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!

[-] Contents

01/23/2019

Let us start with the classical work of Hecke on the integral representation of -functions of modular forms. Let be a cusp form of level , which has a Fourier expansion Let be its -function and completed -function. Hecke showed that the completed -function is equal to the Mellin transform of the modular form and use it to show the analytic continuation and functional equation of . Evaluating at the center of functional equation (which lies outside of the range of convergence of ), we obtain a *central value formula* 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 gives rise to a vector in a cuspidal automorphic representation of , where . Assume is normalized (i.e., ) and a Hecke eigenform. Then the central value formula can be rewritten as (Here the extra 1/2 comes from normalization of measures). The RHS is an *automorphic period* along the subgroup (a real number since is a normalized eigenform).

Recall also that Rankin-Selberg expressed the Petersson inner product in terms of the Rankin-Selberg -function The RHS is equal to an adjoint -value at the edge of critical strip while the LHS is also equal to (again due to normalization of measures) times

Squaring Hecke's central value formula and dividing the Rankin-Selberg identity, we obtain where is the completed Riemann zeta function.

This new identity generalizes to any vector of any cuspidal automorphic representation of over any number field . Let . Then for a sufficiently large finite set of places including the archimedean places, where the local period Here we choose the Tamagawa measures on and , and the local measures are chosen such that and the local period for almost all 's. One can show that is not identically zero and has no pole or zero at , hence the new identity implies the equivalence that

there exists such that .

Waldspurger proved a remarkable generalization by replacing with *any* nontrivial torus in . Such a torus is isomorphic to for a quadratic extension (and the embedding is unique up to conjugation). Let be the quadratic character associated to by global class field theory.

Theorem 1 (Waldspurger)
For every , we have where the local periods are defined similarly via integration over instead of .

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 with integral representations of (i.e., only for the central value — no parameter to vary). Moreover, unlike the split torus case, the local periods can be identically zero: it turns out that

.

Moreover,and for all .

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* of , where is a quaternion algebra with an embedding . We thus have an embedding and we can again consider -period on .

Assume admits a Jacquet-Langlands lift to . In particular, for all places where splits. Then Waldspurger's formula extends to this more general situation by replacing with , and the local periods by (while the -functions stay the *same*).

The following nice theorem gives a criterion of , in terms of local root numbers.

As a consequence, if is a division algebra, then Combining with the classification of quaternion algebra over and the properties of Jacquet-Langlands correspondence, we know that

there exists with for all .

Therefore Waldspurger's formula implies thatthere exists such that .

Moreover, if such exists, it is also unique. Since all the local periods are positive definite, it follows from Waldspurger's formula that as predicted by the Riemann hypothesis.One obvious generalization is to replace the pair by and it leads to the Rankin-Selberg type integral representation of -functions on due to Jacquet—Piatetski-Shapiro—Shalika. We will discuss this important result later.

The other generalization is by noticing the exceptional isomorphism where is the 2-dimensional quadratic space over , and is the 3-dimensional quadratic space over . Similarly, we also have exceptional isomorphisms where is the 1-dimensional -hermitian space , and is the 2-dimensional -hermitian space . In view of these, Gan-Gross-Prasad proposed the generalization of the results of Waldspurger and Saito-Tunnell to any pair of quadratic or hermitian spaces with a non-degenerate hyperplane.

Let (or ) and (or ). Assume that , are cuspidal automorphic representations of and of Ramanujan type.

Similarly, to state a version only about central -values, we need the notion of *pure inner forms*. Let be another quadratic or hermitian space with the same dimension (and discriminant in the quadratic case) as . Let or and or , where and are orthogonal direct sum with the same line . In particular, for almost all .

Definition 1
Say and are *nearly equivalent* if for almost all places. Unlike the case of this is a nontrivial equivalence relation.

Conjecture 2 (Gan-Gross-Prasad)
TFAE:

- There exists and nearly equivalent to respectively such that is nonzero.
- .

Conjecture 3 (Ichino-Ikeda, N. Harris)
Assume . Assume further that , is tempered for all . Let . Then we have an identity Here is an integer related to the -packet/the size of nearly equivalence classes.

Remark 1

- Conjecturally, is tempered for all if and only if is of Ramanujan type.
- Sakellaridis-Venkatesh showed .
- The local Gan-Gross-Prasad conjecture describes when in terms of the local root numbers of the local Langlands parameters for orthogonal or unitary groups.
- Known results for the global conjectures: orthogonal case (Waldspurger), (Ichino-Ikeda); unitary case (W. Zhang, H. Xue, Beuzart-Plessis).
- 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).

01/28/2019

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

Let be a connected reductive group over a number field . Let be the maximal split torus. Assume has finite volume. Let be a unitary character. We denote by the space of measurable functions such that for any and Let be the subspace of cuspidal functions (i.e. its integral along is zero for any the nilpotent radical of a proper parabolic subgroup of ). Then by a theorem of Gelfand and Piatetski-Shapiro, we have a decomposition where runs over irreducible *unitary cuspidal automorphic representations* of .

We recall the definition of *automorphic forms* (smooth, -finite, -finite, and of uniformly moderate growth functions on ). Notice that we don't require -finite condition (so we obtain a larger space of automorphic forms), which is more suited for the analytic theory. Let be the space of automorphic forms such that for any . In particular, acts on (not just a -module at ). One can define a nontrivial topology on which makes it a locally Frechet space. This topology gives the global realization of the -modules, and agrees with the Casselman-Wallach globalization in this case (see Bernstein-Krotz).

Let be the subspace of *cuspidal automorphic forms*. Let . Since every cuspidal form is of rapid decay modulo center (i.e., ), we know that which is dense in . Moreover, where is the subspace of smooth vectors. In particular, any automorphic form generates a finite length representation under the right translation of .

Example 1

- The constant function , but unless has no proper parabolic subgroup (by Borel, Harish-Chandra, this last condition is equivalent to is compact).
- For , , any gives .

We define an *irreducible cuspidal automorphic representation* of to be a topologically irreducible subrepresentation of , or equivalently an irreducible representation of the form where is a unitary cuspidal automorphic representation, together with an embedding . From now on we will work with and forget about , and write for .

