These are notes for two talks in the Beyond Endoscopy Learning Seminar at Columbia, Spring 2018. Our main references are [1] and [2].

Recall that two key constructions are required in the Braverman-Kazhdan program for proving analytic continuation and functional equations for general Langlands $L$-function $L(s,\pi, \rho)$. One is a suitable space of Schwartz functions $\mathcal{S}^\rho(G)$ at each local place, containing a distinguished function encoding the unramified local $L$-factor (known as the basic function $\mathcal{C}_\rho$ after Sakellaridis). The other is a generalized Fourier transform (known as the Hankel transform after Ngo) preserving the Schwartz space and the basic function. With a global Poisson summation formula, one should be able to establish the desired analytic properties of $L(s,\pi,\rho)$ in a way analogous to Godement-Jacquet theory for standard $L$-function on $\GL_n$. Our goal today is to discuss the basic function $\mathcal{C}_\rho$ and to explain its an algebro-geometric interpretation due to Bouthier-Ngo-Sakellaridis, using the $L$-monoid $\bar G_\rho$ appeared in previous talks and its arc space.

[-] Contents
Satake/Langlands parameters
Basic functions
Vinberg's universal monoids
Ngo's $L$-monoids
Arc spaces
IC sheaves and functions
A global model
Geometric Satake

TopSatake/Langlands parameters

Let $F$ be a non-archimedean local field. Let $G$ be a split reductive group over $F$. Let $\hat G=\hat G(\mathbb{C})$ be its dual group. Let $\mathcal{H}=C_c^\infty(G(\mathcal{O})\backslash G(F)/G(\mathcal{O})$ be the spherical Hecke algebra. Recall that the classical Satake transform $$\mathrm{Sat}:\mathcal{H}\rightarrow \mathbb{C}[X _ * (T)],\quad f\mapsto \left( t\mapsto\delta_B(t)^{1/2}\int_{N(F)}f(tn)dn\right),$$ induces an algebra isomorphism onto the $W$-invariants $$\mathcal{H}\xrightarrow{\sim} \mathbb{C}[X _ * (T)]^W.$$ An unramified representation $\pi$ of $G(F)$ corresponds to a 1-dimensional character of $\mathcal{H}$, given by its action on the spherical vector $$\pi(f)v = \int_{G(F)} f(g)\pi(g)vdg,\quad v\in \pi^{G(\mathcal{O})}.$$ Langlands noticed that $\mathbb{C}[X _ * (T)]^W$ is the coordinate ring of the variety $\hat T/W$, so a 1-dimensional character $\mathbb{C}[X _ * (T)]^W$ corresponds to a point $\alpha_\pi\in \hat T/W$, i.e., a semisimple conjugacy class in $\hat G$. In this we obtain a bijection $\pi\mapsto \alpha_\pi$ between unramified representations of $G(F)$ and the Satake (or rather, Langlands) parameters. The Satake transform is then characterized by the identity $$\tr \pi(f)=\mathrm{Sat}(f)(\alpha_\pi), \quad f\in \mathcal{H}.$$ Also notice that the target of the Satake isomorphism can be identified with the representation ring of $\hat G$, and thus with the $\hat G$-invariant regular functions $\mathcal{O}(\hat G)^{\hat G}$ on $\hat G$ (via the trace map).

Remark 1 Notice the Satake isomorphism is of combinatorial nature: both the source and the target of depends only on the root datum of $G$ and the size of the residue field $q$. In fact, the Satake isomorphism can be defined over $\mathbb{Z}[q^{\pm1/2}]$.

TopBasic functions

The importance of the Satake parameter is due to its key role in defining the unramified local $L$-factor $L(s,\pi,\rho)$. Let $\rho: \hat G\rightarrow \GL(V)$ be an irreducible representation. Recall by definition $$L(s,\pi,\rho)=\det (1-\rho(\alpha_\pi)q^{-s})^{-1}.$$ Now if we have a diagonal matrix $A=\diag(\alpha_1,\ldots\alpha_k)$, then $$\det(1-A t)^{-1}=\prod_{i=1}^k (1-\alpha_i t)^{-1}=\prod_{i=1}^k(1+\alpha_i t+ \alpha_i^2 t^2+\cdots)=1+(\tr A)t+(\tr\Sym^2 A)t^2+\cdots.$$ Therefore $$L(s,\pi,\rho)=\sum_{d\ge0} \tr(\Sym^d\rho)(\alpha_\pi) q^{-ds}.$$ To remove the dependence on $\pi$, we are motivated to introduce the following definition.

