chi_E, \psi_E^{\varepsilon_\pm})\int_{E^\times}f_{\pm}(z)\widetilde \chi_E(z)d^\times z=\gamma(\widetilde \chi_E, \psi_E^{\varepsilon_\pm})\int_{E^\times/E^1}\Orb(z,f_{\pm})\widetilde \chi_E(z)d^\times z.$$ The gamma factor $\gamma(\widetilde \chi_E, \psi_E^{\varepsilon_\pm})$ is equal to $\widetilde\chi(\varepsilon_{\pm})/\chi(\varepsilon_{\pm})\cdot \gamma(\widetilde \chi_E,\psi_E)$. Thus the Mellin transform of the LHS is equal to $$\varepsilon(\eta,\psi)\gamma(\widetilde \chi_E,\psi_E)\int_{F^\times}\Orb_a(f)\eta(a)\widetilde \chi(a)d^\times a.$$

It follows that the Mellin transforms of both sides are equal because $f\leftrightarrow f'$.

Remark 23 In the case $n=1$ one can also directly prove the existence of smooth transfer by explicitly computing the orbital integrals. But it is Proposition 16 that plays a role for the proof of the general case of Theorem 21.


We rephrase Proposition 16 as follows for later use.

Corollary 4 Let $E/F$ be a quadratic extension of $p$-adic fields. Let $h:E\times E\rightarrow E$ be a 1-dimensional hermitian form. If $\phi\in C_c^\infty(E)\leftrightarrow \phi'\in C_c^\infty(F^2)$ in the sense $$\int_{E^1}\phi(uw)du=\eta_{E/F}(b)\int_{F^\times}\phi(tb,t^{-1}c)d^\times t,\quad\text{for all } h(w,w)=bc.$$ Then $\eta_{E/F}(\disc h)\varepsilon(\eta_{E/F},\psi) \leftrightarrow \hat \phi'$.
Remark 24 The constants in Theorem 21 are explicitly given by $$c_i^W=
  \eta(\disc(W))\varepsilon(\eta,\psi)^n, & i=1, \\
  \eta(\disc(W))^{n-1}\varepsilon(\eta,\psi)^{n(n-1)/2},& i=2, \\
  \eta(\disc(W))^n\varepsilon(\eta,\psi)^{n(n+1)/2},& i=3.
\end{cases}$$ Notice that $c_3^W=c_2^W\cdot c_1^W$. So to prove Theorem 21, it suffices to prove it for $i=1,2$.

TopInduction for $\mathcal{F}_1$

Our next goal is to prove Theorem 21 for $\mathcal{F}_1$. Let $f\in C_c^\infty(\widetilde{\mathfrak{h}(W)})$ and $f'\in C_c^\infty(\widetilde{\mathfrak{gl}_n})$ match. Consider the usual Fourier transforms $\phi\in \mathbb{C}_c^\infty(W)\mapsto \hat \phi$ and $\phi'\in C_c^\infty(F^n \oplus F_n)\mapsto \hat\phi'$. The following result reduces the general $n$ case to the case of (product of) $n=1$.

Proposition 17 Let $A\in \mathfrak{h}(W)(F)$ and $A'\in \mathfrak{gl}_n(F)$ be regular semisimple elements (in the usual sense). Let $T=\mathrm{Cent}_{\UU(W)}(A)$, and $T'{}=\mathrm{Cent}_{\GL_n}(A')$ (maximal tori).
  1. There exists $\phi=\phi_{f,A}\in C_c^\infty(W)$ such that for every $w\in W$ with $(A,w)\in \widetilde{\mathfrak{h}(W)}_\mathrm{rs}$, $$\Orb((A,w),f)=\Orb^T(w,\phi):=\int_{T(F)}\phi(tw)dt,$$ $$\Orb((A,w),\mathcal{F}_1f)=\Orb^T(w,\hat \phi).$$
  2. There exists $\phi'{}=\phi_{f',A'}\in C_c^\infty(F^n \oplus F_n)$ such that for every $(b,c)\in F^n \oplus F_n$ with $(A', b,c)\in \widetilde{\mathfrak{gl}_{n,\mathrm{rs}}}$, $$\Orb(A', b,c),f')=\Orb^{T'}((b,c),\phi'):=\int_{T'(F)}\phi'(tb, ct^{-1}) \eta(t)dt,$$ $$\Orb((A',b,c),\mathcal{F}_1f')=\Orb^{T'}((b,c),\hat\phi').$$
Proof Let us prove (a). Let $\Omega:=\Ad(\UU(W)). A\subseteq \mathfrak{h}(W)(F)$. It is closed, as $A$ is semisimple. We have two maps $$\xymatrix{& \UU(W) \times W \ar[ld]_p \ar[rd]^{q} & \\ W & & \Omega \times W }$$ where $p$ is the projection onto the second factor, and $q(g,w)=(gAg^{-1}, gw)$. Then $p$ and $q$ are principal bundles under $\UU(W)$ and $T(F) $ respectively for the actions $$\gamma(g,w)=(\gamma g,w),\quad t(g,w)=(gt^{-1}, tw).$$ Hence we obtain surjective maps $$\xymatrix{ & C_c^\infty(\UU(W)\times W) \ar[ld]_{p_*} \ar[rd]^{q_*} & \\ C_c^\infty(W) & & C_c^\infty(\Omega\times W)}$$ where $$(p_*\Phi)(w)=\int_{\UU(W)} \Phi(g,w)dg,$$ $$(q_*\Phi)(gAg^{-1}, gw)=\int_{T(F)}\Phi(gt,t^{-1}w)dt.$$ We choose $\Phi\in C_c^\infty(\UU(W)\times W)$ such that $q_*\Phi=f|_{\Omega\times W}$ (this is possible since $\Omega$ is closed), and define $\phi=p_*\Phi$. If $w\in W$ such that $(A,w)$ is regular semisimple, then $$\Orb((A,w),f)=\int_{\UU(W)}f(gAg^{-1},gw)dg=\int_{\UU(W)}\int_T \Phi(gt^{-1},tw)dtdg,$$ changing the order of integration this is equal to $$\int_T\int_{\UU(W)}\Phi(g, tw)dgdt=\int_T \phi(tw)dt=\Orb^T(w,\phi),$$ which proves the first equality. For the second equality, we notice that if $\mathcal{F}$ is the partial Fourier transform on $W$ of $\UU(W)\times W$. Then $$p_*(\mathcal{F}\Phi)=\hat \phi,\quad q_*(\mathcal{F}\Phi)=\mathcal{F}_1f.$$

The proof of (b) is similar.

Now we can prove Theorem 21 for $\mathcal{F}_1$. Since the orbital integrals are locally constant functions on the regular semisimple locus, it suffices to show the equality of orbital integrals on an open dense subset of regular semisimple locus (e.g., the locus where $A$ is regular semisimple). We may thus assume that $A$ is regular semisimple. Then $A'$ is also regular semisimple and they have the same characteristic polynomial $P:=P_A=P_{A'}$. Write $P=\prod_{i\in I} P_i$ as a product of irreducible polynomials over $F $. Write $F_i=F[T]/(P_i)=F(\alpha_i)$. Let $T, T',\phi, \phi'$ be as in Proposition 17. Then by $f\leftrightarrow f'$, we have $$\Orb^T(w,\phi)=\omega(b)\Orb^{T'}((b,c),\phi'),\quad \omega(b)=\eta(\det(b,\cdots, (A')^{n-1}b)),$$ for every $w\in W$, $(b,c)\in F^n \oplus F_n$ such that $(A,w)$ and $(A', b,c)$ are matching regular semisimple elements. We would like to show $c_1^W\mathcal{F}_1 f\leftrightarrow \mathcal{F}_1f'$. By Proposition 17, it remains to show that $$c_1^W\Orb^T(w, \hat\phi)=\omega(b)\Orb^{T'}((b,c),\hat\phi').$$

There are linear isomorphisms $$F^n\cong \bigoplus_{i\in I}F_i,\quad W\cong \bigoplus_{i\in I} E_i,$$ through which $A'$ and $A$ act by multiplication by $(\alpha_i)_{i\in I}$, and induce $$F_n\cong \bigoplus_{i\in I} F_i,\quad h_i: E_i\times E_i\rightarrow E_i,$$ where $h_i$ is a 1-dimensional hermitian form for $E_i/F_i$, such that $$\langle c,b\rangle=\sum_{i\in I}\Tr_{F_i/F}(c_ib_i),\quad h(w,w')=\sum_{i\in I}\Tr_{E_i/E}h_i(w_i, w_i').$$ Then $$T'\cong \prod_{i\in I}R_{F_i/F}\GL_{1,F_i},\quad T\cong \prod_{i\in I}R_{F_i/F}\UU(1)_{E_i/F_i}.$$ We easily check that

  1. $(A', b,c)\in \widetilde{\mathfrak{gl}_{n,\mathrm{rs}}}$ if and only if $b_ic_i\ne0$ for all $i\in I$.
  2. $(A,w)\in \widetilde{\mathfrak{h}(W)}_\mathrm{rs}$ if and only if $w_i\ne0$ for all $i\in I$.
  3. $(A,w)\leftrightarrow (A',b,c)$ if and only if $h_i(w_i,w_i)=b_ic_i$ for all $i\in I$.
  4. $\omega(b)=c\prod_{i\in I}\omega(b_i)$, where $\omega(b_i)=\eta_{E_i/F_i}(b_i)$ for some constant $c$.

Thus by Corollary 4 , we obtain $$\prod_{i\in I}\eta_{E_i/F_i}(\disc h_i)\varepsilon(\eta_{E_i/F_i}, \psi_{F_i})\Orb^T(w,\hat\phi)=\omega(b)\Orb^T((b,c),\hat\phi').$$ Now we can finish the proof of Theorem 21 for $\mathcal{F}_1$ by the following comparison of constants.

Proposition 18 We have $$\prod_{i\in I}\eta_{E_i/F_i}(\disc h_i)\varepsilon(\eta_{E_i/F_i}, \psi_{F_i})=c_1^W,$$ which is in particular independent of the regular semisimple elements $A'$ and $A$.


Proof Let $d_i=[F_i:F]$. We need to show that $$\eta_{E/F}(\disc \Tr_{E_i/E}\circ h_i)\varepsilon(\eta_{E/F}, \psi)^{d_i}=\eta_{E_i/F_i}(\disc h_i)\varepsilon(\eta_{E_i/F_i},\psi_{F_i}).$$ Let $\disc(F_i/F)\in F^\times/(F^\times)^2$ be the discriminant of the quadratic form $q_{F_i}:(x,y)\in F_i^2\mapsto\Tr_{F_i/F}(xy)$. Then one can easily check that $$\disc(\Tr_{E_i/E}\circ h_i)=N_{F_i/F}(\disc h_i)\disc(F_i/F) .$$ Since $\eta_{E_i/F_i}=\eta_{E/F}\circ N_{F_i/F}$, it follows that $$\eta_{E/F}(\disc \Tr_{E_i/E}\circ h_i)=\eta_{E_i/F_i}(\disc h_i)\eta_{E/F}(\disc(F_i/F)).$$ In this way we can eliminate the dependence one the 1-dimensional hermitian form $h_i$, and it remains to show that $$\varepsilon(\eta_{E_i/F_i},\psi_{F_i})=\eta_{E/F}(\disc F_i/F)\varepsilon(\eta_{E/F}, \psi)^{d_i}.$$

To prove this last identity, we reinterpret the local $\varepsilon$-factor using the Weil index of quadratic forms. Let $q$ be a quadratic form on a $F $-vector space $V$. Weil showed that there exists a unique constant $\gamma_\psi(q)\in \mu_8$ such that $$\int_V f(v)\psi(q(v))dv=\gamma_\psi(q)\int_V \hat f(v)\psi(-q(v))dv,$$ where the Fourier transform is defined to be $$\hat f(v)=\int_V f(w)\psi(2q(v,w))dw.$$ The Weil index $\gamma_\psi(q)$ satisfies the following basic properties:

  1. $\gamma_\psi(q_1 \oplus q_2)=\gamma_\psi(q_1)\gamma_\psi(q_2)$.
  2. $\gamma_\psi(aq)=\gamma_{\psi_a}(q)$, where $\psi_a(x)=\psi(ax)$.
  3. If $K/F$ is a finite extension and $(V,q)$ is a quadratic space over $K  $. Then $\gamma_{\psi_K}(q)=\gamma_\psi(\Tr_{K/F}\circ q)$.
  4. (Jacquet-Langlands) If $V=E$, $q=N_{E/F}$, then $\gamma_{\psi}(N_{E/F})=\varepsilon(\eta_{E/F},\psi)$.

By (d) and (c), we have $$\varepsilon(\psi_{E_i/F_i}, \psi_{F_i})=\gamma_{\psi_{F_i}}(N_{E_i/F_i})=\gamma_\psi(\Tr_{F_i/F}\circ N_{E_i/F_i}).$$ Under $E_i=E \otimes_F F_i$, we have $$\Tr_{F_i/F}\circ N_{E_i/F_i}=N_{E/F} \otimes q_{F_i}.$$ We may decompose $q_{F_i}=a_1q_0 \oplus^{\perp}\cdots \oplus^{\perp} a_{d_i}q_0$ where $q_0(x)=x^2$. Then we have $$N_{E/F} \otimes q_{F_i}=a_1 N_{E/F} \oplus^{\perp} \cdots \oplus^{\perp} a_{d_i} N_{E/F}.$$ Using (a) and (b), we obtain $$\gamma_\psi(\Tr_{F_i/F}\circ N_{E_i/F_i})=\gamma_{\psi_{a_1}}(N_{E/F})\cdots \gamma_{\psi_{a_{d_i}}}(N_{E/F}).$$ By (d) again, this is equal to $$\eta_{E/F}(a_1\cdots a_{d_i}) \varepsilon(\eta_{E/F},\psi)=\disc(F_i/F)\varepsilon(\eta_{E/F},\psi).$$ This completes the proof.