We have the *factorization theorem*: any irreducible cuspidal automorphic representation (or any smooth admissible irreducible representation) of , factorizes as a restricted tensor product 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 on , there exists and a continuous semi-norm such that for any and ), 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).

01/30/2019

We defined the -group, Langlands parameters for unramified representations, and general Langlands -functions. For , 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.

02/04/2019

For , 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 , Kemarsky for ) 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 -function for (Jacquet—Piatetski-Shapiro—Shalika, generalizing Hecke's integral).

As a consequence of the integral representation, we can express the central -value for cuspidal automorphic representations on as an automorphic period over . Moreover, the local periods are not identically zero (since it has an interpretation of a local zeta integral divided by the local -value, and the local -function is the common gcd of local zeta integrals), and thus the central -value is nonvanishing if and only if an automorphic period is nonvanishing. However, we cannot quite formulate this nonvanishing criterion using the *partial* -values like we did for , due to the fact that we do not yet know that the local -function does not have a pole at . Notice the generalized Ramanujan conjecture implies that are tempered, and in this tempered case the estimates on Whittaker functions imply that is holomorphic for . However, current bounds on generalized Ramanujan conjecture only implies that is holomorphic for for small .

02/06/2019

Refs:

- 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.

Let be a quadratic extension of number fields. Let be an hermitian space of dimension . Recall that , where acts via the outer automorphism through its order two quotient , where . We define the base change map on -groups where acts via the order two quotient by permuting the two copies of .

Remark 2
More generally, let be a connected reductive group over . Let be a finite extension. There is a base change map , and the induced map on for Langlands parameters should correspond to the restriction of the conjectural Langlands group to .

The Langlands philosophy predicts a base change map such that (the local base change map) for almost all . Here:

- The local base change for an unramified representation on is given by the unramified representation of such that the Langlands parameter of is given by the base change of the Langlands parameter of via the above base change map of -groups.
- The local base change map at a split place is given by on .

Theorem 3 (Mok, Kaletha-Mingeuz-Shin-White)
Any admits a weak base change. Moreover, any such appears with multiplicity one in .

Remark 3
These works follow the work of Arthur (for split and ), and are still conditional on the weighted fundamental lemma (which is proved by Chaudouard-Laumon for any split groups). This weak base change 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 satisfying:

- there exists a split place such that is supercuspidal.

This implies that is supercuspidal, hence is cuspidal (and thus unique).

Remark 4
Recall the two equivalent characterization of supercuspidal representations for a -adic reductive group :

- Any matrix coefficient is compactly supported modulo center on .
- The Jacquet module is zero for any proper parabolic subgroup .

(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 such that is supercuspidal, then is cuspidal. Indeed, for any proper parabolic subgroup, the constant term factors through for any .

Definition 3
We say a function of the form *essentially a matrix coefficient* of . For such , and an irreducible representation of , we know that if , then , for some unramified character of .

As an application of (a), we see that if is a test function such that is essentially a matrix coefficient of a supercuspidal representation for some , then its action on has image in .

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 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 The automorphic period on LHS is on a nonsplit torus , while the one on RHS is on the split torus . Waldspurger's formula the follows since the RHS is related to by Hecke's integral.

Definition 5
For and , we define the distribution (there are convergence issues which we ignore for the moment).

Formally, we have a *geometric expansion* where the *(relative) orbital integral* is defined to be

Formally, we have a spectral expansion of the kernel function where So we also have a *spectral expansion* where the *relative character* is defined to be (if you replace by the inner product on then becomes the usual character ).

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

The problem is that both expansions are not convergent unless and are compact. We will instead consider *simple* RTF by choosing the test function such that both expansions are absolutely convergent.

Definition 6
We say satisfies the

*geometric condition*(GC) if there exists such that is supported on in the (relative) semisimple elliptic locus. Here we say is*semisimple*, if the orbit is closed; and say is*elliptic*, if the stabilizer has no split center.*spectral condition*(SC) if there exists such that is essentially a matrix coefficient of a supercuspidal representation (so if is not cuspidal).

Proposition 1
Assume satisfies (GC) and (SC). Assume further that:

- , , and ,
- , are quasi-affine.

Then is absolutely convergent, and we have the RTF identity

Here and the orthonormal basis is taken with respect to the Petersson inner product on

Le be a quadratic extension of number fields. Let be hermitian spaces over of dimension and respectively. For simplicity assume , . Consider Let . The Gan-Gross-Prasad conjecture relates the GGP period on , and its -function Jacquet-Rallis propose to attack the GGP conjecture by comparing two RTFs.

The first RTF (on unitary groups) is associated to . For "good" test functions on , we consider the distribution Its spectral expansion is related to the GGP period .

The second RTF (on general linear groups) is associated to , where and For "good" test function on , we consider the distribution Its spectral expansion is related to for on . We saw last time that

- if and only if , and we will see that
- if and only if for some on some inner form .

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

On unitary groups, we can identify and thus the -orbits in are given by -conjugacy classes in .

On general linear groups, we can similarly identify with , so the orbits are identified with the orbits in . Moreover we can identify Therefore orbits in are given by -conjugacy classes in .

02/11/2019

Recall , where . Fix compatible isomorphisms of -spaces and such that the vector gets sent to the last basis vector . We thus obtain an embedding and . So the orbits on both unitary groups and linear groups can be mapped to the -conjugacy classes in .

Definition 7
Say that is *regular semisimple* (with respect to the conjugation action of ) if

- is a basis of , where and
- is a basis of , where

Remark 5
One can prove this definition agrees the more general notion. More generally, if is a reductive group acting on an affine variety . Then is called *semisimple* if the orbit is Zariski closed; *regular* is of minimal dimension. When with acting diagonally this recovers the usual notion for algebraic groups.

Let be the subset of regular semisimple elements. Let and .

Notice that if and only if is nonvanishing, so are all principal Zariski open subsets.

Proposition 2

- The centralizer of any in is trivial.
- are -conjugate if and only if
- (resp. ) are -conjugate if and only if they are -conjugate (resp. -conjugate).