Definition 1 We define $\mathcal{C}_\rho^d(s)$ to be the inverse under the Satake transform of the function $\tr \Sym^d\rho\cdot q^{-ds}$ (so $\mathcal{C}_\rho^d(s_0)\in \mathcal{H}$ for any given $s=s_0$. Define the basic function to be $$\mathcal{C}_\rho(s)=\sum_{d\ge0}\mathcal{C}_\rho^d(s).$$ When $\Re(s)\gg0$, the sum is locally finite and makes sense as a function on $G(F)$.

Even though each $\mathcal{C}_\rho^d$ is compactly supported (with support lies in the $K$-double cosets indexed by dominant coweights of $G$ corresponding to weights of $\Sym^d\rho$), the support gets larger when $d$ increases and $\mathcal{C}_d$ is not longer compactly supported. Moreover, the values of $\mathcal{C}_\rho$ on each $K$-double cosets can be written down in terms of representation theory (related to Kazhdan-Lusztig polynomials) and thus involve quite complicated combinatorial quantities.

Example 1 Take $G=\mathbb{G}_m$, and $\rho=\Std$. Since $G=T$, both the source and target of the Satake isomorphism are identified as functions on $\mathbb{Z}$. The Satake transform sends the characteristic function $\mathbf{1}_{\val=d}$ to $\tr \Sym^d\rho: t\mapsto t^d$. So $\mathcal{C}_\rho^d=\mathbf{1}_{\val=d}$ and the basic function is given by $\mathcal{C}_\rho=\mathbf{1}_\mathcal{O}$ (always viewed as a function on $G(F)$). This generalizes to the standard representation of $G=\GL_n$, in which case $\mathcal{C}_\rho^d=\mathbf{1}_{\mathrm{M}_n(\mathcal{O})_{\val(\det)=d}}$ (this is already a nontrivial computation) and hence $\mathcal{C}_\rho=\mathbf{1}_{\mathrm{M}_n(\mathcal{O})}$.
Example 2 Take $G=\mathbb{G}_m$, and $\rho: \mathbb{G}_m\rightarrow \GL_2(\mathbb{C}), t\mapsto \diag(t,t)$ (i.e., $\rho=\Std \oplus \Std$). Then $\Sym^d\rho$ has dimension $d+1$ given by $t\mapsto (t^{d},\ldots, t^d)$, whose trace is $t\mapsto (d+1)t^d$. So the corresponding basic function is given by $$\mathcal{C}_\rho=\sum_{d\ge0}(d+1)\mathbf{1}_{\val=d}=\val(\cdot)+1.$$ This is no longer the characteristic function of any set. More generally, take $G=\mathbb{G}_m$ and $\rho=\chi_1 \oplus \cdots \oplus \chi_n$ ($\chi_i\ge0$). Then $$\mathcal{C}_\rho=\sum_{d\ge0}\# \{(a_1,\ldots,a_n):\sum a_i\chi_i=d, a_i\ge0\}\cdot \mathbf{1}_{\val=d}.$$ So the value of $\mathcal{C}_\rho$ encodes partition numbers, and can not have simple formula. We also see that the support of $\mathcal{C}_\rho$ is contained in the cone generated by the weights of $\rho$.
Example 3 Take $G=\GL_2$ and $\rho=\Sym^k\Std$. Then computing $\mathcal{C}_\rho$ amounts to decomposing $\Sym^d(\Sym^k\Std)$ into irreducibles, again this is a difficult combinatoric problem. In fact, we have $$\Sym^d\Sym^k\cong \bigoplus_{i=0}^{[d k/2]}(\Sym^{dk- 2i} \otimes \det{}^{dk-i})^{\oplus N(d,k,i)}.$$ Here the multiplicity $N(d,k,i)=p(d,k,i)-p(d,k,i-1)$, and $p(d,k,i)$ is the number of partitions of $i$ into at most $k$ parts, having largest part at most $d$.

I hope these examples illustrate that writing down an explicit formula for the basic function is quite hopeless in general (but see Wen-Wei Li's paper). Instead we would like to focus on finding some natural algebro-geometric object which encodes these combinatoric information. This is the main motivation to introduce the $L$-monoid.