TopInduction for $\mathcal{F}_2$

Finally, let us prove Theorem 21 for $\mathcal{F}_2$. Again since orbital integrals on locally constant functions on the regular semisimple locus, it suffices to prove the theorem for $X=(A,w)$ with $h(w)\ne0$. Then the matching $Y=(A',b,c)$ satisfies $\langle c,b\rangle =h(w)\ne0$. Up to the action of $\GL_n(F)$, we may assume that $b=e_n$ and $c=\lambda\cdot {}^te_n$, where $\lambda=h(w)$.

Let $W'{}=\langle w\rangle^{\perp}$. Then $$\mathrm{Stab}_{\UU(W)}(w)=\UU(W'), \quad\mathrm{Stab}_{\GL_n}(b,c)=\GL_{n-1}.$$ Let $\phi\in C_c^\infty(\mathfrak{h}(W))\mapsto \hat \phi$ and $\phi'\in C_c^\infty(\mathfrak{gl}_n)\mapsto \hat \phi'$ be the total Fourier transforms. Then by the same argument as Proposition 17, we can show the following.

Proposition 19
  1. There exists $\phi\in C_c^\infty(\mathfrak{h}(W))_\mathrm{rs}$, we have $$\Orb((A,w),f)=\Orb(A,\phi):=\int_{\UU(W')}\phi(hAh^{-1})dh,$$ $$\Orb((A,w),\mathcal{F}_2f)=\Orb(A, \hat \phi).$$
  2. There exists $\phi'\in C_c^\infty(\mathfrak{gl}_n)$ such that for $A'\in \mathfrak{gl}_{n,\mathrm{rs}}(F)$, we have $$\Orb((A',b,c),f')=\Orb(A',\phi'):=\int_{\GL_{n-1}(F)}\phi'(hA'h^{-1})dh,$$ $$\Orb((A', b,c),\mathcal{F}_2f')=\Orb(A',\hat\phi').$$

Since $f\leftrightarrow f'$, by Proposition 19 we obtain $$\Orb(A,\phi)=\omega(A',b,c)\Orb(A',\phi'),$$ for every $A\in \mathfrak{h}(W)_\mathrm{rs}\leftrightarrow A'\in \mathfrak{gl}_{n,\mathrm{rs}}(F)$. Moreover, $$\omega(A', b,c)=c\omega(A')$$ for some constant $c$ independent of $A'$, where $\omega(A')$ is the usual transfer factor on $\mathfrak{gl}_{n,\mathrm{rs}}(F)$. Therefore $\phi\leftrightarrow \phi'$. By induction, the total Fourier transform commutes with smooth transfer (Lemma 4). It follows that $$c_3^{W'}\Orb(A,\hat\phi)=\omega(A',b,c)\Orb(A',\hat \phi'),$$ and thus by Proposition 19 we have $$c_3^{W'}\Orb((A,w),\mathcal{F}_2f)=\omega(A', b,c)\Orb((A',b,c),\mathcal{F}_2f').$$ Finally one checks that $c_2^W=c_3^{W'}$, which finishes the proof of Theorem 21 for $\mathcal{F}_2$.

TopThe Jacquet-Rallis fundamental lemma

Let $F $ be a $p$-adic field. Let $E$ be its unique unramified quadratic extension. Let $n\ge1$. There are two isomorphism classes of hermitian spaces $W_{\pm}$ of dimension $n$ with $$\disc(W_+)\in N(E^\times),\quad \disc(W_-)\in F^\times-N(E^\times).$$ The space $W_+$ admits a basis $e_1,\ldots,e_n$ such that $h(e_i,e_j)=\delta_{i,j}$. We obtain a self-dual lattice $$L_+=\mathcal{O}_E e_1 \oplus\ldots \oplus \mathcal{O}_E e_n$$ (i.e., the dual lattice $L^\vee_+:=\{v\in W_+: h(v,w)\in O_E,\forall w\in L_+\}$ is equal to $L_+$). This allows us to define a model of $\UU_n=\UU(W_+)$ over $O_F$, so that $$\UU_n(O_F)=\{g\in \UU_n(F): gL_+=L_+\}.$$ We define similarly a model over $O_F$ of $\UU_{n+1}=\UU(V_+)$ using the lattice $$L_+'{}=L_+{} \oplus \mathcal{O}_E e,\quad V_+=W_+ \oplus^\perp\langle e\rangle.$$ Let $$G=\UU_n\times \UU_{n+1} \hookleftarrow H=\UU_n$$ and $$H_1=R_{E/F}\GL_n\hookrightarrow G'{}=R_{E/F}(\GL_n\times \GL_{n+1})\hookleftarrow H_2=\GL_n\times \GL_{n+1}.$$ We normalize the transfer factor on $G_\mathrm{rs}'(F)$ by choosing the unique unramified extension $\eta'$ of $\eta$ to $E^\times$. So $\eta'(z)=(-1)^{\val(z)}$. We also normalize the Haar measures by $$\Vol(H(\mathcal{O}_F))=\Vol(H_1(\mathcal{O}_F))=\Vol(H_2(\mathcal{O}_F))=1.$$

Conjecture 5 (Jacquet-Rallis fundamental lemma) We have $\mathbf{1}_{G'(\mathcal{O}_F)}\leftrightarrow (\mathbf{1}_{G(O_F)},0)$.

TopReduction to Lie algebras

Consider $$\mathfrak{u}_{n+1}=\Lie \UU_{n+1}\curvearrowleft \UU_n,\quad \mathfrak{s}_{n+1}=\Lie S_{n+1}\curvearrowleft \GL_n.$$ We again normalize the transfer factor by choosing the unramified $\eta'$ and Haar measures such that $$\Vol(\UU_n(\mathcal{O}_F))=\Vol(\GL_n(\mathcal{O}_F))=1.$$

Conjecture 6 (Jacquet-Rallis fundamental lemma, Lie algebra version) We have $\mathbf{1}_{\mathfrak{s}_{n+1}(\mathcal{O}_F)} \leftrightarrow (\mathbf{1}_{\mathfrak{u}_{n+1}(\mathcal{O}_F)}, 0)$
Proposition 20 (Yun) When $p\ne2$, Conjecture 6 implies Conjecture 5.
Proof Recall (from the proof of Theorem 18): for $f\in C_c^\infty(G(F))$, we may descend it to $\tilde f\in C_c^\infty(\UU_{n+1}(F))$; and for $\phi\in C_c^\infty(\mathfrak{u}_{n+1}(F))$, we may lift it to $c_{\xi,*}\phi\in C_c^\infty(\UU_{n+1}(F))$ using the Cayley transform. (Warning: as $c_\xi$ is only a birational isomorphism, $c_{\xi,*}\phi$ is not necessarily compactly supported. However, its restriction to each orbit is compactly supported, so its orbital integrals are still well-defined).

Similarly, for $f'\in C_c^\infty(G'(F))$, we may descend it to $\tilde f'\in C_c^\infty(S_{n+1}(F))$; and for $\phi'\in C_c^\infty(\mathfrak{s}_{n+1}(F))$, we may lift it to $c_{\xi,*}\phi'\in C_c^\infty(S_{n+1}(F))$.

We have proved that

  1. $f\leftrightarrow f'$ if and only if $\tilde f\leftrightarrow \tilde f'$.
  2. $\phi\leftrightarrow \phi'$ if and only if $c_{\xi,*}\phi\leftrightarrow \eta'(\det(\xi+\cdot))^nc_{\xi,*}\phi'$. (Warning: there was an extra constant $\alpha_\xi$, but a direct computation shows that $\alpha_\xi=\eta'(2\xi)^{n(n-1)/2}=1$ as $p\ne2$).


Now we finish the proof when $|k_F|>n$.

By our of normalization of measures, we see that $$\widetilde{\mathbf{1}}_{G(O_F)}=\mathbf{1}_{\UU_{n+1}(O_F)},\quad \widetilde{\mathbf{1}}_{G'(O_F)}=\mathbf{1}_{S_{n+1}(O_F)}.$$ By (a), it remains to show that $$\mathbf{1}_{\mathfrak{s}_{n+1}(O_F)}\leftrightarrow (\mathbf{1}_{\mathfrak{u}(n+1)(O_F)}, 0)\Longrightarrow \mathbf{1}_{S_{n+1}(O_F)}\leftrightarrow (\mathbf{1}_{\UU_{n+1}(O_F)}, 0)$$ Let $s\in S_{n+1,\mathrm{rs}}(F)$, if the characteristic polynomial of $s$ is not integral, then $\GL_n(F).s \cap S_{n+1}(O_F)=\varnothing$, and hence $\Orb(s, \mathbf{1}_{S_{n+1}(O_F)})=0$. For $g\in \UU_{n+1,\mathrm{rs}}(F)$ matching $s$, the characteristic polynomial of $g$ is equal to that of $s$, and hence by the same argument we know that $\Orb(g, \mathbf{1}_{\UU_{n+1}(O_F)})=0$.

Thus we may assume that the characteristic polynomial of $s$ is integral. Since the characteristic polynomial has degree $n+1$, and since the image of $\ker N_{E/F}$ in the residue field is of size $|k_F|+1$ which is $>n+1$ by assumption, we know there exists $\xi\in \ker N_{E/F}$ such that $$\det (s+\xi)\in O_E^\times.$$ In this case we claim that $$\Orb(s, \mathbf{1}_{S_{n+1}(O_F)})=\Orb(s, c_{\xi,*}\mathbf{1}_{\mathfrak{s}_{n+1}}(O_F)).$$ In fact, let $Y=c_\xi^{-1}(s)=\frac{s-\xi}{s+\xi}$. Then $$hYh^{-1}=\frac{hsh^{-1}-\xi}{hsh^{-1}+\xi},\quad hsh^{-1}=\xi\frac{ hYh^{-1}+1}{-hYh^{-1}+1},\quad h\in \GL_n(F).$$ Moreover, $\det(1-Y)=\det(2\xi/(s+\xi))\in O_E^\times$. So $hsh^{-1}\in S_{n+1}(O_F)$ if and only if $hYh^{-1}\in \mathfrak{s}_{n+1}(O_F)$, which implies the claim.

Similarly, if $g\in \UU_{n+1,\mathrm{rs}}(F)$ matches $s$, then for the same choice of $\xi $, we have (by the same argument) $$\Orb(g, \mathbf{1}_{\UU_{n+1}(O_F)})=\Orb(g, c_{\xi,*}\mathbf{1}_{\mathfrak{u}_{n+1}(O_F)}).$$

The desired result then follows from (b), by noticing that since $\det (s+\xi)\in O_E^\times$, the extra factor $\eta'(\det(s+\xi))^n=1$ as well.

TopProof by induction

Now let us reformulate the Lie algebra version to a version more suitable for induction. Assume $p\ne2$, then $E=F[\sqrt{\tau}]$ for some $\tau\in O_F^\times$. We have $$\mathfrak{s}_{n+1}\cong \mathfrak{gl}_{n+1}=\widetilde{\mathfrak{gl}_n}\oplus F, \quad \sqrt{\tau}Y \mapsto Y$$ and $$\mathfrak{u}_{n+1}\cong \mathfrak{h}_{n+1}=\widetilde{\mathfrak{h}}_n \oplus F,\quad \sqrt{\tau}X\mapsto X.$$ Moreover, $\mathbf{1}_{\mathfrak{s}_{n+1}(O_F)}$ gets sent to $\mathbf{1}_{\widetilde{\mathfrak{gl}_n}(O_F)} \otimes \mathbf{1}_{O_F}$, and $\mathbf{1}_{\mathfrak{u}_{n+1}(O_F)}$ gets sent to $\mathbf{1}_{\widetilde{\mathfrak{h}_n}(O_F)} \otimes \mathbf{1}_{O_F}$. So Conjecture 6 is equivalent to

Theorem 23 (Beuzart-Plessis) We have $\mathbf{1}_{\widetilde{\mathfrak{gl}_n}(O_F)}\leftrightarrow (\mathbf{1}_{\widetilde{\mathfrak{h}_n}(O_F)}, 0)$. Namely
  1. for any $Y\in \widetilde{\mathfrak{gl}_{n,\mathrm{rs}}}(F) \leftrightarrow X\in \widetilde{\mathfrak{h}_{n,\mathrm{rs}}}(F)$, we have $$\Orb(X, \mathbf{1}_{\widetilde{\mathfrak{h}_n}(O_F)})=\omega(Y)\Orb(Y, \mathbf{1}_{\widetilde{\mathfrak{gl}_n}(O_F)}).$$
  2. for any $Y\in \widetilde{\mathfrak{gl}_{n,\mathrm{rs}}}(F)$ not matching any $X\in  \widetilde{\mathfrak{h}_{n,\mathrm{rs}}}(F)$, we have $$\Orb(Y, \mathbf{1}_{\widetilde{\mathfrak{gl}_n}(O_F)})=0.$$

We prove Theorem 23 by induction on $n$. For $X=(A,w)\in \mathfrak{h}_n(F)$, we set $q(X)=h_n(w)$. For $Y=(A,b,c)\in \widetilde{\mathfrak{gl}_n}(F)$, we set $q(Y)=cb$. Notice that if $X, Y$ are matching regular semisimple elements, then $q(X)=q(Y)$.