Proof

- If , where is regular semisimple and . Then for all . As form a basis, we know that .
- If , where . Then for all . Conversely, there exists a unique sending to , Moreover also sends the dual basis to because by assumption. Since and , we know that . To check that , we just need to show that for (because and then acts in the same way on the basis ). For , this is true by the construction of . For , we check that for , we have which is equal to Thus as desired.
- Let us show the assertion about unitary groups (the symmetric space is similar). Assume are -conjugate. By (a), there exists a unique such that . Let be the adjoint map with respect to the hermitian form on . Then Therefore . By the uniqueness of we know that , and hence . ¡õ

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

Proposition 3
Let be a field. Let so that .

- Any is -conjugate to an element of . Therefore we have an embedding Here the actions are by conjugation.
- We have a bijection Here runs over all -dimensional -hermitian spaces up to isomorphism, and .

Remark 6
In the split case , then there is only one isomorphism class of -dimensional -hermitian spaces. In this case and similarly So the proposition is clear in this split case (even without regular semisimple condition).

Proof

- Let be the matrix representing the hermitian form on . Let . Then . So By (a) and (b) of Proposition 2, we know that there exists a unique such that Then , so , hence by the uniqueness of . By Hilbert 90, there exists such that . We check that so that .
- Let . We check as in (a) that and are -conjugate. Thus there exists a unique such that . Taking adjoint we obtain . By the uniqueness of , we know that i.e., is hermitian. Then for the hermitian space . This shows the surjectivity of the map. To see the injectivity, assume , , and for a hermitian matrix. Then thus and so by the uniqueness of we have , i.e., as hermitian spaces. ¡õ

Remark 7
There is another way to look at the bijection in (b). There exists an isomorphism of varieties Taking -points, on the RHS we obtain While on the LHS we obtain a disjoint union

In fact, if is a reductive group acting freely on an affine variety . Then classifies isomorphism classes of pairs , where is a -torsor over , and is an -equivariant morphism. Since -torsors over up to isomorphism is in bijection with , we have where is the -torsor corresponding to , , and (with -action).

We recover the previous result by noticing that and is in bijection with isomorphism classes of -dimensional -hermitian spaces, and , and .

Definition 8

- Say and
*match*if their orbits correspond by the bijection in Proposition 3 (b). - Say and
*match*if their images in and match. Here , for .

Definition 9
Say is *good* if and

- (GC'): there exists such that is supported in (in particular, in semisimple elliptic locus).
- (SC): there exists such that is essentially a matrix coefficient of a supercuspidal representation.

Define good similarly.

For good functions and , we have a simple for , and a simple for . 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 where runs over all -hermitian spaces of dimension , it is better to

- compare (rather than a single hermitian space ) with .

For good test functions and , we would like to show for matching with . As the orbital integrals are product of local orbital integrals, we would like the global matching to come from a local one Here the *transfer factor* is necessary as the RHS does not only depends on the orbit of , due to the character . Notice the transfer factor should satisfy Moreover, for , for almost all , and we have a product formula . We will construct such transfer factor explicitly later (it will also satisfy for split ).

02/13/2019

(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 over local and global fields.

- If is a nonsplit archimedean place, then there are isomorphism classes of -hermitian spaces, classified by its signature .
- If is a nonsplit nonarchimedean place, then there are 2 isomorphism classes , , with .
- If is a split place, then there is only one isomorphism class.
- Globally, the isomorphism classes of -hermitian spaces is determined by its localization satisfying for almost all , and .

To construct sufficiently many matching global test functions, we need

Theorem 4 (Fundamental lemma, Yun, Gordon, Beuzart-Plessis)
For any global , we have for almost all .

Theorem 5 (Smooth transfer, W. Zhang for -adic places, H. Xue for archimedean places)
Every matches some 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 is split in . Choosing an isomorphism , and a place of above , we obtain

Proof
By definition, matches with . Then by definition where . Expanding definition, we obtain making a change of variables, we obtain which is precisely as desired (as is trivial).
¡õ

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

Fix two split places . Choose local test functions and such that

- For , and , for almost all .
- For , is essentially a matrix coefficient of a supercuspidal representation, and .
- For , , and .

Let , and for a global hermitian space , let . Notice for almost all by (a). Then and are good test functions, and comparing the geometric expansions we obtain It follows from the spectral expansion that Fix , we would like to isolate in the RHS. Notice both sides are a prior *infinite* sum.

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 there is a stronger version: a set of places of density is enough).

Theorem 6 (Ramakrishnan)
Let such that for almost all places of lying above split places of . Then .

Let be the set of places of such that is split in and , and , where If is not unramified for some , then both sides of the desired equality are zero. So we may assume is unramified. Consider the spherical Hecke algebras at , By Lemma 1, we have a smooth transfer map For , we have Here is a scalar.

Then for unramified, we have . Moreover, Lemma 1 implies that and 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 Moreover, the contributing and satisfy is supercuspidal, and is supercuspidal by our choice of and (hence cuspidal).

The Satake isomorphism identifies as the ring of regular functions on the space of unramified representations of . In this way we can view as a functions on . We now use the fact that the space *unitary* unramified representations is compact in , and thus the restriction of to it is a separating self-adjoint algebra of continuous functions. By the Stone-Weierstrass theorem, is dense in the space of all continuous functions on . This allows us to *separate* the contribution according to a given . So we obtain on the LHS a sum over such that , while on the RHS a sum over such that . By Ramakrishnan's Theorem 6, the RHS is thus equal to , and LHS is equal to the sum of over such that (as given by local base change at all degree one primes).
¡õ

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

Theorem 7 (W. Zhang)
Assume the fundamental lemma (Theorem 4) and the smooth transfer (Theorem 5). Let be an -dimensional -hermitian space. Let and assume that is supercuspidal at two split places . Then the followings are equivalent:

- There exists of dimension and such that and .
- .