TopVinberg's universal monoids

Let $G$ be a split reductive group over a field $k$ (later $k$ will be the residue field of the local field $F$). Assume $G$ has a nontrivial map to $\mathbb{G}_m$, denoted by $\det: G\rightarrow \mathbb{G}_m$. Assume $G' = \ker(\det)$ is semisimple and simply-connected. Our first goal is to construct Vinberg's universal monoid $\bar G$. It is a normal affine variety $\bar G$ fitting into a commutative diagram $$\xymatrix{ G^+ \ar@{^(->}[r] \ar[d] & \bar G \ar[d]\\ \mathbb{G}_m^r \ar@{^(->}[r] & \mathbb{A}^r.}$$ This monoid is universal in the sense that every reductive monoid with derived group equal to $G'$ can be obtained by base change from $\bar G$ (in fact the construction of $\bar G$ will only depend on $G'$).

Let $T'\subseteq G'$ be a maximal torus of $G'$. Let $G^+=( T'\times G')/\Delta(Z(G'))$. Let $r$ be the semisimple rank of $G'$. Let $\{\omega_1,\ldots,\omega_r\}$ be the set of fundamental weights of $G'$ (dual to the coroots). Let $\rho_i$ be the fundamental representation of $G'$ associated to $\omega_i$. We extend $\rho_i$ from $G'$ to $G^+$ by $$\rho_i^+: G^+=( T'\times G')/\Delta(Z(G'))\rightarrow \GL(V_i), \quad (t, g)\mapsto\omega_i(w(t^{-1})) \rho_i(g).$$ Here $w\in W$ is the longest element in the Weyl group. We also extend the simple roots $\alpha_i$ from $T'$ to $G^+$ by $$\alpha_i^+: G^+\rightarrow \mathbb{G}_m,\quad (t,g)\mapsto \alpha_i(t).$$ These extensions together give a homomorphism $$(\alpha^+,\rho^+): G^+\rightarrow \mathbb{G}_m^r \times \prod_{i=1}^r \GL(V_i).$$

Definition 2 We define $\bar G$ to be the closure of the image of $G^+$ in $$\mathbb{A}^r\times \prod_{i=1}^r \End(V_i).$$
Example 4 Consider $G=\GL_2$. Then $G' = \SL_2$, $T'\cong\mathbb{G}_m$, $G^+=(T'\times \SL_2)/\mu_2$. We have $r=1$, $\rho_1=\Std$, $w(\diag(t,t^{-1}))=\diag(t^{-1},t)$, $\omega_1(\diag(t,t^{-1}))=t$, $\alpha_1(\diag(t,t^{-1}))=t^2$. So $$(\alpha^+, \rho^+): (\diag(t,t^{-1}),g)\mapsto (t^2, \diag(t, t)\cdot g).$$ So $\bar G=\{ (t, g)\in \mathbb{A}^1\times \mathrm{M}_2: t=\det g\}.$ In other words, this is a monoid in $\mathbb{A}^5$ defined by the equation $t=ac-bd$ (which is smooth).

TopNgo's $L$-monoids

Now let $\rho: \hat G\rightarrow \GL(V)$ be an irreducible representation. Let $T^\mathrm{ad}=T'/Z(G')$ be a maximal torus in the adjoint group of $G'$. The highest weight of $\rho$ defines a cocharacter $\lambda_\rho: \mathbb{G}_m \rightarrow T$, hence a cocharacter of $\lambda_{\rho, \mathrm{ad}}: \mathbb{G}_m\rightarrow T^\mathrm{ad}$. We identify $$T^\mathrm{ad}\cong\mathbb{G}_m^r,\quad t\mapsto (\alpha_1(t),\ldots, \alpha_r(t)),$$ using a choice of simple roots. Then $$\lambda_{\rho, \mathrm{ad}}: \mathbb{G}_m\rightarrow \mathbb{G}_m^r$$ can be extended to a morphism of monoids $$\bar\lambda_{\rho,\mathrm{ad}}: \mathbb{A}^1\rightarrow \mathbb{A}^r.$$