Proposition 21 Theorem 23 holds when $|q(X)|\ge1$ and $|q(Y)|\ge1$.
Proof Since $q$ is an invariant polynomial taking integral values on $\widetilde{\mathfrak{h}_n}(O_F)$ and $\widetilde{\mathfrak{gl}_n}(O_F)$, we know when $|q(X)|>1$, we have $\Orb(X, \mathbf{1}_{\widetilde{\mathfrak{h}_n}(O_F)})=0$, and similarly we have $|q(Y)|>1$ when $\Orb(Y, \mathbf{1}_{\widetilde{\mathfrak{gl}_n}(O_F)})=0$.

Now we assume $|q(X)|=|q(Y)|=1$. Up to the action of $\UU_n(F)$ and $\GL_n(F)$, we may assume that $X=(A, \lambda e_n)$ for some $\lambda \in O_E^\times$ (since $N_{E/F}$ is surjective on units), and $Y=(A', e_n,\nu e_n^*)$ for some $\nu\in O_F^\times$. Since $X\leftrightarrow Y$ if and only if they are $\GL_n(E)$-conjugate, we have

  1. if $X\leftrightarrow Y$ then $q(X)=N(\lambda)=\nu=q(Y)$, and $A\in \mathfrak{h}_{n,\mathrm{rs}}(F)\leftrightarrow A'\in \mathfrak{gl}_{n,\mathrm{rs}}(F)$.
  2. if $Y$ does not match any element of $\widetilde{\mathfrak{h}_{n, \mathrm{rs}}}(F)$, then $A'$ does not match any element of $\mathfrak{h}_{n,\mathrm{rs}}(F)$.

By induction, we have correspondingly in each case

  1. $\Orb(A, \mathbf{1}_{\mathfrak{h}_n(O_F)})=\omega(A')\Orb(A', \mathbf{1}_{\mathfrak{gl}_n(O_F)})$
  2. $\Orb(A', \mathbf{1}_{\mathfrak{gl}_n(O_F)})=0$.

Notice $\omega(Y)=\eta'(\det(e_n,\ldots,(A')^{n-1}e_n))=\omega(A')$. It remains to show that $$\Orb(X, \mathbf{1}_{\widetilde{\mathfrak{h}_n}(O_F)})=\Orb(A, \mathbf{1}_{\mathfrak{h}_n(O_F)}),\quad \Orb(Y, \mathbf{1}_{\widetilde{\mathfrak{gl}_n}(O_F)})=\Orb(A', \mathbf{1}_{\mathfrak{gl}_n(O_F)}).$$

Let us show the first identity. By definition $$\Orb(X, \mathbf{1}_{\widetilde{\mathfrak{h}_n}(O_F)})=\int_{\UU_n(F)}\mathbf{1}_{\widetilde{\mathfrak{h}_n}(O_F)}(gAg^{-1}, \lambda g e_n)dg$$ Notice that if the integrand is nonzero, then $ge_n\in O_E^n$ and $h(ge_n)=h(e_n)=1$. Since $\UU_n(O_F)$ acts transitively on $\{v\in O_E^n: h(v)=1\}$ and $\mathrm{Stab}_{\UU_n}(e_n)=\UU_{n-1}$, it follows that $g\in \UU_n(O_F) \UU_{n-1}(F)$, and thus $$\Orb(X, \mathbf{1}_{\widetilde{\mathfrak{h}_n}(O_F)})=\int_{\UU_n(O_F)\times \UU_{n-1}(F)} \mathbf{1}_{\widetilde{\mathfrak{h}_n}(O_F)}(kgAg^{-1}k^{-1}, \lambda kge_n)dkdg.$$ Since $\mathbf{1}_{\widetilde{\mathfrak{h}_n}(O_F)}$ is invariant under $\UU_n(O_F)$, we know this integral is equal to $$\int_{\UU_{n-1}(F)}\mathbf{1}_{\widetilde{\mathfrak{h}_n}(O_F)}(gAg^{-1}, \lambda e_n)dg=\Orb(A, \mathbf{1}_{\mathfrak{h}_n(O_F)})$$ as desired. The proof of the second identity is similar.

Let $f\in C_c^\infty(\widetilde{\mathfrak{gl}_n}(F))$ be any function matching $\mathbf{1}_{\widetilde{\mathfrak{h}_n}(O_F)}$. To finish the proof of Theorem 23, we would like to show that $$\Orb(Y,f)=\Orb(Y, \mathbf{1}_{\widetilde{\mathfrak{gl}_n}(O_F)}),$$ for any $Y\in \widetilde{\mathfrak{gl}_{n,\mathrm{rs}}}(F)$. As orbital integral are locally constant on regular semisimple locus, it suffices to prove it when $q(Y)\ne0$. By Proposition 21, this equality is true when $|q(Y)|\ge1$.


Now we need an extra ingredient, namely the Weil representation (ref: Bump, Section 4.8). Let $(V,q)$ be a quadratic space of even dimension over $F $. Let $\psi: F\rightarrow S^1$ be a character. Then there exists a unique representation $\omega$ of $\SL_2(F)$, known as the Weil representation or the oscillator representation on $C_c^\infty(V)$ such that for any $\Phi\in C_c^\infty(V)$, we have $$(\omega\left(\begin{smallmatrix} 1 & x \\ & 1\end{smallmatrix}\right) \Phi)(v)=\psi(x q(v))\Phi(v),\quad\forall x\in F$$ and $$\omega\left(\begin{smallmatrix} & 1 \\ -1 & \end{smallmatrix}\right) \Phi=\gamma_\psi(q)\hat\Phi,\quad \hat\Phi(v)=\int_V \Phi(w)\psi(2q(v,w))dw.$$

We apply this to $(F^n \oplus F_n, q)$ and $\psi$ unramified. We obtain the Weil representation $\omega$ of $\SL_2(F)$ on $C_c^\infty(F^n \oplus F_n)$ and we can extend it to a representation on $$C_c^\infty(\widetilde{\mathfrak{gl}_n}(F))=C_c^\infty(\mathfrak{gl}_n(F))\otimes C_c^\infty(F^n \oplus F_n)$$ by letting $\SL_2(F)$ act trivially on $C_c^\infty(\mathfrak{gl}_n(F))$. Concretely, we have $$\left(\omega\left(\begin{smallmatrix} 1 & x \\ & 1\end{smallmatrix}\right) f'\right)(Y)=\psi(xq(Y)) f'(Y), \quad \omega \left(\begin{smallmatrix} & 1\\-1 & \end{smallmatrix}\right)f'{}=\mathcal{F}_1f',$$ where $\mathcal{F}_1$ is the partial Fourier transform on $F^n \oplus F_n$. Notice that since $q$ is a sum of hyperbolic planes, the Weil index $\gamma_\psi(q)=1$ for any $\psi$.

We have a surjection $$C_c^\infty(\widetilde{\mathfrak{gl}_n}(F))\twoheadrightarrow \Orb(\widetilde{\mathfrak{gl}_n}(F)),\quad f'\mapsto \Orb(-, f').$$

Lemma 5 The representation $\omega$ descends to a representation of $\SL_2(F)$ on $\Orb(\widetilde{\mathfrak{gl}_n}(F))$.
Proof We need to check that the action of $\left(\begin{smallmatrix} 1 & x \\ & 1\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix} & 1 \\ -1 &\end{smallmatrix}\right)$ descend to $\Orb(\widetilde{\mathfrak{gl}_n}(F))$. It is clear that $$\Orb(Y, \omega\left(\begin{smallmatrix} 1& x\\ & 1\end{smallmatrix}\right)f')=\psi(x q(Y))\Orb(Y,f').$$ Moreover, $\Orb(-,f')=0$ if and only if $f'\leftrightarrow (0,0)$, if and only if $\mathcal{F}_1f'\leftrightarrow (0,0)$ (since $\mathcal{F}_1$ preserves the smooth transfer), if and only if $\Orb(-, \mathcal{F}_1f')=0$. It follows that the kernel of $C_c^\infty(\widetilde{\mathfrak{gl}_n}(F))\twoheadrightarrow \Orb(\widetilde{\mathfrak{gl}_n}(F))$ is stable under $\mathcal{F}_1$, and hence the action of $\left(\begin{smallmatrix} & 1 \\ -1 &\end{smallmatrix}\right)$ descends as well.

Now we can finish the proof of Theorem 23. Set $$\Phi=\Orb(-,f)-\Orb(-, \mathbf{1}_{\widetilde{\mathfrak{gl}_n}(O_F)})\in \Orb(\widetilde{\mathfrak{gl}_n}(F)).$$ We need to show that $\Phi(Y)=0$ if $q(Y)\ne0$.

We notice that $\Phi$ is fixed by $\left(\begin{smallmatrix}1 & x \\ & 1\end{smallmatrix}\right)$ for any $x\in \mathfrak{p}_F^{-1}$. In fact, if $|q(Y)|\ge1$, then $\Phi(Y)=0$ by Proposition 21; if $|q(Y)|<1$, then $\psi(x q(Y))=1$. In both cases we have $\psi(xq(Y))\Phi(Y)=\Phi(Y)$.

We also notice that $\Phi$ is fixed by $\left(\begin{smallmatrix} & 1 \\ -1 &\end{smallmatrix}\right)$. Indeed since $\psi$ is unramified, we have $$\mathcal{F}_1 \mathbf{1}_{\widetilde{\mathfrak{gl}_n}(O_F)}=\mathbf{1}_{\widetilde{\mathfrak{gl}_n}(O_F)},\quad \mathcal{F}_1 \mathbf{1}_{\widetilde{\mathfrak{h}_n(O_F)}}= \mathbf{1}_{\widetilde{\mathfrak{h}_n(O_F)}}.$$ It follows that if $f\leftrightarrow (\mathbf{1}_{\widetilde{\mathfrak{h}_n(O_F)}},0)$, then $\mathcal{F}_1 f\leftrightarrow (\mathbf{1}_{\widetilde{\mathfrak{h}_n(O_F)}},0)$ as well. Thus $$\Orb(-, \mathcal{F}_1f)=\Orb(-,f)$$ (notice the constant $c_1=1$, as everything is unramified), and $\Phi$ is fixed by $\left(\begin{smallmatrix} & 1 \\ -1 &\end{smallmatrix}\right)$.

It follows that $\Phi$ is fixed by $\SL_2(F)$, and in particular $\Phi$ is fixed by $\left(\begin{smallmatrix} 1 & x \\ & 1\end{smallmatrix}\right)$ for $x\in F$. But if $q(Y)\ne0$, then there exists $x\in F$ such that $\psi(x(q(Y))\ne1$. This implies that $\Phi(Y)=0$, and completes the proof of Theorem 23.

TopLocal GGP for unitary groups


  • Kottwitz, Harmonic analysis on reductive p-adic groups and Lie algebras
  • Beuzart-Plessis, The local Gan-Gross-Prasad conjecture,
  • Beuzart-Plessis, A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the Archimedean case (including the $p$-adic case)

TopGeneral formulation

Let $E/F$ be a quadratic extension of local fields of characteristic 0 (i.e., $\mathbb{C}/\mathbb{R}$ or $p$-adic). Let $W\subseteq V$ be hermitian spaces over $E$, with $Z=W^\perp$ odd dimensional and $\UU(Z)$ is quasi-split (automatic in the $p$-adic case). Equivalently, there exists a basis $(z_r,\ldots, z_0, z_{-1},\ldots, z_{-r})$ of $Z$ such that $h(z_i,z_j)\ne0$ if and only if $i=-j$. Gan-Gross-Prasad construct a triple $(G, H, \xi)$ (unique up to $G(F)$-conjugacy), consisting of

  • $G=\UU(W)\times \UU(V)$,
  • $H=\UU(W)\ltimes N\subseteq G$,
  • $\xi: H(F)\rightarrow \mathbb{C}^\times$.


  • $N $ is the unipotent radical of the parabolic subgroup $$P=\mathrm{Stab}_{\UU(V)}(\langle  z_r\rangle\subsetneq \langle z_r,z_{r-1}\rangle\subsetneq\cdots \langle z_r, \cdots, z_1\rangle).$$ Notice $\UU(W)\subseteq P$ and we have the embedding $$H\hookrightarrow G,\quad (g_W, u)\mapsto (g_W, g_W u).$$
  • The character $\xi: H(F)\twoheadrightarrow N(F)\xrightarrow{\xi_N}\mathbb{C} ^\times$, where we choose the non-degenerate character $$\xi_N(u)=\psi_E\left(\sum_{i=0}^{r-1} h(uz_i,z_{-i-1})\right),\quad \psi_E:E\rightarrow S^1.$$ Notice that $\xi_N $ is $\UU(W)$-invariant and it is generic with that property.
Example 5
  1. (codimension 1 case) When $\dim W=\dim V-1$. We have $H=\UU(W)\hookrightarrow G=\UU(W)\times \UU(V)$, $\xi=1$.
  2. (Whittaker case) When $\dim W=0$, we have $H=N\hookrightarrow G=\UU(V)$ is a maximal unipotent subgroup and $\xi $ is a non-degenerate character.
Remark 25 The case we are considering is known as the Bessel case of GGP. There is also GGP conjecture when $\dim W^\perp$ is even (the Fourier-Jacobi case), where $\xi $ is a certain Weil representation. The proof of local GGP in the Fourier-Jacobi case is completely different and we will not discuss it.
Definition 48 Let $\Irr(G)$ be the set of isomorphism classes of irreducible smooth admissible representations of $\mathbb{C}$-representations of $G(F)$. Recall that in the archimedean case, smooth admissible means the Casselman-Wallach globalization of a Harish-Chandra module.
Definition 49 For $\pi\in\Irr(G)$, we define its multiplicity $$m(\pi)=\dim \Hom_H(\pi, \xi),$$ the dimension of the space of continuous $(H,\xi)$-equivariant linear forms on $\pi$.
Theorem 24 (Aizenbud-Gourevitch-Rallis-Schiffman ($p$-adic, codimension 1), Gan-Gross-Prasad ($p$-adic, general), Jiang-Sun-Zhu (archimedean)) We have $m(\pi)\le1$.
Remark 26
  • The usage of this multiplicity one result in the proof of local GGP can be avoided (finite multiplicity suffices).
  • In the archimedean case, we can define similarly the algebraic multiplicity $m(V):=\Hom_\mathfrak{h}(V, d\xi)$ for a Harish-Chandra module $V$. But in general we do not have $m(V)\le1$ (for example in the Whittaker case the multiplicity can be as high as the size of the Weyl group), but it is conjectured to be true in the codimension 1 case.