02/18/2019

Proof of (a) (b) By replacing with , we may assume that . Write , . 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 ,

As a consequence, the global GGP period always factorizes as and thus the global relative character also factorizes as

By assumption, , and so . We apply the global comparison (Prop. 4) by choosing the local test functions and as follows:

- For , unless , and so that (which is the unit function at almost all places ).
- For , is supercuspidal, we choose to be essentially a matrix coefficient of such that . (For example, we can take a nonzero matrix coefficient, and take , where . Then Moreover, there exists essentially a matrix coefficient of such that . Set and .
- For , is supercuspidal, we choose such that . The existence of follows from the following theorem of Ichino-Zhang (as is of measure 0). Moreover, there exists such that . Set and .

The global comparison (Prop. 4) now gives By the factorization as and the compatibility of local base change at split places. By the multiplicity one theorem (Theorem 3), we know that the in the LHS are all distinct, and thus the nonzero 's are linearly independent (as one can always choose a test function acting as 1 on one and zero on the other 's). So we can choose such that if and only if . For such a choice, we have the LHS is nonzero, and hence the RHS . Hence and . And is equivalent to by the Rankin-Selberg integral.

Proof of (b) (a) As a corollary of previous proof,

To prove (b) (a), we need to show that and . This can be proved by combining Flicker-Rallis (relating the period with poles of Asai -functions) and Mok (relating poles of Asai -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 be supercuspidal at two split places . Then there exists supercuspidal at such that . In particular, by the previous corollary.

Proof
Take any supercuspidal representations and of and respectively. Since and are supercuspidal, we have by the Rankin-Selberg theory that (in fact this is true for any generic representations ). Now we apply the following more general globalization of distinguished representations (applied to , , ) to construct the desired .
¡õ

Proposition 6
Let be connected reductive group over . Assume . Let . Let be a finite set of nonarchimedean places. Let be a supercuspidal representations of , and . Assume Then there exists such that

- , i.e., there exists such that .
- For all , is an unramified twist of .

Proof
By assumption, surjects onto . As is supercuspidal and hence a projective module when restricted to , we obtain an embedding Take be a nonzero vector in the image of this embedding, and take a nonzero vector of the form . Then is a quotient of . The same holds for replaced by for any . But is dense in by weak approximation, we may assume that . Since the restriction of to is supercuspidal, we know that is a *nonzero cusp form*. So there exists not perpendicular to , i.e, for some . For such , (a) is satisfied. Moreover, the inner product gives an -invariant paring between and , and thus Therefore, and hence is an unramified twist of .
¡õ

Now we can finish the proof of (b) (a). By the Rankin-Selberg integral, we have . By Proposition 5 applied to and , we obtain . Thus for some . We can again modify at and such that are as before, and use the global comparison to conclude that for some and some .

02/20/2019

Refs:

- 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

Let be a connected reductive group over a number field . Let be a nonzero -invariant volume form on . For example for , we may take . Let be a nontrivial character (these all conjugate under ).

Weil associated to a Haar measure on as follows. Locally in the analytic topology, write then we define where is the -self-dual measure on , i.e., the Fourier transform preserves the -norm. The definition of does not depend on the choice of coordinates and hence give rise to a measure on .

For , we have Therefore formally the product does not depend on the choice of . However, the product may diverge. In fact, by Weil-Steinberg, we have at unramified places (where comes from and of conductor 1), where is the Artin-Tate motive over associated to .

Since 0 may be outside the range of convergence of , we modify the local measures and define

From now on, we will always take global measure to be the Tamagawa measure, and the local measure to be . So by definition,

Let be a quadratic extension of number fields.

Theorem 10 (Rallis, Flicker)
Let be a sufficiently large finite set of places. The partial Asai -function has meromorphic continuation to with at most a simple pole at . For every , we have where the Whittaker function factors as and

This theorem is an analogue to (by letting ):

Theorem 11 (Jacquet-Shalika)
Let such that . Then has meromorphic continuation to with at most a simple pole at . For every and , we have here we normalize the Petersson inner product

Remark 10
Notice the local functionals at appearing in the period formula are always convergent and *nonzero* (as the Kirillov model ). Therefore

- has a pole at if and only if ,
- has a pole at if and only if .

When there is a pole at , these local functionals are -invariant a priori.

Proof (Sketch of Theorem 11)
The proof uses the Rankin-Selberg method for .

Definition 14
Let be a *Schwartz function*, i.e.,

- is a smooth rapidly decay function for archimedean,
- for nonarchimedean,
- for almost all .

Definition 15
Define It converges when . This function can be viewed as a section of a parabolic induction. In fact, let be the mirabolic subgroup , and be a parabolic subgroup of type . Then

Proof
We use Poisson summation as in Tate's thesis (but now for ). Using we can rewrite This is equal to depending on or . The first term is homomorphic in . We then apply the Poisson summation to obtain the last two terms give respectively.
¡õ

By Lemma 2, the global zeta integral has meromorphic continuation to with at most a simple pole at (the residue is ). Now unfolding the global zeta integral we obtain It now factorizes as product of *local zeta integrals* where The local zeta integrals are convergent for , and computation at unramified places shows that Thus It follows that has meromorphic continuation to with at most simple pole at . Moreover, comparing the residue at of we obtain Now we compute which is equal to Notice the outer integral is over and is the Haar measure on . Now we done by choosing and so .
¡õ

Remark 11
Theorem 10 is proved similarly using a generalization of the Rankin-Selberg method to .

Remark 12
Let and . Twisting everything by and apply Theorem 10, we obtain Therefore

- has a pole at if and only if .

Let be the adjoint representation of on . Let be the adjoint representation on on . Let be the base change map. Then one can check Therefore for , we have and When is cuspidal, by Theorem 11 (as ) we know that *exactly one* of has a pole at and the other is nonzero at . Moreover, we have the following theorem.

Theorem 12 (Mok, Kaletha-Mingeuz-Shin-White)
Let . Then is in the image of base change for some if and only if has a pole at .

Thus we obtain the following corollary.

02/25/2019

Let be a local field. Let be the Weil group and be the Weil-Deligne group ( or depending on is nonarchimedean or archimedean).

Definition 18
Let be a connected reductive group. A *-parameter* is a homomorphism such that

- is continuous and semisimple (i.e., is semisimple for all ).
- The composition is the natural inclusion.