Definition 3 The $L$-monoid $\bar G_\rho$ is defined by base changing the universal monoid $\bar G\rightarrow \mathbb{A}^r$ along $\bar\lambda_{\rho, \mathrm{ad}}$. So we have a commutative diagram $$\xymatrix{ \bar G_\rho^\times \ar@{^(->}[r] \ar[d] & \bar G_\rho \ar[d]\\ \mathbb{G}_m \ar@{^(->}[r] & \mathbb{A}^1.}$$
Example 5 Again take $G=\GL_2$, $\rho=\Sym^n(\Std)$. Then $\lambda_{\rho,\mathrm{ad}}: \mathbb{G}_m\rightarrow \mathbb{G}_m, t\mapsto t^n$. So we have $$\bar G_\rho=\{(t,g)\in \mathbb{A}^1\times\mathrm{M}_2: t^n=\det g\}.$$ Notice that the unit group is $\GL_2$ when $n$ is odd and $\mathbb{G}_m \times \SL_2$ when $n$ is even (the derived group is $\SL_2$ in both cases). Notice that this monoid is singular at the origin when $n>1$, which reflects the fact that the basic function are more complicated than the $n=1$ case.
Remark 2 Assuming that $\mathbb{G}_m\xrightarrow{\hat\det}\hat G\xrightarrow{\rho}\GL(V)$ is the identity map (e.g., $n=1$ in the previous example) ensures that the unit group of $\bar G_\rho$ is $G$ and we obtain a commutative diagram $$\xymatrix{G \ar@{^(->}[r] \ar[d]^{\det } & \bar G_\rho \ar[d]\\ \mathbb{G}_m \ar@{^(->}[r] & \mathbb{A}^1.}$$ The construction of $\bar G_\rho$ can be characterized in terms of toric varieties: it is the unique reductive monoid with unit group $G$ such that the closure of any maximal torus $T$ in $\bar G_\rho$ is the toric variety associated to $T$ and the cone generated by the weights of $\rho$.

TopArc spaces

Directly comes from the construction of $\bar G_\rho$ one sees that $\bar G_\rho(\mathcal{O})\cap G(F)$ is exactly supported on the $K$-double cosets associated dominant weights generated by the weights of $\rho$. So the basic function $\mathcal{C}_\rho$ can be viewed as a function on $\bar G_\rho(\mathcal{O})\cap G(F)$. Now take $F=k((t))$. Then we have the advantage of endowing $\bar G_\rho(\mathcal{O})$ an algebro-geometric structure over the residue field $k$.

Definition 4 Let $X$ be an algebraic variety over a field $k$. We define its $n $-th jet space $\mathcal{L}_n(X)$ to be the functor sending a $k$-algebra $R$ to the set $X(R[t]/t^{n+1})$. If $X$ is affine, then $\mathcal{L}_n(X)$ is also representable by an affine $X$-scheme of finite type. In particular, $\mathcal{L}_n(X)(k)=X(k[t]/t^{n+1})=\Hom(k[t]/t^{n+1}, X)$ consists of order-$n$ arcs in $X$. When $n=1$ we exactly recover the tangent bundle of $X$. For more general $n$, $\mathcal{L}_n(X)$ contains information about the singularities of $X$.
Example 6 $\mathcal{L}_n(\mathbb{A}^1)=\mathbb{A}^{n+1}$.
Example 7 Notice that if $X$ is defined by $f(x,y)=0$, then $\mathcal{L}_n(X)$ is defined by the equation $f(x+a_1 t+\cdots a_{n}t^n, y+b_1 t+\cdots b_{n}t^n)=0 \pmod{t^n}$ with extra variables $a_i, b_i$. Take $X=\{xy=0\}\subseteq \mathbb{A}^2$. Then $\mathcal{L}_n(X)$ is given by $$(x_0+x_1t+\cdots x_nt^n)(y_0+y_1t+\cdots y_nt^n)=0\pmod{t^{n+1}}.$$ One can find $\mathcal{L}_n(X)$ exactly has $n+2$ irreducible components, each isomorphic to $\mathbb{A}^{n+1}$ given by the first $k$ of the $x$-coordinates are 0 and first $\ell$ of the $y$-coordinates equal to zero, where $k+\ell=n+1$. The component with $k=0$ maps to the line $x=0$, and the component with $k=n+1$ maps to the other line $y=0$. All the rest $n-1$ components maps to the singularity (the origin).
Example 8 Take $X=\{x^3+y^3+z^3=0\}\subseteq \mathbb{A}^3$. Then $\mathcal{L}_n(X)$ has one irreducible component of dimension $2(m+1)$ which dominates $X$, and has one extra component of the same dimension mapping to the origin when $m\equiv2\pmod{3}$.

If $X$ is smooth, then the natural map $\mathcal{L}_n(X)\rightarrow X$ is smooth and surjective. In general, if $X$ is not smooth, then $\mathcal{L}_n(X)\rightarrow X$ may fail to be surjective, and the transition maps can be rather complicated.