TopPure inner forms and local Langlands correspondence

Recall that $$H^1(F, H)\cong H^1(F, \UU(W))$$ is bijection with the set of isomorphism classes of hermitian spaces $W'$ with $\dim W'{}=\dim W$. We denote by $W_\alpha$ the hermitian space associated to $\alpha\in H^1(F, H)$. Similarly, $$H^1(F, G)\cong H^1(F, \UU(W))\times H^1(F, \UU(V))$$ is bijection with the set of isomorphism classes of pairs of hermitian spaces $(W', V')$ with $\dim W'{}=\dim W$ and $\dim V'{}=\dim V$. We denote by $(W_\beta, V_\beta)$ the pair of hermitian spaces associated to $\beta\in H^1(F, G)$. The natural embedding $$H^1(F, H)\hookrightarrow H^1(F, G)$$ is then given by explicitly $$W_\alpha\mapsto (W_\alpha, V_\alpha),\quad V_\alpha=W_\alpha \oplus^\perp Z.$$

Definition 50
  1. A pure inner form of $G$ is a group of the form $G_\beta=\UU(W_\beta)\times \UU(V_\beta)$ for $\beta\in H^1(F,G)$.
  2. A pure inner form of $(G, H,\xi)$ is a GGP triple $(G_\alpha, H_\alpha, \xi_\alpha)$ associated to the pair $(W_\alpha, V_\alpha)$ for some $\alpha\in H^1(F,H)$.
Remark 27 In the $p$-adic case there are only 2 pure inner forms of $(G, H,\xi)$. In the archimedean case, there are $d_W+1$ different pure inner forms of $(G, H,\xi)$, determined by the signature of $\UU(W_\beta)$.


Recall that in the local Langlands correspondence (Theorem 13), the bijection $$\coprod_{\beta\in H^1(F,G)}\Pi^{G_\beta}(\phi)\cong \hat S_\phi,\quad \pi(\phi, \chi)\mapsfrom \chi$$ depends on the choice of a Whittaker datum.

Definition 51 A Whittaker datum (for the family of pure inner forms $\{G_\beta: \beta\in H^1(F,G)\}$) is a triple $(G_\beta,N_\beta,\psi_\beta)$ (up to $G_\beta$-conjugacy), where
  • $\beta\in H^1(F,G)$ such that $G_\beta$ is quasi-split.
  • $N_\beta\subseteq G_\beta$ is a maximal unipotent subgroup.
  • $\psi_\beta: N_\beta(F)\rightarrow S^1$ is generic (i.e. nondegenerate).
Remark 28
  1. A unitary group of even rank $2k$ has unique quasi-split pure inner form ($U(k,k)$ in the archimedean case), but there are two conjugacy classes of Whittaker data on this pure inner form.
  2. A unitary group of odd rank $2k+1$ has two quasi-split pure inner forms ($U(k, k+1)$, $U(k+1,k)$ in the archimedean case), and each of these support a unique conjugacy class of Whittaker data.
  3. Therefore there are 4 different choices of Whittaker data for $\{G_\beta: \beta\in H^1(F, G)\}$. Among them there is a unique quasi-split inner form $G_\alpha$, which supports two different choices of Whittaker data. Gan-Gross-Prasad proved that there is a natural bijection between the conjugacy classes of Whittaker data on $G_\alpha$ and the $N(E^\times)$-orbits of nontrivial characters $\psi_E: E/F\rightarrow S^1$. So fixing $\psi_E$ allows us to fix the local Langlands correspondence for $G$ and its pure inner forms.
Definition 52 An $L$-parameter $\phi: \mathcal{L}_F\rightarrow {}^LG$ is generic if $L(s,\phi,\Ad)$ is regular at $s=1$. It is known that $\phi$ is generic if and only if $\coprod_\beta \Pi^{G_\beta}(\phi)$ contains generic representations (i.e. admitting a Whittaker model). If $\phi$ is generic, then $\pi(\phi,\mathbf{1})\in \Pi^{G_\beta}(\phi)$ is the unique representation in the packet admitting a Whittaker model of type $(N_\beta,\psi_\beta)$ (i.e., $\Hom_{N_\beta}(\pi(\phi,\mathbf{1}),\psi_\beta)\ne0$).

TopLocal root numbers and the local GGP conjecture

Let $\phi: \mathcal{L}_F\rightarrow {}^{L}G$ be an $L$-parameter. By base change (Proposition 7), $\phi$ can be identified with a pair $(\phi_W,\phi_V)$ where

  • $\phi_W$ is a $(-1)^{d_W+1}$-conjugate-dual representation of $\mathcal{L}_E$ of dimension $d_W$.
  • $\phi_V$ is a $(-1)^{d_V+1}$-conjugate-dual representation of $\mathcal{L}_E$ of dimension $d_V$.

Let $B_W$ be the nondegenerate $(-1)^{d_W+1}$-conjugate-dual form on the space of $\phi_W$ (and similarly for $B_V$). Then $$S_\phi\cong S_{\phi_W}\times S_{\phi_V}=\pi_0(\Aut(\phi_W, B_W))\times \pi_0(\Aut(\phi_V, B_V)).$$ More precisely, if $$\phi_W=\bigoplus_{i\in I} n_i\phi_i \oplus \bigoplus_{j\in J} 2m_j \phi_j \oplus \bigoplus_{k\in K}l_k(\phi_k \oplus \tilde\phi_k^\sigma)$$ is the decomposition into $(-1)^{d_W+1}$-conjugate-dual, $(-1)^{d_W}$-conjugate-dual and non-conjugate-dual irreducible representations. Then $S_{\phi_W}\cong (\mathbb{Z}/ 2 \mathbb{Z})^I$ (and similarly $S_{\phi_V}\cong (\mathbb{Z}/2 \mathbb{Z})^{I'}$).

Let $\psi_E:E/F\rightarrow S^1$ be nontrivial. Then the standard properties of local root numbers implies that if $\phi_0$ is a conjugate-dual representation of $\mathcal{L}_E$, then $\varepsilon(\phi_0,\psi_E)^2=1$.

Definition 53 We define characters $$\chi: S_{\phi_W}\mapsto\{\pm1\}, \quad e_i\in (\mathbb{Z}/ 2 \mathbb{Z})^I\mapsto \varepsilon(\phi_i \otimes \phi_V, \psi_E)$$ and similarly for $\chi': S_{\phi_V}\rightarrow \{\pm1\}$. We define the GGP character of $\phi$ to be $$\chi_\phi:=\chi \boxtimes  \chi'\in \hat S_\phi.$$ Notice that it depends on the choice of $\psi_E$.

Now we can state more precisely the local GGP conjecture. Fix a nontrivial $\phi_E:E/F\rightarrow S^1$, and use it to normalize the local Langlands correspondence and the GGP character.

Conjecture 7 (Gan-Gross-Prasad) Let $\phi:  \mathcal{L}_F\rightarrow {}^{L}G$ be a generic $L$-parameter. Then there exists a unique $\pi\in\coprod_{\alpha\in H^1(F,H)}\Pi^{G_\alpha}(\phi)$ such that $m(\pi)=1$. More precisely, we have $$m(\pi(\phi,\chi))=
  1, & \chi=\chi_\phi,\\
  0, & \chi\ne \chi_\phi.
Remark 29
  1. In a series of 5 papers, Waldspurger and Moeglin-Waldspurger have established the analogous conjecture for $p$-adic orthogonal groups.
  2. This conjecture was proved by similar techniques for $p$-adic unitary groups (Beuzart-Plessis, the tempered case; Gan-Ichino, the general case).
  3. Some results are known in the archimedean case (Beuzart-Plessis, the tempered case; Hongyu He, the discrete case). The method for the tempered case should be enough to prove the discrete case (not written yet).

TopAn integral formula for the multiplicity

Assume for the moment that $G(F)$ is compact (in the $p$-adic case this implies that $d_V\le 2$), then by the orthogonality relations of characters we have $$m(\pi)=\int_{H(F)} \theta_\pi(h) dh,$$ here $\pi\in\Irr(G)$, $\theta_\pi(g)=\tr(\pi(g))$ is the character of $\pi$, and we normalize the Haar measure so that $\Vol(H(F))=1$. A striking discovery of Waldspurger is that there is a similar formula in the noncompact case as well.

Let $\pi\in\Irr(G)$. For $f\in C_c^\infty(G)$, we have an operator on $\pi$ defined by $$\pi(f):=\int_{G}f(g)\pi(g)dg.$$ It is of finite rank in the $p$-adic case and is of trace-class in the archimedean case. We define a distribution $$\Theta_\pi(f):=\tr \pi(f).$$

Theorem 25 (Harish-Chandra, Barbasch—Vogan)
  1. There exists $\Theta_\pi\in L^1_\mathrm{loc}(G)$, called the character of $\pi$, which is smooth on the regular semisimple locus $G_\mathrm{reg}$ such that for any $f\in C_c^\infty(G)$, $$\Theta_\pi(f)=\int_{G}f(g)\Theta_\pi(g)dg.$$
  2. Let $x\in G_\mathrm{ss}(F)$ be a semisimple element. Then there is a local character expansion in $\omega\subseteq \mathfrak{g}_x=\Lie G_x$ a small neighborhood of 0, $$\Theta_\pi(x e^X)=\sum_{\mathcal{O}\in \mathrm{Nil}_\mathrm{reg}(\mathfrak{g}_x)} c_{\pi, \mathcal{O}}(x)\hat j(\mathcal{O},X)+o(|X|^{-\delta_x/2}),\quad \forall X\in \omega,$$ where
    • $\mathrm{Nil}_\mathrm{reg}(\mathfrak{g}_x)$ is the set of regular nilpotent orbits in $\mathfrak{g}_x(F)$ (which is empty when $G_x$ is not quasi-split).
    • $\hat j(\mathcal{O},-)$ is the Fourier transform of the orbital integral over $\mathcal{O}$, i.e., the unique locally integral function on $\mathfrak{g}_x(F)$ (smooth on the regular semisimple locus) such that $$\int_{\mathfrak{g}_x}\phi(X)\hat j(\mathcal{O},X)dX=\int_\mathcal{O}\hat \phi(N)dN, \quad \forall \phi\in C_c^\infty(\mathfrak{g}_x(F)),$$ where the Fourier transform is defined using a $G_x$-invariant form $\langle\cdot,\cdot \rangle$ (e.g. the Killing form) $$\hat \phi(X)=\int_{\mathfrak{g}_x(F)}\phi(Y)\psi(\langle X,Y\rangle) dY.$$
    • $c_{\pi,\mathcal{O}}(x)\in \mathbb{C}$.
    • $\delta_x=\dim G_x-\rk G_x$.
Definition 54 We define the regularized character $c_\pi: G_\mathrm{ss}(F)\rightarrow \mathbb{C}$ by $$c_\pi(x)=
  \displaystyle\frac{1}{|\mathrm{Nil}_\mathrm{reg}(\mathfrak{g}_x)|}\sum_{\mathcal{O}\in \mathrm{Nil}_\mathrm{reg}(\mathfrak{g}_x)} c_{\pi, \mathcal{O}}(x), & G_x\text{ is quasi-split}, \\
  0, & \text{otherwise}.
\end{cases}$$ Then $c_\pi(x)=\Theta_\pi(x)$ when $x\in G_\mathrm{reg}(F)$.


Definition 55 Let $\Gamma_\mathrm{ss}(H)=\Gamma_\mathrm{ss}(\UU(W))$ be the set of semisimple $H(F)$-conjugacy classes. We define $$\Gamma(G,H):=\{x\in \Gamma_\mathrm{ss}(\UU(W)): \exists W=W' \oplus^\perp W'', x\in \UU(W')_\mathrm{reg,ell}\}.$$ Here $x\in \UU(W')$ is regular elliptic if $\UU(W')_x$ is a compact torus. Concretely, let $\mathcal{W}$ be the set of representatives of the isomorphism classes (i.e., $\UU(W)$-orbits by Witt's theorem) of nondegenerate subspaces of $W$, then $$\Gamma(G,H)=\coprod_{W'\in \mathcal{W}}\coprod_{T\in \mathcal{T}_\mathrm{ell}(\UU(W'))} T_\mathrm{reg}(F)/W(\UU(W'),T).$$ Here $\mathcal{T}_\mathrm{ell}(\UU(W'))$ is a set of representatives of conjugacy classes of elliptic maximal torus in $\UU(W')$, and $W(\UU(W'),T)=N_{\UU(W')}T/T$ is the Weyl group relative to $T$.
Definition 56 We equip $\Gamma(G, H)$ with the unique Borel measure such that for any $\phi\in C_c(\Gamma(G,H))$, $$\int_{\Gamma(G,H)}\phi(x)dx=\sum_{W'\in \mathcal{W}}\sum_{T\in \mathcal{T}_\mathrm{ell}(\UU(W'))} |W(\UU(W'),T)|^{-1}\int_{T(F)} \phi(t) D^H(t)dt.$$ Here we normalize the Haar measure on $T(F) $ by $\Vol(T(F))=1$, and $$D^H(t):=|\det(1-\Ad(t))|_{ \mathfrak{h}/{\mathfrak{h} }_t}|$$ is the Weyl discriminant. Notice that $1\in \Gamma(G,H)$ with measure 1, so the contribution of 1 to the multiplicity formula will not be negligible.
Definition 57 For $\pi \in \Irr(G)$, we define $$m_\mathrm{geom}(\pi):=\int_{\Gamma(G,H)}^*c_\pi(x)dx:=\lim_{s\rightarrow 0^+}\int_{\Gamma(G,H)}c_\pi(x)\Delta(x)^sdx,$$ where $$\Delta(x)=|\det(1-x)|_{W/\ker(1-x)}|.$$
Theorem 26 (Beuzart-Plessis)
  1. This expression makes sense, i.e., the integral converges for $\Re(s)>0$ and the limit exists.
  2. If $\pi$ is tempered, then $m(\pi)=m_\mathrm{geom}(\pi)$.