Two -parameters are *equivalent* if they are -conjugate.

Let be a quadratic extension of local fields. Fix .

Definition 19
A complex representation is called *-conjugate-dual* for , if there exists an isomorphism (where , and ) such that is equal to (notice ). Equivalently, there exists a nondegerarte pairing such that and Such is called an *-conjugate-dual form* for .

Proposition 7

- We have a commutative diagram

- Let be the inverse of the top arrow. Then an irreducible admissible is -conjugate-dual if and only if has nonzero fixed vector. If is moreover unitary, then this is further equivalent to has a pole at .

Proof

- The commutativity is clear. The inverse of the top horizontal arrow is given by , where and For the bottom horizontal arrow, if is an -parameter with . Then , where , is a nondegerarte -conjugate-dual form for . Conversely, if has a nondegerarte -conjugate-dual form , then extends to an -parameter with .
- is the representation of on given by and A fixed vector of its dual representation is nothing but a -conjugate-dual form on . Moreover, as is irreducible, then is nonzero implies that is nondegerarte by Shur's lemma. The second claim follows from a general property of local -functions: if is unitary, then if and only if has a pole at . ¡õ

Remark 14
Let be the set of isomorphism classes of -parameters . For , we define its *component group* .

If is a nondegerarte -conjugate-dual form on , then Moreover, if we decompose where is irreducible and -conjugate-dual, is irreducible -conjugate-dual, and is irreducible but not conjugate-dual. Then and thus

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*) Setting the *-packet* Then there are bijections (depending on some auxiliary choice of the Whittaker datum) Here runs over all isomorphism classes of -dimensioanl hermitian spaces.

Moreover, these are compatible with unramified local Langlands correspondence: if and are unramified, then is trivial on and and given by the unramified Langlands parameter and . It also satisfies local-global compatibility and endosocpic relations which characterize this correspondence.

Definition 20
Let , by the local Langlands correspondence for , the representation gives . We denote by .

By Proposition 7, determines and thus can serve as a *substitute* for it. For example, if is generic, we can recover as follows. By Bernstein-Zelevinsky, we may write uniquely as a parabolic induction (an *isobraic sum*) wehre are irreducible essentially square-integrable representations. Rewriting it according to the shape of the , where are square-integrable and -conjugate-dual (equivalently, has a pole at ), and similarly for and . Then Let be -conjugate-dual, correpsonding to a Langlands parameter of . Then which is compatible with the previous identification , and we may recover an element of by and .

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 be a quadratic extension of global fields.

The hypothetical global Langlands correpsondence gives a map from to global *discrete* (i.e., with finite centralizer) -parameters These discrete -parameters (in view of Proposition 7) should be the same as -dimensional -conjugate-dual admissible representations such that is a direct sum of distinct irreducible -conjugate-daul representations (notice , , is the only way to get finite centralizer). These , by the hypothetical global Langlands correpsondence, should in turn be in bijection with automorphic representations where each is unitary and -conjugate-dual, i.e., has a pole at (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)

- For each , is of the form where 's are distict unitary cuspidal automorphic representations of such that each has a pole at . Moreover, for all .
- Conversely, let be of the form satisfying the conditions as in (a). Set . Then for all , is -conjugate-dual and we have a natural morphism (the identity if splits). If is an admissible irreducible representations of with , then if and only if

Remark 15

- This theorem can be restated as where is as in (b), and .
- Generalized Ramanujan conjecture for predicts that any is tempered everywhere, i.e, is tempered for all . 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 to the Ramanujan conjecture for (which is still open).

03/01/2019

Let a quadratic extension of number fields. Let be a hermitian space over of dimension . Let with . Let . Let . Assume is tempered for all (in particular, is of Ramanujan type: is tempered at one place, which implies it is cuspidal and hence generic). Fix a -invariant inner product on , recall the local periods The integral is convergent since is tempered. Then is an -invariant sesquilinear form on .

Conjecture 4 (Ichino-Ikeda, N. Harris)
For nonzero, we have where

- is a sufficiently large finite set of places,
- the component group if .
- .

Remark 16

- is well defined and nonzero (Corollary 2).
- Unramified computation (N. Harris) shows that for almost all .
- If is cuspidal, then .
- This conjecture can be
*reformulated*in terms of the relative character Namely, for , we have an identity where is the local relative character. This identity can be thought of as a regularized version of the factorization "". - Both sides of the conjectural Ichino-Ikeda formula can be seen as -invariant sesquilinear forms on , hence are necessarily proportional by local multiplicity one.

Theorem 15 (W. Zhang, H. Xue, Beuzart-Plessis)
Let be tempered at all places, and supercuspidal at two split places. Then the Ichino-Ikeda conjecture holds for .

We may assume that , otherwise by the global Gan-Gross-Prasad conjecture and thus both sides vanish. By multiplicity one, it suffices to show the relative character identity for for *one* test function such that . Choose , and such that

- and are good.
- for all .
- (we have seen this is possible in the proof the global GGP).

By the global comparison we have For in the LHS, we have by Theorem 14. Now we need the following theorem.

Theorem 16 (Weak local GGP, Beuzart-Plessis)
For any and any tempered, there is at most one such that

Consequently, there is only one nonzero term on the LHS of the global comparison, and hence Now we can use the factorization of the RHS to obtain the desire factorization for the LHS.

Write , then where the orthogonormal basis is taken with respect to where the volume factor is equal to 4 under the Tamagawa measure. Using the explicit factorization of (Jacquet—Piatetski-Shapiro—Shalika), (Theorem 10) and (Theorem 11) in terms of the Whittaker model of , we have where Here is the local Whittaker model with respect to , and . Then where , and . The orthogonormal basis is take with respect to the scalar product By this factorization, it suffices check the following local relative character relations.

Theorem 17 (Local relative character relations, Beuzart-Plessis)
There exists explicit constants such that , and for every tempered with , and , we have

Remark 17

- This was proved by W. Zhang for unramified or supercuspidal representations (and easy at split places).
- We can show that this gives a
*spectral characterization*of transfers. Namely, if and only if the local relative character relation holds for every tempered such that . 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:

- the existence of smooth matching,
- the fundamental lemma,
- the local GGP conjectures,
- the local relative character relation.