Definition 5 We define the (formal) arc space to be $\mathcal{L}(X)=\varprojlim_n\mathcal{L}_n(X)$. In particular, $\mathcal{L}(X)(k)=X(k[ [t] ])=X(\mathcal{O})$, which consists of (formal) arcs $\mathbb{D}\rightarrow X$ of $X$ (here $\mathbb{D}=\Spf k[ [t] ]$ is the formal disc).

Again if $X$ is smooth then $\mathcal{L}(X)\rightarrow X$ is formally smooth and surjective. A theorem of John Nash says that the inverse image of $X_\mathrm{sing}$ in $\mathcal{L}(X)$ has only finitely many irreducible components, each corresponds to a component in the inverse image of $X_\mathrm{sing}$ in any resolution of singularities of $Y\rightarrow X$.

Definition 6 Let $X^\circ\subseteq X$ be a smooth open dense subvariety. We define $\mathcal{L}^\circ(X)\subseteq \mathcal{L}(X)$ to be the space of non-degenerate arcs in $X^\circ$. Namely for a $k$-algebra $R$, $\mathcal{L}^\circ(X)(R)$ consists of arcs $\phi: \mathbb{D}_R\rightarrow X$ such that inverse image $\phi^{-1}(X^\circ)$ is open in $\mathbb{D}_R$ and surjects to $\Spec R$. In particular, we have $$\mathcal{L}^\circ(X)(k)=X(\mathcal{O})\cap X^\circ(F).$$

If one has a $\ell$-adic sheaf $\mathcal{F}$ on $\mathcal{L}(X)$, then taking the Frobenius trace gives us a function $$\mathcal{C}_\mathcal{F}: \mathcal{L}(X)(k)=X(\mathcal{O})\rightarrow \overline{\mathbb{Q}_\ell}, \quad x\mapsto \Tr(\Frob_x: \mathcal{F}_x).$$ (if $\mathcal{F}$ is a complex, then take alternating trace on the cohomology groups). Similarly, if we only have a sheaf on $\mathcal{L}^\circ(X)$, we can still obtain a function on $X(\mathcal{O})\cap X^\circ (F)$. When specializing to $X=\bar G_\rho$ and $X^\circ = G$, we can obtain a function on $\bar G_\rho(\mathcal{O})\cap G(F)$ as desired. Our next goal is then to construct a canonical sheaf on $\mathcal{L}^\circ(\bar G_\rho)$, whose associated function gives the basic function $\mathcal{C}_\rho$.

TopIC sheaves and functions

If $X$ is a variety over $k$, there is a canonical sheaf associated to $X$, i.e., its IC sheaf which generalizes the constant sheaf and encodes the singularities of $X$.

Definition 7 Let $j: X^\circ\hookrightarrow X$ be a smooth open dense subvariety. We define $$\mathrm{IC}_X:=j _ {! * } \mathbb{Q} _ \ell=\im (j _ ! \overline{\mathbb{Q} _ \ell}\rightarrow Rj _ * \overline{\mathbb{Q} _ \ell}),$$ to be the middle extension of the constant sheaf on $X^\circ$ (so $\mathrm{IC}_X$ a complex of sheaves in the derived category of $X$). It is independent of the choice of $X^\circ$ and measures the singularities of $X$ along the boundary. The shift $\mathrm{IC}_X[\dim X]$ serves as the dualizing sheaf for the Poincare(-Verdier) duality for singular varieties, and is a basic example of a perverse sheaf.

However, because the arc space $\mathcal{L}(X)$ is infinite type over $k$, there is no good theory of IC sheaves/perverse sheaves on $\mathcal{L}(X)$. Fortunately, the singularities of $\mathcal{L}(X)$ have a finite dimensional model.

Definition 8 A finite dimensional formal model of $\mathcal{L}(X)(k)$ at $x\in \mathcal{L}(X)(k)$ is a formal scheme $Y_y$ (the subscript means taking formal completion), where $Y$ is a finite type $k$-scheme and $y\in Y(k)$ a point such that $$\mathcal{L}(X)_x\cong Y_y\times \mathbb{D}^\infty.$$
Theorem 1 (Drinfeld (2002), generalizing Grinberg—Kazhdan (2000) for $\Char k=0$) Finite dimensional model exists at each point $x\in\mathcal{L}^\circ(X)(k)$.