TopFirst application to the local GGP conjecture

We use the integral formula to prove the following application to the local GGP conjecture.

Theorem 27 (Beuzart-Plessis) Let $\phi:\mathcal{L}_F\rightarrow {}^{L}G$ be a tempered $L$-parameter. Then $$\sum_{\alpha\in H^1(F,H)}\sum_{\pi\in\Pi^{G_\alpha}(\phi)} m(\pi)=1.$$
Remark 30 An $L$-parameter $\phi$ is tempered if and only if $\phi(W_F)$ is bounded, if and only if $\Gamma^G(\phi)$ contains only tempered representations. Moreover, we have a parametrization of the tempered dual of $G(F)$ by $$\mathrm{Temp}(G)=\coprod_{\phi: \mathcal{L}_F\rightarrow {}^{L}G}\Pi^G(\phi).$$
Proof Without loss of generality we may assume that $G$ is quasi-split. We will use the following three proprieties of the $L$-packets$\Pi^{G_\alpha}(\phi)$:
  1. (stability) For any $\alpha\in H^1(F,H)$, the sum $$\Theta_{\alpha,\phi}=\sum_{\pi\in \Pi^{G_\alpha}(\phi)}\Theta_\pi\in C^\infty(G_{\alpha,\mathrm{reg}}(F))$$ is stable, i.e., constant on $G_\alpha(\bar F)$-conjugacy classes).
  2. (transfer) For any $\alpha\in H^1(F,H)$, $\Theta_{\alpha,\phi}$ is the transfer of $e(G_\alpha)\Theta_{\phi}$, where $e(G_\alpha)\in\{\pm1\}$ is the Kottwitz sign (in the $p$-adic case, there are only two pure inner forms with $e(G)=1$ and $e(G')=-1$). Namely, for every $x\in G_{\alpha, \mathrm{reg}}(F)$ and $y\in G_\mathrm{reg}(F)$ which are conjugate inside $G_\alpha(\bar F)=G(\bar F)$, we have $\Theta_{\alpha,\phi}(x)=e(G_\alpha)\Theta_\phi(y)$.
  3. (whittaker) For every Whittaker datum $(U,\psi)$ of $G$, there exists a unique $\pi\in\Pi^G(\phi)$ such that $\Hom_U(\pi,\psi)\ne0$.

There is a natural bijection $$\mathrm{Nil}_\mathrm{reg}(\mathfrak{g})\cong\{\text{Whittaker data } (U,\psi)\text{ of } G\}/\text{conj.}, \quad\mathcal{O}\mapsto (U_\mathcal{O},\psi_\mathcal{O}).$$ Moreover we have (F. Rodier, $p$-adic case; H. Matumoto, archimedean case) $$c_{\pi, \mathcal{O}}(1)=\dim \Hom_{U_\mathcal{O}}(\pi, \psi_\mathcal{O}).$$ It follows from (whittaker) and the uniqueness of Whittaker model that $$c_\phi(1):=\sum_{\pi\in \Pi^G(\phi)}c_\pi(1)=1.$$

By Theorem 26, we have $$\sum_{\pi\in\Pi^{G_\alpha}(\phi)} m(\pi)=\int_{\Gamma(G_\alpha, H_\alpha)} c_{\alpha,\phi}(y)dy,\quad c_{\alpha,\phi}:=\sum_{\pi\in\Pi^{G_\alpha}(\phi)}c_\pi.$$ Let $\Gamma_\mathrm{stab}(G_\alpha,H_\alpha):=\Gamma(G_\alpha, H_\alpha)/\text{stable conj.}$ be the space of stable conjugacy classes, and let $p_\alpha: \Gamma(G_\alpha, H_\alpha)\rightarrow \Gamma_\mathrm{stab}(G_\alpha, H_\alpha)$ be the natural projection. Then by (stability) we obtain $$\sum_{\pi\in\Pi^{G_\alpha}(\phi)} m(\pi)=\int_{\Gamma_\mathrm{stab}(G_\alpha,H_\alpha)}^* |p_\alpha^{-1}(y)| c_{\alpha,\phi}(y)dy.$$ Since every $y\in \Gamma(G_\alpha, H_\alpha)$ is stably conjugate to an $x\in\Gamma(G, H)$, by (transfer) we obtain $$\sum_{\alpha\in H^1(F,H)}\sum_{\pi\in \Pi^{G_\alpha}(\phi)}m(\pi)=\int_{\Gamma_\mathrm{stab}(G,H)}^*c_\phi(x)\sum_{\alpha\in H^1(F,H)}\sum_{y\in\Gamma(G_\alpha, H_\alpha)\atop y\sim_\mathrm{stab}x}e(G_\alpha) dx.$$

Write $$\Sigma(x):=\sum_{\alpha\in H^1(F,H)}\sum_{y\in\Gamma(G_\alpha, H_\alpha)\atop y\sim_\mathrm{stab}x}e(G_\alpha)$$ for short. One can show that there exists an isotropic torus $T_x\subseteq H$ such that $$\coprod_{\alpha\in H^1(F,H)}\{y\in \Gamma(G_\alpha, H_\alpha): y\sim_\mathrm{stab}x\}\cong H^1(F, T_x),\quad y\mapsfrom \beta,$$ such that $\beta=\iota(\alpha)$ under the natural map $\iota: H^1(F, T_x)\rightarrow H^1(F,H)$. It follows that $$\Sigma(x)=\sum_{\beta\in H^1(F, T_x)} e(G_{\iota(\beta)}).$$ Moreover, the composition $$H^1(F, T_x)\xrightarrow{\iota} H^1(F,H)\rightarrow H^1(F, G)\xrightarrow{e} H^1(F, \mu_2)\cong\{\pm1\}$$ is a group homomorphism which is nontrivial when $x\ne1$. Thus $\Sigma(x)=0$ unless $x=1$, and so $$\sum_{\alpha\in H^1(F,H)}\sum_{\pi\in\Pi^{G_\alpha}(\phi)} m(\pi)=c_\phi(1)=1.$$ This completes the proof.


TopProof via the local trace formula

For $\pi\in\Irr(G)$, we would like to show that $$m(\pi)=m_\mathrm{geom}(\pi):=\int_{\Gamma(G,H)}^* c_\pi(x)dx,$$ where $\Gamma(G,H)$ is a space of conjugacy classes, and $c_\pi$ is the regularization of the Harish-Chandra character of $\pi$. Our goal is to compute $m(\pi)$ using local trace formula.

Definition 58 We say $f\in C_c^\infty(G)$ is strongly cuspidal if for all proper parabolic subgroup $P=MU\subsetneq G$, we have $$\int_{\UU(F)}f(mu)du=0$$ for all $m\in M(F)$. We denote the space of strongly cuspidal functions by $C_{c,\mathrm{cusp}}^\infty(G)$.
Theorem 28 If $f\in C_{c,\mathrm{cusp}}^\infty(G)$, then $$J(f):=\int_{H\backslash G}K_f(x,x)dx$$ converges.
Proof It suffices to show that $x\in X=H\backslash G\mapsto K_f(x,x)$ is compactly supported (rapidly decreasing in the archimedean case). For simplicity, assume that we are in the codimension 1 case. Sakellaridis-Venkatesh, using a certain equivariant compactification of $X$, constructed a boundary degeneration $X_\Theta$ (where $\Theta$ is a $G$-orbit at $\infty$; or a "spherical root"). Concretely, $X_\Theta=H_\Theta\backslash G$ where $\mathfrak{h}_\Theta=\Lie H_\Theta$ is a limit of conjugates of $\mathfrak{h}$ in the Grassmannian of $\mathfrak{g}$ (a certain point in the closure of the conjugates of $\mathfrak{h}$ in the Grassmannian). Then one can show that nary $\Theta$ (after identifying a neighborhood of $\Theta$ in $X$ and $x_\Theta$) $$K_f(x,x)=K_f^\Theta(x,x),\quad K_f^\Theta(x,y)=\int_{H_\Theta}f(x^{-1}h_\Theta y)d h_{\Theta}$$ is equal to the kernel function of $f$ acting on $C^\infty(X_\Theta)$. (In the archimedean case, the difference between $K_f(x,x)$ and $K_f^\Theta(x,x)$ is rapidly decreasing).

Now the crucial point is that $X_\Theta$ is parabolic induced, i.e., there exists $P=MU\subsetneq G $ and $H_\Theta^M\subseteq M$ such that $H_\Theta=H_\Theta^M\ltimes U$ (this is true for most spherical varieties). So if $f$ is strongly cuspidal, then $K_f^\Theta(x,x)=0$ identically. Hence $K_f(x,x)=0$ near the boundary.

Our next goal is to expand $J(f)$ both geometrically and spectrally.

TopArthur's local trace formula

We replace the GGP triple $(G, H,\xi)$ by $(G=H\times H, H,\xi=1)$, where $H $ is any connected reductive group over $F $. Then the action of $G=H\times H$ on $C^\infty(H\backslash G)\cong C^\infty(H)$ is given by right and left translations. Let $f=\phi \otimes \phi'\in C_c^\infty(G)= C_c^\infty(H) \otimes C_c^\infty(H)$. The action of $f$ on $C^\infty(H)$ is given by the kernel function $$K_{\phi,\phi'}(x,y)=\int_H \phi(x^{-1} hy)\phi'(h)dh.$$ We assume that $Z(H(F))$ is compact.

Theorem 29 If $\phi$ is strongly cuspidal, then $$J^A(\phi,\phi'):=\int_H K_{\phi,\phi'}(x,x)dx$$ converge.
Proof The proof is similar to Theorem 28. Here the boundary degenerations are of the form $$X_\Theta=M^\mathrm{diag}\ltimes (U\times \bar U)\backslash H\times H,$$ where $P=MU\subsetneq H$ is a parabolic subgroup, and $\bar P=M\bar U$ is the opposite parabolic subgroup. In particular, integrating along $U$ is already 0 since $\phi$ is strongly cuspidal.

TopThe geometric expansion

Definition 59 Let $x\in H_\mathrm{reg}(F)$ be a regular semisimple element. Let $\phi\in C_c^\infty(H)$. We define the orbital integral $$\Phi(x,\phi):=\int_{H_x(F)\backslash H(F)} \phi(h^{-1}xh)dh.$$
Definition 60 Let $M\subseteq H$ be a Levi subgroup. Let $x\in H_\mathrm{reg}(F)\cap M(F)$. Let $K\subseteq H(F)$ be a special maximal compact subgroup (all we need is $ K$ satisfies the Iwasawa decomposition $H(F)=P(F)K$ for any parabolic $P $). Associated to these data, we define the weighted orbital integral $$\Phi_M(x,\phi)=\int_{H_x(F)\backslash H(F)}\phi(h^{-1}xh)v_M(h)dh,$$ where $v_M(h)$ is Arthur's weight function.
Remark 31
  1. The weight $v_M$ depends on the choice of $ K$.
  2. When $M=H$, the weight $v_M=1$ and $\Phi_H(x,\phi)=\Phi(x,\phi)$.
  3. Let $\mathcal{A}_M=\Hom(X^*(M), \mathbb{R})$. We have a morphism $$H_M: M(F)\rightarrow \mathcal{A}_M,\quad \langle H_M(m), \chi\rangle=\log |\chi(m)|.$$ For any parabolic subgroup $P=MU\subseteq H$, we define $$H_P: H(F)\rightarrow \mathcal{A}_M,\quad H_P(muk)=H_M(m).$$ Then $v_M(h)$ is the volume of the convex hull of $H_P(h)$ for all $P=MU\subseteq H$.
Definition 61 Let $\Gamma_\mathrm{reg}(H)$ be the set of regular semisimple conjugacy classes in $H_\mathrm{reg}(F)$. We equip $\Gamma_\mathrm{reg}(H)$ with the unique measure such that $$\int_{\Gamma_\mathrm{reg}(H)} f(x)dx=\sum_{T\in \mathcal{T}(H)} |W(H,T)|^{-1}\int_{T(F)} D^H(t)f(t) dt,$$ where $\mathcal{T}(H)$ is a set of representatives of conjugacy classes of maximal tori in $H $.
Theorem 30 (Arthur) For $\phi\in C_{c,\mathrm{cusp}}^\infty(H)$, $\phi'\in C_c^\infty(H)$, we have $$J^A(\phi,\phi')=\int_{\Gamma_\mathrm{reg}(H)}(-1)^{a_M(x)} \Phi_{M(x)}(x,\phi)\Phi_H(x,\phi')dx.$$ Here $M(x)=\mathrm{Cent}_H(A_{H_x})$ is the minimal Levi containing $x$, and $a_{M(x)}=\dim A_{M(x)}$.
Remark 32 Since $\phi$ is strongly cuspidal, the weighted orbital integral $\Phi_{M(x)}(x,\phi)$ only depends on the orbit of $x$ and it does not depend on the choice of $ K$. Moreover, $\Phi_M(x,\phi)=0$ for all other Levi subgroups $M\ne M(x)$.
Theorem 31 (Harish-Chandra) Let $x\in H_\mathrm{reg}(F)$, $\phi\in C_{c,\mathrm{cusp}}^\infty(H)$, $K\subseteq H(F)$ a compact open subgroup. Then the function $$h\in H(F)\mapsto \int_K \phi(h^{-1}k^{-1} x kh)dk$$ is compactly supported, and $$(-1)^{a_{M(x)}} \Phi_{H(x)}(x,\phi)=\int_{H(F)}\int_K \phi(h^{-1}k^{-1} x k h)dkdh.$$

TopThe spectral expansion.