Now we come back to the local situation. Let be a quadratic extension of -adic fields. Let hermitian spaces over of dimension and , with , . Let , .

Recall that and Recall that to be the inverse image of defined by the conditions in Definition 7. Fix extending (when is unramified, we may choose to be the unramified quadratic character).

Definition 22
For , we define the *transfer factor* to be where . Different choices of define transfer factors differing from each other by a smooth function.

It is easy to check that In the global setting (relative to ), if we pick extending and construct using for all , then

03/04/2019

Fix an identification , which in turn gives an identification by mapping to , and an inclusion

Definition 24
We say is *regular semisimple* if and are basis of and respectively. We denote the set of regular semisimple elements by . We define , and similarly .

For and , they have trivial stabilizers and closed orbits. Analogous to the group case, we introduce the following definitions.

Definition 26
Say and *match* if they are -conjugate in , or equivalently if for all . Like the group case, this matching induces a bijection

We first reduce Theorem 5 to a transfer statement between and (inhomogeneous version). For , we define its projection Similarly for , we define its projection where is chosen such that . We have the surjective projections and with and

So Theorem 5 reduces to

Theorem 19 (Smooth transfer, inhomogeneous version)
For every , there exists such that for all . Here .

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 , we define the *Cayley map* (resp. ) to (resp. ) by This induces (resp. ) equivariant isomorphisms and

Since varying we obtain *coverings* we know that any or can be written as a finite sum (as there are only only finitely many eigenvalues) where , and the pushforward function is defined as

We can easily check the following lemma.

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 for some and . By Theorem 18, there exists matching . Since , there exists such that By Lemma 3 (d), we have also matches . So we may assume that .

We set , then Lemma 3 (b) and (c) imply that which is equal to Therefore by Lemma 3 (a), matches . ¡õ

Let be an affine variety over a field with an action by a reductive group over . We define the *GIT quotient* . The GIT quotient is a categorical quotient, namely any -invariant morphism factors uniquely through . The geometric points of the GIT quotient in general are *not* in bijection with the -orbits in . However, for , contains a unique *closed* -orbit (called a *semisimple* orbit), giving a bijection This is not true over : for , might be empty, and even if it is not empty can split into more than one -orbit (e.g., the difference between stable conjugacy and usual conjugacy in the group case).

To bypass this difficulty, assume there is a -invariant nonempty open subset such that every has closed orbit and *trivial* stabilizer. Then we define which is a geometric quotient. For , is either empty or *exactly one* -orbit in . Therefore we obtain an open embedding in the analytic topology that

Proposition 9
There is a (necessarily unique) isomorphism such that its restriction to the regular semisimple loci gives back the *matching of orbits* as

Proof
Notice that the action of on and on both become the action of on after base change to . Therefore where , are involutions of with fixed points respectively. Explicitly, where represents the hermitian form on .

It remains to check for that the actions of and are the same on the GIT quotients. It is enough to check this for an open dense subset of the GIT quotients, namely for in general position, under -action. Equivalently, we need to check that , i.e., . But as in the group case (Proposition 2 (b)), for , then under if and only if for all . Applying this criterion to it follows that as desired. ¡õ

03/06/2019

Definition 30
Let . Let . Then Let , be the projections. Let . Then for any , we have is one regular semisimple -orbit in some , and is one regular semisimple -orbit.

Definition 31
For and , we define for any , where is such that . Similarly define for any . We define spaces of functions on ,

Then Theorem 18 can be reformulated as the identity

Proposition 10
A function belongs to ( or ) if and only if it is compactly supported in and it coincides near every with a function in .

Proof
Necessity is clear. Conversely, assume that is compactly supported and locally an orbital integral. Then there exists a finite covering of by such that and . Up to refining the cover, we may assume that 's are disjoint. Then Hence is an orbital integral.
¡õ

Definition 32
We say that the *transfer exists near * if there exists an open neighborhood of such that . By Proposition 10, to prove that the transfer exists if suffices to show that the transfer exists near every .

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

Let be a connected smooth affine variety over with an action of a reductive group over .

Definition 33
We say is *semisimple* if is Zariski closed in (equivalently, is closed in the analytic topology in ). Then by Matsushima's criterion, the stabilizer is also reductive. It acts on , the normal space to at , called the *slice representation*.

Theorem 20 (etale Luna's slice)
There exists a locally closed -invariant smooth affine subvariety containing and a -morphism sending to 0, such that we have the following Cartesian diagram with etale horizontal maps where with diagonal -action given by . In particular, we have .

Definition 34
Such is called an *etale slice* (think: cut every -orbit in a single -orbit; the -action around can be described in terms of -action on , which can be further linearized to the slice representation ). We may thus *localize and linearize* the orbital integrals to the etale slice .

Taking -points, we obtain Cartesian diagrams with etale (i.e. local isomorphism) horizontal maps

Definition 36
Let be an open neighborhood of such that and are both open embeddings. Then is a -invariant open neighborhood of , and are both open embeddings. In this way can be viewed as a -invariant open neighborhood of and is a -invariant open neighborhood of . Such is called an *analytic slice*.

We have a commutative diagram It will help us compare -orbital integrals on with -orbital integral on .

Assume that the regular semisimple locus is nonempty (hence open and dense), and assume that for any , we have (hence the same holds for and , and moreover and is equal to the preimage of ).

Definition 37
Given a character , we choose transfer factors such that and such that We assume that and coincide on .

Then we have Both subspaces can be considered as spaces of functions on .

Proposition 11 (Harish-Chandra's semisimple descent)
We have . In other words, the orbital integral locally around can be identified as orbital integral locally around .

Proof
The result follows from the commutative diagram of spaces of functions,

where are given by which are both surjective. ¡õ

Remark 18
Let . Then We have where 's are representatives of the semisimple -orbits in , and the is the analytic slices for . Moreover, we have and is the preimage of . By Proposition 11, we have The latter is also equal to

03/11/2019

Definition 39
Consider the space It is the most singular locus in (maximal stabilizers), and consists of elements of the form , where . Similarly, the most singular locus in is given by consists of elements of the form , where . Moreover, both spaces and are mapped isomorphically to , called the *center* of .