Bouthier-Ngo-Sakellaridis [1] show that the stalk $\mathrm{IC}_{Y,y}$ of the IC sheaf of $Y$ does not depend on the choice of the finite dimensional formal model $Y_y$. It now makes sense to define the IC function on the non-degenerate arcs by $$\mathrm{IC}_{\mathcal{L}(X)}: \mathcal{L}^\circ(X)(k)\rightarrow \overline{\mathbb{Q}_\ell}, \quad x\mapsto \tr(\Frob_y: \mathrm{IC}_{Y,y}).$$ It is a numerical invariant encoding the singularities of $X$. By taking $X^\circ= G$ and $X=\bar G_\rho$, we obtain $$\mathrm{IC}_\rho: \bar G_\rho(\mathcal{O})\cap G(F)\rightarrow \overline{\mathbb{Q}_\ell}.$$

Now we can state the main theorem of [1].

Theorem 2 (Bouthier-Ngo-Sakellaridis (2016)) Let $\nu_G$ be the half sum of all positive roots. Then $$\mathrm{IC}_{\rho}=\mathcal{C}_\rho(-\langle\nu_G, \lambda_\rho\rangle).$$
Example 9 When $G=\GL_n$ and $\rho=\Std$, we have $\langle\nu_G, \lambda_\rho\rangle=(n-1)/2$. So we recover $$\tr (\pi \otimes |\det|^s|)(\mathrm{IC}_\rho)=L(s-(n-1)/2, \pi).$$ The shift $(n-1)/2$ matches with the Godement-Jacquet zeta integral as well.

TopA global model

To prove the main theorem, we need a concrete construction of the finite dimensional model of $\mathcal{L}(X)$ at non-degenerate arcs. To do so we make use of a global smooth projective curve $C/k$. From now on, let $X=\bar G_\rho$ (with left and right $G$-actions).

Definition 9 Recall that an $S$-point of the quotient stack $[X/G]$ consists of a principal $G$-bundle $\mathcal{E}$ over $S$ together with a $G$-equivariant map $\phi: \mathcal{E}\rightarrow X$. Consider the stack $\Map(C, [X/G])$, whose $k$-points consists of maps $\phi: C\rightarrow [X/G]$, namely a principal $G$-bundle $\mathcal{E}$ over $C$ together a $G$-equivariant homomorphism $\mathcal{E}\rightarrow X$. We now add the non-degeneracy and define $M$ to be the open substack of $\Map(C, [X/G])$ such that $\phi:\mathcal{E}\rightarrow X$ factors through $\phi:\mathcal{E}|_U\rightarrow G$ for a open subset $U\subseteq C$. Then one can show that $M$ is an algebraic space locally of finite type.
Definition 10 To relate $M$ to $\mathcal{L}(X)$, we fix a $k$-point $v\in C(k)$. Define $\tilde M$ to be the stack classifying a point $(\mathcal{E},\phi)\in M$ together with $\theta: \mathbb{D}_v\times G\cong \mathcal{E}_v$, a trivialization of $\mathcal{E}$ on the formal disc $\mathbb{D}_v$. Then we have a canonical projection $$\tilde M\rightarrow M,$$ which is a torsor under $\mathcal{L}(G)$, hence is formally smooth. On the other hand, given a point $(\mathcal{E},\phi,\theta)\in \tilde M$, we obtain an arc by the composite map $$\mathbb{D}_v\rightarrow \mathbb{D}_v\times G\xrightarrow{\theta} \mathcal{E}_v\hookrightarrow \mathcal{E}\xrightarrow{\phi} X.$$ Moreover, this arc is non-degenerate (by the non-degenerate requirement when defining $M$). Thus we obtain a morphism $$\tilde M\rightarrow \mathcal{L}^\circ(X).$$

The following essentially says that there is no obstruction for deforming $G$-bundles while fixing the induced formal arc.

Proposition 1 Let $x\in \mathcal{L}^\circ(X)(k)$. Let $y\in \tilde M(k)$ be a point such that $\phi|_{C\backslash v}$ lies in the smooth locus of $X$, and such that its image in $\mathcal{L}^\circ(X)(k)$ is $x$ (such $y$ always exists by Beauville-Laszlo patching the trivial $G$-bundle). Then $\tilde M_y\rightarrow (\mathcal{L}^\circ X)_x$ is formally smooth.