Definition 62 Let $\pi\in \Irr(H)$. For $\phi\in C_c^\infty(H)$, we define the character $\Phi(\pi,\phi)=\tr\pi(\phi)$.
Definition 63 Let $M\subseteq H$ be a Levis subgroup. Let $\sigma\in \mathrm{Temp}(M)$ be a tempered representation of $M$. Let $K\subseteq H(F)$ be a special maximal compact subgroup. Let $\pi=i_P^H(\sigma)$. We define the weighted character to be $$\Phi_M(\pi,\phi)=\tr R_M(\sigma)\pi(\phi),$$ where $R_M(\sigma)$ is Arthur's weight, defined using standard intertwining operators.
Remark 33
  1. The weight $R_M(\sigma)$ depends on the choice of $ K$.
  2. When $M=H$, the weight $R_H(\sigma)=1$ and $\Phi_H(\pi, \phi)=\Phi(\pi,\phi)$.
Definition 64 Let $\Rep_\mathrm{temp}(H)$ be the category of finite length tempered representations. For $P=MU\subseteq H$, we have a normalized parabolic induction functor $$i_P^H:\Rep_\mathrm{temp}(M)\rightarrow \Rep_\mathrm{temp}(H).$$ It has a left adjoint $$\Rep_\mathrm{temp}(H)\rightarrow \Rep_\mathrm{temp}(M),\quad \pi\mapsto \pi_P^w,$$ known as the weak Jacquet module, cut out from the the usual Jacquet module (which does not preserve the temperedness) by only keeping the unitary exponents of $A_M$ (i.e., generalized eigenspaces corresponding to unitary characters of $A_M$).
Definition 65 Let $R_\mathrm{temp}(H)$ be the Grothendieck group of $\Rep_\mathrm{temp}(H) \otimes \mathbb{C}$. Then by the exactness, the above two functors descends to functors on the Grothendieck groups $\sigma \mapsto i_M^H(\sigma)$ (only depends on the Levi $M $) and $\pi\mapsto \pi_P^w$. We define $$R_\mathrm{ell}(H):=\{\pi \in R_\mathrm{temp}(H), \forall P\subsetneq H, \pi_P^w=0\},$$ and $$R_\mathrm{ind}(H):=\sum_{P=MU\subsetneq H} i_M^H(R_\mathrm{temp}(M)).$$ Then it is a fact that $$R_\mathrm{temp}(H)=R_\mathrm{ell}(H) \oplus R_\mathrm{ind}(H).$$ Moreover, Arthur constructed a basis $T_\mathrm{ell}(H)$ of $R_\mathrm{ell}(H)$, whose elements are called elliptic representations.
Remark 34
  1. $T_\mathrm{ell}(H)$ contains $\Pi_2(H)$, the isomorphism classes of square-integral representations of $H(F)$, but it is usually bigger (unless $H=\GL_n$). For example, if $H=\SL_2$, $\chi: F^\times\rightarrow \{\pm1\}$ is nontrivial, and $i_B^H(\chi)= \pi^+ \oplus \pi^-$, then $\pi^+-\pi^-\in T_\mathrm{ell}(H)$.
  2. $T_\mathrm{ell} (H)$ is invariant under unramified unitary twists. So the group of unramified unitary characters $X_\mathrm{unr}^{unit}(H)$ acts on $T_\mathrm{ell}$.


(Announcement: the remaining two lectures will be on 04/22 and 04/29)

Definition 66 Let $$\mathcal{X}(H):=\{(M,\sigma): \text{Levi }M\subseteq H, \sigma\in T_\mathrm{ell}(M)\}/\text{conj.}$$ Then we have an embedding $$\mathcal{X}(H)\hookrightarrow R_\mathrm{temp}(H),\quad (M,\sigma)\mapsto i_M^H\sigma.$$ We equip $\mathcal{X}(H)$ with a measure such that $$\int_{\mathcal{X}(H)}f(\pi)d\pi=\sum_{M\in \mathcal{M}}\sum_{\mathcal{O}\in T_\mathrm{ell}(H)/X_\mathrm{unr}^\mathrm{unit}(M)} |W(G,M)|^{-1}\int_\mathcal{O}f(i_M^H(\sigma))d\sigma.$$ Here $\mathcal{M}$ is a set of representatives of conjugacy classes of Levi subgroups, and $d\sigma$ comes from a Haar measure on $X_\mathrm{unr}^\mathrm{unit}(M)$ (of which $\mathcal{O}$ is a quotient by a finite subgroup).
Theorem 32 (Arthur) For $\phi\in C_{c,\mathrm{cusp}}^\infty(H)$, $\phi'\in C_c^\infty(H)$, we have $$J^A(\phi,\phi')=\int_{\mathcal{X}(H)}(-1)^{a_{M(\pi)}} \Phi_{M(\pi)}(\pi, \phi)\Phi_H(\pi^\vee,\phi')d\pi.$$ Here $M(\pi)$ is any Levi from which $\pi$ is induced, i.e., $\pi\in i_{M(\pi)}^H(T_\mathrm{ell}(M(\pi)))$.
Remark 35 A priori the distribution $\Phi_{M(\pi)}(\pi,\phi)$ depends on the the choice of $ K$ and $M(\pi)$ but as $\phi$ is strongly cuspidal, it actually does not depend on these choices.

TopThe final identity

Let $\phi\in C_{c,\mathrm{cusp}}^\infty(H)$.

Definition 67 For $x\in H_\mathrm{reg}(F)$, we define $$\Theta_\phi(x)=(-1)^{a_{M(x)}} \Phi_{M(x)}(x,\phi).$$
Definition 68 For $\pi\in\mathcal{X}(H)$, we define $$\hat \Theta_\phi(\pi)=(-1)^{a_{M(\pi)}}\Phi_{M(\pi)}(\pi^\vee,\phi).$$

Now we can state Arthur's local trace formula identity.

Theorem 33 (Arthur) Let $\phi\in C_{c,\mathrm{cusp}}^\infty(H)$, $\phi'\in C_c^\infty(H)$, we have $$\underbrace{\int_{\Gamma_\mathrm{reg}(H)}\Theta_\phi(x)\Phi_H(x,\phi')dx}_{\text{geometric side}}=J^A(\phi,\phi')=\underbrace{\int_{\mathcal{X}(H)}\hat\Theta_\phi(\pi)\Phi_H(\pi^\vee,\phi')d\pi}_{\text{spectral side}}.$$

By Weyl's integration formula, the geometric side is equal to $$\int_{H(F)}\Theta_\phi(h)\phi'(h)dh.$$ By the definition of Harish-Chandra's character $\Theta_\pi$, the spectral side is equal to $$\int_{\mathcal{X}(H)}\hat\Theta_\phi(\pi)\int_H\phi'(h)\Theta_{\pi^\vee}(h)dhd\pi=\int_{H(F)}\phi'(h)\int_{\mathcal{X}(H)}\hat\Theta_\phi(\pi)\Theta_{\pi^\vee}d\pi d h.$$ Therefore we obtain a equivalent form of Theorem 33, $$\Theta_\phi(h)=\int_{\mathcal{X}(H)}\hat\Theta_\phi(\pi)\Theta_{\pi^\vee}(h)d\pi,$$ justing the notation $\hat\Phi_\phi(\pi)$ as a Fourier transform. This implies that the weighted orbital integral $\Theta_\phi$ behaves like a character. In particular, it has the same kind of local germ expansion as characters. More precisely, for $x\in H_\mathrm{ss}(F)$, we have in a neighborhood $\omega\subseteq \mathfrak{h}_x(F)$ of 0, $$\Theta_\phi(xe^X)=\sum_{\mathcal{O}\in \mathrm{Nil}_\mathrm{reg}(\mathfrak{h}_x)}c_{\phi,\mathcal{O}}(x)\hat j(\mathcal{O},X)+o(|X|^{-\delta_x/2}),\forall X\in \omega.$$

Define $c_\phi: H_\mathrm{ss}(F)\rightarrow \mathbb{C}$ using the same formula as Definition 54 (replacing $c_{\pi,\mathcal{O}}$ by $c_{\phi,\mathcal{O}}$). Then by comparing the local expansion of both sides of Arthur's local trace formula, we obtain $$c_\phi(x)=\int_{\mathcal{X}(H)}\hat\Theta_\phi(\pi)c_{\pi^\vee}(x)d\pi.$$

TopA local trace formula for GGP triples

Definition 69 Let $(G, H,\xi)$ be a GGP triple. For $f\in C_c^\infty(G)$, define $$J(f)=\int_{H\backslash G}K_f(x,x)dx=\int_{H\backslash G}\int_H f(x^{-1} hx)\xi(h)dhdx.$$
Theorem 34 (local trace formula for GGP triples) For $f\in C_{c,\mathrm{cusp}}^\infty(G)$, we have $$\underbrace{\int_{\Gamma(G,H)}^*c_f(x)dx}_{\text{geometric side}}=J(f)=\underbrace{\int_{\mathcal{X}(G)}\hat\Theta_f(\pi)m(\pi^\vee)d\pi}_{\text{spectral side}}.$$ Here $\pi\mapsto m(\pi)$ is extended by linearity to all virtual representations.
Remark 36 Unlike Arthur's local trace formula, the geometric expansion in the GGP setting contains genuine singular orbital integrals, which causes essential difficulty, and one cannot apply Arthur's method directly. On the other hand, the spectral expansion works similarly as Arthur's local trace formula (the only difference is to obtain the multiplicity of $H $-distinguished representations, via local periods).

Now we deduce Theorem 26 from Theorem 34. Recall that $$m_\mathrm{geom}(\pi)=\int_{\Gamma(G,H)}^*c_\pi(x)dx.$$ By Arthur's local trace formula, the geometric side of $J(f)$ is equal to $$\int_{\mathcal{X}(G)} \hat\Theta_f(\pi)m_\mathrm{geom}(\pi^\vee)d\pi.$$ So Theorem 34 is equivalent to $$\int_{\mathcal{X}(G)}\hat\Theta_f(\pi)(m(\pi^\vee)-m_\mathrm{geom}(\pi^\vee))d\pi=0.$$ One can show that for every $\pi\in \mathcal{X}(G)$, there exists $f\in C_{c,\mathrm{cusp}}^\infty(G)$ such that $\hat\Theta_f(\pi)\ne0$. Then using standard techniques (action of the Bernstein center/center of universal enveloping algebra + Stone-Weierstrass), we obtain for $\pi\in \mathcal{X}(G)$, $$m(\pi^\vee)=m_\mathrm{geom}(\pi^\vee).$$ As $\mathcal{X}(G)$ forms a basis of $R_\mathrm{temp}(G)$, we have proved Theorem 26.

Finally, let us explain Arthur's method and the difficulty in the GGP setting which needs to be overcome to prove Theorem 34. Let $(\Omega_N)_{N\ge1}$ be an increasing sequence of compact subsets of $H\backslash G$ such that $\cup_N\Omega_N=H\backslash G$. Let $\kappa_N=\mathbf{1}_{\Omega_N}$. Then $$J(f)=\lim_{N\rightarrow\infty} J_N(f),\quad J_N(f)=\int_{H\backslash G} \kappa_N(x) K_f(x,x)dx.$$ Using Weyl's integration formula on $H $, we have $$J_N(f)=\sum_{T\in \mathcal{T}(H)}|W(H,T)|^{-1}\int_T D^H(t)\int_{T'\backslash G}f(g^{-1}tg)\kappa_{N,T}(g)dgdt.$$ Here $T'{}=\mathrm{Cent}_G(T)$ is a maximal torus in $G$ and $\kappa_{N,T}(g)=\int_{T\backslash T'}\kappa_N(tg)dt$ can be thought of as a weight. Arthur proves that the limit of these $\kappa_{N,T}$-weighted orbital integrals is equal to the desired weighted orbital integral. However, the error term explodes near $T- T_\mathrm{reg}$ in the GGP case and $D^H(t)$ does not vanish sufficiently fast to control it (i.e., we cannot apply dominant convergence theorem).

The strategy to overcome this difficulty:

  1. Use the semisimple descent to localize near $x\in H_\mathrm{ss}(F)$. If $x\ne1$ we are left with a similar expression for the triple $(G_x, H_x, \xi_x=\xi|_{H_x})$, which turns out to be a product of a smaller GGP triple and an Arthur triple), and we may use induction to conclude.
  2. Near $x=1$, we use the exponential map to descent to the Lie algebra, and perform a Fourier transform, after which the expression converges better. Then we are able to apply Arthur's techniques to control the error term and perform an inverse Fourier transform back at the end.


TopAn integral formula for $\varepsilon$-factors of pairs

Our next goal is to relate the multiplicity $m(\pi)$ to local root numbers as predicted by the local GGP conjecture. To do so, we will need a multiplicity formula for representations of twisted groups (motivation: base change for unitary groups is a twisted endoscopy lift for general linear groups; stable characters on unitary groups transfer to twisted characters on general linear groups).