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

Proposition 12
The transfer exists locally near every , i.e., there exits an open neighborhood of such that .

By Harish-Chandra's semisimple descent, we have a decomposition of both orbital integral spaces in terms of slice representations. Let , and be the projections. Let be a set of representatives of semisimple orbits. We denote be a representative of the unique semisimple orbit. Then we have the slice representations Let and By Harish-Chandra's semisimple descent, if is small enough, then be comes an open neighborhood of 0 in and , and So it remains to compare the orbital integrals on the slice representations and .

Remark 19
Two warnings are in order (which we will ignore):

- the identification of as open neighborhoods of and are not canonical and one needs to check the choice made are compatible.
- the orbital integrals on and are twisted by and we need to check the transfer factors are also compatible.

We will see that *essentially* (i.e. *up to the center*) there exists finite extensions of and such that we have a "decomposition" where . And similarly So by induction on (assuming the transfer exists near all for smaller ), we obtain the desired comparison between the orbital integral spaces on and . Thus our remaining goal is to prove such decompositions.

Let us first describe the semisimple orbits. Let . Then we have Define Similarly define Then we have identifications and compatible with the action of and respectively. Moreover, Define and Then

Define The local transfer now becomes a comparison between orbital integral spaces on and . ;

Proposition 13 (Rallis-Schiffmann)

- is semisimple if and only if is nondegenerate and is semisimple (in the usual sense), where .
- is semisimple if and only if and are in perfect duality, and is semisimple (in the usual sense), where .
- , have the same image in if and only if
- and , have matching orbits.
- , have the same characteristic polynomial.

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

Proof
By Proposition 13 (c), and that there exists with the same image in , we know that and match the same regular semisimple element, hence they are conjugate. Similarly and are semisimple elements with the same characteristic polynomial, hence they are also conjugate.
¡õ

Let , and . Let be the characteristic polynomial of . Let be the unique hermitian space over such that matches . Then by Proposition 13 (c), the orbits in is in bijection with conjugacy classes of pairs where

- is an hermitian space over , and
- such that .

In this way these semisimple orbits are further in bijection with -tuple of hermitian spaces over of dimension , where , and , given by This finishes the description of the semisimple orbits.

Now we can also compute the centralizers and

Finally we can describe the slice representations. Let be the slice representation at inside , which is a representation of the *trivial* group (as is regular semisimple). Then as a representation of . Similarly, let be the slice representation at in , which is a representation of the trivial group. Then as a representation of .

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

03/13/2019

Definition 40
We say is *regular semisimple* if is a basis of . We say is *regular semisimple* if is a basis of and is a basis of .

Then we have a matching of orbits given by the condition for all , or equivalently, and are in the same -orbit in .

Definition 42
Define , be the preimage of the center , known as the *nilpotent cone* (up to the center). Then (resp. ) if and only if the closure of its orbit orbit meets the center (resp. ) of (resp. ). Define .

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

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 be the partial Fourier transform with respect to or . Namely, for , we define For , we define Here we fix a nontrivial character and the Haar measures are chosen to be the self-dual ones (i.e. preserves the -norms).

The following basic properties are easy to check.

Proposition 15

- (equivariance of Fourier transforms) Let . Then for for any and . Similarly statement holds for .
- .

Theorem 21 (W. Zhang, Fourier transforms preserve smooth transfer)
There exists explicit constants such that if , then for .

Consequently, if is transferable, then is also transferable; and if is transferable, then 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.

Remark 20

- Here and 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 . - Aizenbud proved the dual statement: there is no -invariant distribution on such that is supported on for (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 Let be the total Fourier transform on or . The following is an easy exercise.

Lemma 4
Theorem 21 for is equivalent to the following: if , then .

Today we will explain the proof of Theorem 21 when . In this case There are two isomorphism classes of hermitian lines where , and Then

- is regular semisimple if and only if ,
- is regular semisimple if and only if .
- they match if and only if (by convention ).

Notice the center and does not play a role, and is the Fourier transform with respect to the center, we know easily that commutes with transfer. Hence it is enough to consider .

For , it has orbital integrals For , it has orbital integral Then if and only if

Theorem 21 in this case can be made more precise as

Remark 22
We will use Tate's thesis to prove this result. For any character , we write such that For , recall the *Tate integral* is defined to be the Mellin transform which converges absolutely for . Recall that *Tate's theorem* says that for , we have where is Tate's -factor, is the usual Fourier transform, and . Let be the -factor for . Recall that the inductivity of -factors in degree 0 says that

Proof (Proposition 16)
Assume . Let . We would like to show that We show this by computing the Mellin transform of both sides. For , the Mellin transform of the RHS is given by which is a Tate integral in two variables, and hence converges. If , where , then it is easy to see that . Therefore by Tate's theorem we have when , the Mellin transform of the RHS is equal to which is the same as

Using , we know the Mellin transform of the LHS is equal to Notice which by Tate's theorem (when ) is equal to The gamma factor is equal to . Thus the Mellin transform of the LHS is equal to

It follows that the Mellin transforms of both sides are equal because . ¡õ

Remark 23
In the case 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.

03/26/2019

We rephrase Proposition 16 as follows for later use.

Corollary 4
Let be a quadratic extension of -adic fields. Let be a 1-dimensional hermitian form. If in the sense Then .

Remark 24
The constants in Theorem 21 are explicitly given by
Notice that . So to prove Theorem 21, it suffices to prove it for .

Our next goal is to prove Theorem 21 for . Let and match. Consider the usual Fourier transforms and . The following result reduces the general case to the case of (product of) .

Proposition 17
Let and be regular semisimple elements (in the usual sense). Let , and (maximal tori).