It follows that $$M_y\times \mathbb{D}^\infty\cong \tilde M_y\times \mathbb{D}^\infty\cong (\mathcal{L}^\circ X)_x\times \mathbb{D}^\infty,$$ and hence $M_y$ can serve as a finite dimensional formal model of $\mathcal{L}^\circ(X)$ at $x$. In particular, we obtain $$\mathrm{IC}_M(y)\cong \mathrm{IC}_{\rho}(x)$$

TopGeometric Satake

Let $y\in M(k)$. From the fixed map $\det: G\rightarrow \mathbb{G}_m$ one naturally associates to $y$ a line bundle on $C$. Using the trivialization of $\mathcal{E}|_U$ induced from $\phi: \mathcal{E}|_U\rightarrow G$, we also obtain a generic section of this line bundle, hence a divisor $D$ on $C$.

Let $D=\sum n_iv_i$ and $M_D\subseteq M$ be the substack whose associated divisor is $D$. By the Beauville-Laszlo patching, the data of a $G$-bundle $\mathcal{E}$ and a trivialization away from $D$ is the same as giving $G$-bundles $\mathcal{E}_i$ on the formal disc $\mathbb{D}_{v_i}$ together with a trivialization on the punctured formal disc $\mathbb{D}_{v_i}^*$. Then we obtain a map into the affine Grassmannians $\Gr$ (whose $k$-points are $G(F)/G(\mathcal{O})$) at $v_i$'s, $$M_D\rightarrow\prod_{i=1}^m \Gr_{v_i}.$$ Moreover, a trivialization of $\mathcal{E}|_U$ actually comes from a $G$-equivariant map $\mathcal{E}\rightarrow X=\bar G_\rho$ if and only if $\mathcal{E}_i$ has invariant $\le n_i\lambda_\rho$ for each $i$. Thus we obtain an isomorphism $$M_D\cong \prod_{i=1}^m \Gr_{v_i,\le n_i\lambda_\rho}.$$ Notice each term on the right is indeed a projective variety (a Schubert variety), which models singularity of $\mathcal{L}^\circ(X)(k)$ when $n_i\rightarrow\infty$. Varying $D$, we obtain an isomorphism $$M(k)\cong \sideset{}{'}\prod_{v\in |C|} (\bar G_\rho(\mathcal{O}_v)\cap G(F_v))/G(\mathcal{O}_v).$$ Using this isomorphism and a fixed $v\in C(k)$, we can choose the point $y\in M(k)$ explicitly corresponding to a point $x\in \mathcal{L}^\circ (X)(k)$ such that $\mathrm{IC}_\mathcal{\rho}(x)$ is the $v$-component of $\mathrm{IC}_M(y)$.

Now recall the geometric Satake correspondence.

Theorem 3 (Mirkovic-Vilonen (2007)) Let $K_\rho$ be the IC sheaf of the Schubert variety $\Gr_{\le \lambda_\rho}$ shifted by its dimension $\langle2\nu_G, \lambda_\rho\rangle$. Then the map $\rho\mapsto K_\rho$ gives an equivalence of tensor categories between the finite dimensional representations of $\hat G$ and $\mathcal{L}(G)$-equivariant perverse sheaves on $\Gr$ (the tensor structure given the convolution product).

Bouthier-Ngo-Sakellaridis show that $$\mathrm{IC}_{M_D}\cong \boxtimes_{i=1}^m K_{v_i, \Sym^{n_i}(\rho)}[-n_i\langle 2\nu_G, \lambda_\rho\rangle](-n_i\langle \nu_G,\lambda_\rho\rangle).$$ (The symmetric power essentially comes from looking at the map $C^{n_i}\rightarrow \Sym^{n_i}C$). Hence by the geometric Satake we have $$\mathrm{IC}_M=\prod_{v\in|C|}\sum_{d\ge0} \mathcal{C}_{\rho,v}^d(-\langle\nu_G,\lambda_\rho\rangle).$$ The main theorem now follows by taking the $v$-component.

Last Update: 05/05/2018. Copyright © 2015 - 2018, Chao Li.

References

[1]Bouthier, A. and Ngô, B. C. and Sakellaridis, Y., On the formal arc space of a reductive monoid, Amer. J. Math. 138 (2016), no.1, 81--108.

[2]Ngo, Bao Chau, Hankel transform, Langlands functoriality and functional equation of automorphic L-functions, http://math.uchicago.edu/~ngo/takagi.pdf.