Definition 70 A reductive twisted group over $F $ is a pair $(G, \tilde G)$ where
  1. $G$ is a connected reductive group over $F $.
  2. $\tilde G$ is $F $-variety which is a $G$-bi-torsor (with two commuting left and right actions of $G$ on $\tilde G$) subject to the condition $\tilde G(F)\ne\varnothing$ (i.e. the torsors are trivial).
Remark 37
  1. $\tilde G$ acts on $G$ by $\tilde \gamma\mapsto\ad_{\tilde \gamma}\in\Aut(G)$ defined by $\ad_{\tilde \gamma}(g)\tilde \gamma=\tilde \gamma g$ of any $g\in G$.
  2. If $\theta\in\Aut(G)$ of finite order, then $\tilde G=G\theta\subseteq G^+=G\rtimes \langle\theta\rangle$ is a twisted group with $\ad_\theta=\theta$. Conversely, if $\theta_{\tilde G}=\ad_{\tilde \gamma}|_{Z(G)}$ is of finite order, then $\tilde G$ is of this form. In particular, when $G$ is adjoint, one can always write $\tilde G$ as a semi-direct product of $G$ and $\pi_0(\tilde G)$).
Definition 71
  1. A parabolic subgroup of $\tilde G$ is the normalizer $\tilde P$ of a parabolic subgroup $P $ of $G$ such that $\tilde P(F)=\varnothing$.
  2. A Levi component of $\tilde P$ is the normalizer $\tilde M$ in $\tilde P$ of a Levi component $M$ of $P $.

If $P=MU$, then $\tilde P=\tilde MU$.

Example 6 Let $E/F$ be a quadratic extension. Let $V_n$ be an $E$-vector space of dimension $n$. Let $G_n=R_{E/F}\GL(V_n)$. Let $$\tilde G_n=\mathrm{Sesq}^*(V_n):=\{\tilde \gamma: V_n\times V_n\rightarrow E\text{ nondegenerate sesquilinear form}\}.$$ Then $(G_n,\tilde G_n)$ is a twisted group via the action $$g\tilde \gamma g'{}=\tilde \gamma(g^{-1}\cdot, g'\cdot).$$ Notice that $\tilde G_n\cong G_n \theta_n$ where $\theta_n(g)={}^t{\bar g}^{-1}$.
Definition 72 Let $F $ be a $p$-adic field. A smooth representation of $\tilde G(F)$ is a pair $(\pi, \tilde \pi)$ where
  1. $\pi:G(F)\rightarrow \GL(V_\pi)$ is a smooth representation.
  2. $\tilde \pi: \tilde G(F)\rightarrow\GL(V_\pi)$ is such that $$\tilde \pi(g\tilde \gamma g')=\pi(g)\tilde \pi(\tilde \gamma)\pi(g'),\quad\tilde g,g'\in G(F),\tilde \gamma\in \tilde G(F).$$

We say $(\pi,\tilde \pi)$ is irreducible if $\pi$ is so.

Remark 38 If $\tilde G=G\theta$, then $\pi$ extends to a representation of $\tilde G(F)$ if and only if $\pi\cong \pi\circ\theta$, as the condition on $\tilde \pi$ exactly gives an intertwining operator between $\pi$ and $\pi\circ\theta$. For $\tilde G_n$, this condition means $\pi\cong\pi^\vee\circ c$, i.e., $\pi$ is conjugate self-dual.
Definition 73 The smooth contragredient $(\pi^\vee,\tilde \pi^\vee)$ of $(\pi, \tilde \pi)$ is defined by by $\tilde\pi^\vee(\tilde \gamma)={}^t\tilde\pi(\tilde \gamma)^{-1}$.
Remark 39
  1. If $\pi$ is irreducible, then by Schur's lemma an extension $\tilde \pi$ to $\tilde G(F)$, if exists, must be unique up to a scalar $\lambda\in \mathbb{C}^\times$.
  2. If $\tilde\pi$ is replaced by $\lambda\tilde \pi$, then $\tilde \pi^\vee$ is replaced by $\lambda^{-1}\tilde \pi^\vee$.
Definition 74 We say $(\pi, \tilde \pi)$ and $(\pi', \tilde \pi')$ are equivalent if $\pi\cong \pi'$ (so $\tilde\pi'\cong\lambda\tilde\pi$ in the irreducible case).
Definition 75 Let $\tilde \pi\in\Irr(\tilde G)$. Then like the group case, the distribution $\tilde f\in C_c^\infty(\tilde G(F))\mapsto \tr \tilde \pi(\tilde f)$ is again represented by a function $\Theta_{\tilde \pi}\in L^1_\mathrm{loc}(\tilde G(F))\cap C^\infty(\tilde G_\mathrm{reg}(F))$, known as the twisted character of $\tilde \pi$. The twisted character also has a local expansion around $\tilde x\in \tilde G_{\mathrm{ss}}(F)$ (reference: Clozel). Let $\mathfrak{g}_{\tilde x}$ be the fixed points of $\ad_{\tilde x}$ in $\mathfrak{g}=\Lie (G_{\tilde x})$, then there exists $\omega\subseteq \mathfrak{g}_{\tilde x}$ a neighborhood of 0 such that $$\Theta_{\tilde \pi}(\tilde x e^X)=\sum_{\mathcal{O}\in \mathrm{Nil}_\mathrm{reg}(\mathfrak{g}_{\tilde x})}c_{\tilde \pi,\mathcal{O}}(\tilde x)\hat j(\mathcal{O}, X)+o(|X|^{-\delta_{\tilde x}/2}),\quad\forall X\in \omega.$$
Definition 76 We define a triple $(\tilde G, \tilde H,\tilde \xi)$ where $(H, \tilde H)\hookrightarrow (G,\tilde G)$ are twisted groups and $\tilde \xi: \tilde H(F)\rightarrow \mathbb{C}^\times$ a twisted character, where
  • $\tilde G=\tilde G_d\times \tilde G_m$ ($d>m$ of different parity).
  • $\tilde H=\tilde G_m N$. Here we choose $V_d\cong V_m \oplus Z$, where $Z=\langle z_r,\ldots, z_0, z_{-1},\ldots, z_{-r}\rangle$. Let 
  P=\mathrm{Stab}_{G_d}(&\langle z_r\rangle\subsetneq \langle z_r,z_{r-1}\rangle\subsetneq\cdots \subsetneq\langle z_r,\ldots, z_1\rangle\subsetneq\\&\langle z_r,\ldots, z_0\rangle \oplus V_m\subsetneq \langle z_r,\ldots,z_0,z_{-1}\rangle \oplus V_m\subsetneq\ldots),
and $N $ be the unipotent radical of $P $.
  • We have an embedding $$\tilde G_m\hookrightarrow \tilde G_d,\quad \tilde \gamma\mapsto\tilde \gamma+ h_Z, \quad h_Z(z_i, z_j)=\delta_{i,-j},$$ which gives the embedding $\tilde H\hookrightarrow \tilde G$.
  • Define the character $\xi$ and the twisted character $\tilde \xi$ by $$\xi(u)=\psi(\sum_{i=-r}^{r-1} h_Z(uz_i,z_{-i-1})),\quad\tilde \xi(\tilde \gamma u)=\xi(u).$$

These are the twisted analogue of GGP triples.

Definition 77 For $\pi\in\Irr(\tilde G)$ generic, by Jacquet—Piatetski-Shapiro—Shalika, we have $\dim\Hom_H(\pi, \xi)\le1$. For $\tilde h\in \tilde H(F)$, the map $$\ell\in \Hom_H(\pi,\xi)\mapsto \tilde \ell:= \tilde \xi(\tilde h)^{-1}\cdot\ell\circ \tilde \pi(\tilde h)\in \Hom_H(\pi,\xi)$$ is an automorphism independent of the choice of $\tilde h$. We define the twisted multiplicity $m(\tilde \pi)\in \mathbb{C}^\times$ such that $\tilde \ell=m(\tilde \pi)\ell$ (in situations without multiplicity one, one takes the trace of this automorphism).
Definition 78 We say $\tilde f\in C_c^\infty(\tilde G)$ is strongly cuspidal if for any proper parabolic subgroup $\tilde P=\tilde MU\subsetneq \tilde G$, $$\int_U \tilde f(\tilde mu)du=0,\quad \forall\tilde m\in \tilde M(F).$$

Let $\tilde f\in C_c^\infty(\tilde G)$, then we have an action $\tilde R(\tilde f)$ of $\tilde G (F)$ on $C^\infty(H\backslash G,\xi)$ with the kernel function $$K_{\tilde f}(x,y)=\int_{\tilde H}\tilde f(x^{-1}\tilde h y)\tilde \xi(\tilde h)d\tilde h.$$

Theorem 35 (local twisted trace formula) If $\tilde f\in C_{c,\mathrm{cusp}}^\infty(\tilde G)$, then $$J(\tilde f)=\int_{H\backslash G} K_{\tilde f}(x,x)dx$$ converges, and $$\int_{\Gamma (\tilde G, \tilde H)}^*c_{\tilde f}(\tilde x)d\tilde x=J(\tilde f)=\int_{\mathcal{X}(\tilde G)}\hat \Theta_{\tilde f} (\tilde \pi)m (\tilde\pi^\vee)d \tilde \pi.$$ Here
  1. $\Gamma(\tilde G,\tilde H)$ is the set of semisimple conjugacy classes $\tilde x\in \Gamma_\mathrm{ss}(\tilde G_m)$, where there exists $V_m=V_n \oplus V_n'$ such that $\tilde x\in \tilde G_n(F)_\mathrm{ell,reg}$ via the embedding $\tilde G_n\hookrightarrow \tilde G_m$ given by a hermitian form on $V_n'$.
  2. $\mathcal{X}(\tilde G)=\{\tilde \pi=i_{\tilde P}^{\tilde{G}}(\tilde \sigma): \tilde \sigma\in T_\mathrm{ell}(\tilde M)\}/\text{equiv.}$
  3. $c_{\tilde f}$ and $\hat \Theta_{\tilde f}$ are defined as in the group case using the twisted weighted orbital integral and the twisted character instead.
Corollary 5 For $\tilde \pi\in \mathrm{Temp}(\tilde G)$ we have $$m(\tilde \pi)=\int_{\Gamma(\tilde G,\tilde H)}^* c_{\tilde \pi}(\tilde x)d\tilde x.$$

Finally, let us link the twisted multiplicity $m(\tilde \pi)$ to local root numbers.

Definition 79 Let $(\tilde G, \tilde N, \tilde\psi_{d,m})$ be a Whittaker triple, where
  • $N=N_d\times N_m\hookrightarrow G=G_d\times G_m$.
  • $\tilde N=\tilde \gamma N$, $\tilde \gamma=(\tilde \gamma_d,\tilde \gamma_m)$ (standard anti-diagonal hermitian forms).
  • $\psi_{d,m}=\psi_d \boxtimes \psi_m^{-1}$ (standard generic characters).
  • $\tilde \psi_{d,m}(\tilde \gamma u)=\psi_{d,m}(u)$.
Definition 80 For $\tilde \pi=\tilde \pi_d \boxtimes \tilde\pi_m\in \mathrm{Temp}(\tilde G)$, we have $\dim\Hom_N(\pi, \psi_{d,m})=1$ by the uniqueness of Whittaker models. We similarly define the twisted multiplicity $m_\mathrm{Whitt}(\tilde \pi)\in \mathbb{C}^\times$.
Proposition 22 For $\tilde \pi=\tilde \pi_d \boxtimes \tilde\pi_m\in \mathrm{Temp}(\tilde G)$, we have $$m(\tilde \pi)=m_\mathrm{Whitt}(\tilde \pi)\varepsilon(\pi_d\times \pi_m,\psi)\omega_{\pi_m}(-1)^{d-1}.$$ Here $\omega_{\pi_m}$ is the central character of $\pi_m$.


Proof This follows from the local functional equation of Jacquet—Piatetski-Shapiro—Shalika. For simplicity assume $d=m+1$. By scaling we may assume that $m_\mathrm{Whitt}(\tilde \pi)=1$. Let $\lambda_d\in \Hom_{N_d}(\pi_d,\psi_d)$ and $\lambda_m\in\Hom_{N_m}(\pi_m,\psi_m^{-1})$ be nonzero linear forms. Then $\tilde \pi=\tilde \pi_d \boxtimes \tilde \pi_m$ with $\lambda_k \circ \tilde \pi_k(\tilde \gamma_k)=\lambda_k$ for $k=m,d$. Consider the Whittaker model $$e\in \pi_d\mapsto W_e(g)=\lambda_d(\pi_d(g)e)\in C^\infty(N_d\backslash G_d,\psi_d),$$ and similarly $$e'\in \pi_m\mapsto W_{e'}\in C^\infty(N_m\backslash G_m,\psi_m^{-1}).$$ Consider the linear form $$\mathcal{L}(e \otimes e'):=\int_{N_m\backslash G_m} W_e(h)W_{e'}(h)dh.$$ By Jacquet—Piatetski-Shapiro—Shalika, this integral converges and define a nonzero linear form in $\Hom_{H}(\pi, \mathbb{C})$. Since the image of $\tilde \gamma_m w_m$ by $\tilde G_m\hookrightarrow \tilde G_d$ is $\tilde \gamma_d w_d$ ($w_k$ is the matrix with 1's on the anti-diagonal), by definition of $m(\tilde \pi)$ we have the identity $$\mathcal{L}\circ (\tilde \pi_d(\tilde \gamma_d w_d) \otimes \tilde \pi_m(\tilde \gamma_mw_m))=m(\tilde \pi)\mathcal{L}.$$ Now it is easy to check for $k=d,m$, $$W_{\tilde \pi_k(\tilde \gamma_kw_k)e}=W_e^\vee, \quad W_e^\vee(g)=W_e(w_k^{-1})^t\bar g^{-1}).$$ Hence the left hand side of the identity evaluated at $e \otimes e'$ is equal to $$\int_{N_m\backslash G_m} W_e^\vee(h)W_{e'}^\vee(h)dh$$ which by Jacquet—Piatetski-Shapiro—Shalika's local functional equation is equal to $$\gamma(1/2, \pi_d\times \pi_m,\psi)\omega_{\pi_m}(-1)^{d-1}\mathcal{L}(e \otimes e').$$ Since $\pi_d\times \pi_m$ is conjugate self-dual, we have $\gamma(1/2, \pi_d\times \pi_m,\psi)=\varepsilon(\pi_d\times\pi_m,\psi)$. This finishes the proof.

TopEndoscopy and the refined local GGP conjecture

Our final goal is to use the relation between twisted multiplicities and $\varepsilon$-factors to prove the full local GGP conjecture (for tempered representations). To do so, we will use the endoscopic relations for characters.

We have the following version of local Langlands parametrization classification (Theorem 13) for tempered representations.

Theorem 36 There is a partition $$\coprod_{V }\mathrm{Temp}(\UU(V))=\coprod_{\phi\subseteq\Phi_\mathrm{temp}(n)}\Pi(\phi).$$ Here $V$ runs over all isomorphism classes of $n$-dimensional hermitian spaces over $E$, and $\Phi_\mathrm{temp}(n)$ is the set of tempered $n$-dimensional $(-1)^{n+1}$-conjugate-dual admissible representations $\mathcal{L}_E\rightarrow\GL(M)$. Moreover, we have a bijection $$\Pi(\phi)\cong \hat S_\phi,\quad \pi(\phi,\chi)\mapsfrom \chi, \quad \pi\mapsto \chi_\pi,$$ which depends on the choice of the Whittaker datum (hence on $\psi:E/F\rightarrow \mathbb{C}^\times$).

The bijection $\Pi(\phi)\cong\hat S_\phi$ is characterized using the endoscopic relations for characters (and the local Langlands correspondence for $\GL_n$).

Definition 81 Let $\Pi^V(\Phi)=\Pi(\Phi)\cap \mathrm{Temp}(\UU(V))$. For $s\in S_\phi$, We define $$\Theta_{\phi,s}^V:=\sum_{\pi\in\Pi^V(\phi)}\chi_\pi(s)\Theta_\pi.$$ Then $\Theta_{\phi}^V:=\Theta_{\Phi,1}^V$ is stable (i.e., constant on regular stable conjugacy classes).

Next we will describe the classical endoscopy relations (relating $\Theta_{\phi,s}^V$ to stable characters for endoscopy groups), and the twisted endoscopy relations (relating $\Theta_{\phi}^V$ to twisted characters for general linear groups).

TopClassical endoscopy relations

The elliptic endoscopy groups of $\UU(V)$ are of the form a product of quasi-split unitary groups $\UU(V_+)\times \UU(V_-)$, where $V_{\pm}$ are hermitian spaces of dimension $n_{\pm}$ such that $n=n_++n_-$. An endoscopy datum (which we do not define precisely here) gives a homomorphism $$^L(\UU(V_+)\times \UU(V_-))\rightarrow {}^L\UU(V).$$ It depends on the choice of $(-1)^{n_{\mp}}$-conjugate-dual character $\mu_{\pm}:\mathcal{L}_E\rightarrow \mathbb{C}^\times$. Notice that a $+1$-conjugate-dual character restricts to $\mu_{F^\times}=\mathbf{1}$, and $(-1)$-conjugate-dual restricts to $\mu_{F^\times}=\eta_{E/F}$.

From the above homomorphism, we obtain a map $$\Phi_\mathrm{temp}(n_+)\times\Phi_\mathrm{temp}(n_-)\rightarrow \Phi_\mathrm{temp}(n),\quad (\phi_+,\phi_-)\mapsto \mu_+\phi_+ \oplus \mu_-\phi_-.$$ Moreover, we obtain a correspondence (not bijective) between stable conjugacy classes $$\UU(V_+)_\mathrm{reg}/\text{stab}\times \UU(V_-)_\mathrm{reg}/\text{stab}\leftrightarrow \UU(V)_\mathrm{reg}/\text{stab}, $$ where $(y_+, y_-)\leftrightarrow x$ if and only if we have an identity of characteristic polynomial $P_x=P_{y_+}P_{y_-}$. We have the Langlands-Shelstad transfer factor (normalized by the Whittaker datum) $$\Delta:(\UU(V_+)_\mathrm{reg}/\text{stab}\times \UU(V_-)_\mathrm{reg}/\text{stab})\times \UU(V)_\mathrm{reg}/\text{conj}\rightarrow \mathbb{C},$$ such that $\Delta(y_+,y_-,x)\ne0$ if and only if $(y_+,y_-)\leftrightarrow x$ (explicit formulas given by Waldspurger).

Definition 82 We say $\Theta: \UU(V)_\mathrm{reg}/\text{conj}\rightarrow \mathbb{C}$ is the transfer of $\Theta': \UU(V_+)_\mathrm{reg}/\text{stab}\times\UU(V_-)_\mathrm{reg}/\text{stab}\rightarrow \mathbb{C}$ if $$\Theta(x)=\sum_y \Delta(y,x)\Theta'(y),\quad\forall x\in \UU(V)_\mathrm{reg}.$$
Theorem 37 (classical endoscopy relations) Let $\phi_\pm\in \Phi_\mathrm{temp}(n_{\pm})$. Let $\phi=\mu_+\phi_+ \oplus \mu_- \phi_-\in \Phi_\mathrm{temp}(n)$. Assume $B=B_+ \oplus B_-$. Let $s$ be the image of $\Id \oplus -\Id$ in $S_\phi$. Then $\Theta_{\phi,s}^V$ is the transfer of $\Theta_{\phi_+}^{V_+}\times \Theta_{\phi_-}^{V_-}$.
Remark 40 Notice that any element $s\in S_{\phi}$ is of this form (i.e., lifts to an element of order 2 in $\Aut(\phi, B)$).

TopTwisted endoscopy relations

Assume $\UU(V)$ is quasi-split. Then $\UU(V)$ is an endoscopic group of $\tilde G_n=R_{E/F}\GL_n\cdot \theta$ with $\theta(g)={}^t\bar g^{-1}$. An endoscopic datum gives an map $$^L\UU(V)\rightarrow {}^L\GL_{n,E}$$ fixed by $\hat \theta$. It depends on a conjugate self-dual character $\mu: \mathcal{L}_E\rightarrow \mathbb{C}^\times$.

From the above homomorphism, each $\phi\in\Phi_\mathrm{temp}(n)$ gives conjugate self-dual parameter $\mu\phi$, which by local Langlands corresponds to a conjugate self-dual representation $\pi(\mu\phi)$. Let $\tilde \pi(\mu\phi)\in \mathrm{Temp}(\tilde G_n)$ be the associated twisted representation, normalized so that $m_\mathrm{Whitt}(\tilde \pi(\mu\phi))=1$. We also have a bijective correspondence $$\UU(V)_\mathrm{reg}/\text{stab}\leftrightarrow \tilde G_{n,\mathrm{reg}}(F)/\text{stab},\quad g\leftrightarrow \tilde \gamma.$$ We have the Kottwitz—Shelstad transfer factor $$\Delta: \UU(V)_\mathrm{reg}/\text{stab}\times \tilde G_{n,\mathrm{reg}}(F)/\text{conj}\rightarrow \mathbb{C}$$ such that $\Delta(y,\tilde x)\ne0$ if and only if $y\leftrightarrow \tilde x$ (explicit formulas given by Waldspurger).

Definition 83 We say $\tilde \Theta: \tilde G_{n,\mathrm{reg}}/\text{conj}\rightarrow \mathbb{C}$ is a transfer of $\Theta: \UU(V)_\mathrm{reg}/\text{stab}\rightarrow \mathbb{C}$ if $$\tilde\Theta(\tilde x)=\sum_y\Delta(y,\tilde x)\Theta(y).$$
Theorem 38 (twisted endoscopy relations) $\Theta_{\tilde \pi(\mu\phi)}$ is a transfer of $\Theta_{\phi}^V$.

In this case one can recover $\Theta_{\phi}^V$ from $\Theta_{\tilde \pi(\mu\phi)}$ since the regular stable conjugacy classes are in bijection.


TopProof of the refined local GGP conjecture

Finally let us prove Conjecture 7 for $p$-adic fields in the tempered case. Let $n=\dim V$ and $m=\dim W$. Let $\phi\in \Phi_\mathrm{temp}(n)$ and $\phi'\in \Phi_\mathrm{temp}(m)$. Recall (Definition 53) we have defined the GGP character $\chi_\mathrm{GGP}=\chi \boxtimes \chi'\in \hat S_\phi\times S_{\phi'}$, given by $$\chi(s)=\varepsilon(\phi^{s=-1} \otimes \phi',\psi),\quad\chi'(s)=\varepsilon(\phi \otimes (\phi')^{s'{}=-1},\psi).$$ Notice that there is unique different pair $(W',V')$ such that $\dim V'=n$, $\dim W'=m$ and $V'{}=W' \oplus^\perp Z$.

Theorem 39 (Beuzart-Plessis) There exists a unique $$(\pi,\sigma)\in \Pi^V(\phi)\times \Pi^W(\phi')\coprod \Pi^{V'}(\phi)\times \Pi^{W'}(\phi')$$ such that $m(\pi \boxtimes \sigma)=1$. Moreover, $$\chi_\pi=\chi,\quad \chi_\sigma=\chi'.$$
Remark 41 A reality check (done by GGP): the dependence on the additive character $\psi$ is canceled.
Proof We define $$m(\phi,s,\phi',s')=\sum_{(\pi,\sigma)}\chi_\pi(s)\chi_\sigma(s')m (\pi \boxtimes \sigma).$$ By Fourier inversion, it remains to show that $$m(\phi,s,\phi',s')=\chi(s)\chi'(s').$$ Fix $(s,s')\in S_\phi\times S_{\phi'}$ such that $s^2=s'^2=1$. Write $$\phi=\phi^{s=1} \oplus \phi^{s=-1}=\mu_+\phi_+ \oplus \mu_-\pi_-,\quad \phi_{\pm}\in\Phi_\mathrm{temp}(n_{\pm}),$$ and similarly for $\phi'$. Using the integral formula $m(\pi \boxtimes \sigma)=m_\mathrm{geom}(\pi \boxtimes \sigma)$ (Theorem 26) and the classical endoscopy relations (Theorem 37), we can express $m(\phi,s,\phi,s')$ in terms of stable characters $\Theta_{\phi_+}^{V_+}\times\Theta_{\phi_-}^{V_-}$ and $\Theta_{\phi'{}_+}^{W_+}\times\Theta_{\phi'{}_-}^{W_-}$ (the stabilization of multiplicities): $$m(\phi,s,\phi',s')=m_{\mu_+\mu_-'}^\mathrm{stab}(\phi_+,\phi_-')m_{\mu_-\mu_+'}^\mathrm{stab}(\phi_-,\phi_+'),$$ where $\dim V_\pm=n_\pm$ and $\dim W_\pm=n_\pm'$, and the RHS is defined by integrals similar to that for $m_\mathrm{geom}(\pi \boxtimes \sigma)$, but replacing $c_{\pi \boxtimes \sigma}$ by $c_\phi$.

On the other hand, let $\varphi\in \Phi_\mathrm{temp}(d)$, $\varphi'\in\Phi_\mathrm{temp}(l)$ and $\mu,\mu'$ conjugate self-dual characters. Using the integral formula for $\varepsilon(\pi(\mu\varphi)\times \pi(\mu'\varphi'),\psi)$ (Proposition 22) and the twisted endoscopy relations (Theorem 38), we can express (the stabilization of local root numbers) $$\varepsilon(\mu\varphi \otimes \mu'\varphi',\psi)=
  m_{\mu\mu'}^\mathrm{stab}(\varphi,\varphi'), & \sgn(\mu\varphi \otimes \mu\varphi')=-1, \\
  m_\mu^\mathrm{stab}(\varphi,0)m_{\mu'}^\mathrm{stab}(\varphi',0), & \sgn(\mu\varphi \otimes \mu\varphi')=+1.
\end{cases}$$ as the relevant elliptic endoscopy group of $\tilde G_n$ is $\UU(n)\times \UU(0)$. Notice that $m_\mu^\mathrm{stab}(\varphi,0)m_{\mu'}^\mathrm{stab}(\varphi',0)=c_\varphi(1)c_{\varphi'}(1)=1$ (alternatively, the local root number for orthogonal Galois representations is always 1 by Deligne). Strictly speaking, the stabilization of local root number is only valid when $d,l$ have different parity, nevertheless one can deduce the same formula for the general case by the trick of adding a character.

Finally, for any $s,s'\in S_\phi\times S_{\phi'}$, we obtain 
  m(\phi,s,\phi',s)&=m_{\mu_+\mu_-'}^\mathrm{stab}(\phi_+, \phi_-')m_{\mu_-\mu_+'}^\mathrm{stab}(\phi_-, \phi_+')\\
  & =\varepsilon(\mu_+\phi_+ \otimes \mu_-'\phi_-',\psi)\varepsilon(\mu_-\phi_- \otimes \mu_+'\phi_+',\psi)\\
  & = \varepsilon(\phi \otimes (\phi')^{s'=-1},\psi)\varepsilon(\phi^{s=-1} \otimes \phi',\psi)\\
  & =\chi(s)\chi'(s').
This finishes the proof.

Last Update: 05/01/2019. Copyright © 2015 - 2019, Chao Li.