- There exists such that for every with ,
- There exists such that for every with ,

Proof
Let us prove (a). Let . It is closed, as is semisimple. We have two maps where is the projection onto the second factor, and . Then and are principal bundles under and respectively for the actions Hence we obtain surjective maps where We choose such that (this is possible since is closed), and define . If such that is regular semisimple, then changing the order of integration this is equal to which proves the first equality. For the second equality, we notice that if is the partial Fourier transform on of . Then

The proof of (b) is similar. ¡õ

Now we can prove Theorem 21 for . 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 is regular semisimple). We may thus assume that is regular semisimple. Then is also regular semisimple and they have the same characteristic polynomial . Write as a product of irreducible polynomials over . Write . Let be as in Proposition 17. Then by , we have for every , such that and are matching regular semisimple elements. We would like to show . By Proposition 17, it remains to show that

There are linear isomorphisms through which and act by multiplication by , and induce where is a 1-dimensional hermitian form for , such that Then We easily check that

- if and only if for all .
- if and only if for all .
- if and only if for all .
- , where for some constant .

Thus by Corollary 4 , we obtain Now we can finish the proof of Theorem 21 for by the following comparison of constants.

03/28/2019

Proof
Let . We need to show that Let be the discriminant of the quadratic form . Then one can easily check that Since , it follows that In this way we can eliminate the dependence one the 1-dimensional hermitian form , and it remains to show that

To prove this last identity, we reinterpret the local -factor using the Weil index of quadratic forms. Let be a quadratic form on a -vector space . Weil showed that there exists a unique constant such that where the Fourier transform is defined to be The *Weil index* satisfies the following basic properties:

- .
- , where .
- If is a finite extension and is a quadratic space over . Then .
- (Jacquet-Langlands) If , , then .

By (d) and (c), we have Under , we have We may decompose where . Then we have Using (a) and (b), we obtain By (d) again, this is equal to This completes the proof. ¡õ

Finally, let us prove Theorem 21 for . Again since orbital integrals on locally constant functions on the regular semisimple locus, it suffices to prove the theorem for with . Then the matching satisfies . Up to the action of , we may assume that and , where .

Let . Then Let and be the total Fourier transforms. Then by the same argument as Proposition 17, we can show the following.

Since , by Proposition 19 we obtain for every . Moreover, for some constant independent of , where is the usual transfer factor on . Therefore . By induction, the total Fourier transform commutes with smooth transfer (Lemma 4). It follows that and thus by Proposition 19 we have Finally one checks that , which finishes the proof of Theorem 21 for .

Let be a -adic field. Let be its unique unramified quadratic extension. Let . There are two isomorphism classes of hermitian spaces of dimension with The space admits a basis such that . We obtain a self-dual lattice (i.e., the dual lattice is equal to ). This allows us to define a model of over , so that We define similarly a model over of using the lattice Let and We normalize the transfer factor on by choosing the unique *unramified* extension of to . So . We also normalize the Haar measures by

Consider We again normalize the transfer factor by choosing the unramified and Haar measures such that

Proof
Recall (from the proof of Theorem 18): for , we may descend it to ; and for , we may lift it to using the Cayley transform. (Warning: as is only a birational isomorphism, is not necessarily compactly supported. However, its restriction to each orbit is compactly supported, so its orbital integrals are still well-defined).

Similarly, for , we may descend it to ; and for , we may lift it to .

We have proved that

- if and only if .
- if and only if . (Warning: there was an extra constant , but a direct computation shows that as ).

04/01/2019

Now we finish the proof when .

By our of normalization of measures, we see that By (a), it remains to show that Let , if the characteristic polynomial of is not integral, then , and hence . For matching , the characteristic polynomial of is equal to that of , and hence by the same argument we know that .

Thus we may assume that the characteristic polynomial of is integral. Since the characteristic polynomial has degree , and since the image of in the residue field is of size which is by assumption, we know there exists such that In this case we claim that In fact, let . Then Moreover, . So if and only if , which implies the claim.

Similarly, if matches , then for the same choice of , we have (by the same argument)

The desired result then follows from (b), by noticing that since , the extra factor as well. ¡õ

Now let us reformulate the Lie algebra version to a version more suitable for induction. Assume , then for some . We have and Moreover, gets sent to , and gets sent to . So Conjecture 6 is equivalent to

We prove Theorem 23 by induction on . For , we set . For , we set . Notice that if are matching regular semisimple elements, then .

Proposition 21
Theorem 23 holds when and .

Proof
Since is an invariant polynomial taking integral values on and , we know when , we have , and similarly we have when .

Now we assume . Up to the action of and , we may assume that for some (since is surjective on units), and for some . Since if and only if they are -conjugate, we have

- if then , and .
- if does not match any element of , then does not match any element of .

By induction, we have correspondingly in each case

- .

Notice . It remains to show that

Let us show the first identity. By definition Notice that if the integrand is nonzero, then and . Since acts transitively on and , it follows that , and thus Since is invariant under , we know this integral is equal to as desired. The proof of the second identity is similar. ¡õ

Let be any function matching . To finish the proof of Theorem 23, we would like to show that for any . As orbital integral are locally constant on regular semisimple locus, it suffices to prove it when . By Proposition 21, this equality is true when .

04/03/2019

Now we need an extra ingredient, namely the Weil representation (ref: Bump, Section 4.8). Let be a quadratic space of even dimension over . Let be a character. Then there exists a unique representation of , known as the *Weil representation* or the *oscillator representation* on such that for any , we have and

We apply this to and unramified. We obtain the Weil representation of on and we can extend it to a representation on by letting act trivially on . Concretely, we have where is the partial Fourier transform on . Notice that since is a sum of hyperbolic planes, the Weil index for any .

We have a surjection

Proof
We need to check that the action of and descend to . It is clear that Moreover, if and only if , if and only if (since preserves the smooth transfer), if and only if . It follows that the kernel of is stable under , and hence the action of descends as well.
¡õ

Now we can finish the proof of Theorem 23. Set We need to show that if .

We notice that is fixed by for any . In fact, if , then by Proposition 21; if , then . In both cases we have .

We also notice that is fixed by . Indeed since is unramified, we have It follows that if , then as well. Thus (notice the constant , as everything is unramified), and is fixed by .

It follows that is fixed by , and in particular is fixed by for . But if , then there exists such that . This implies that , and completes the proof of Theorem 23.

Refs:

- Kottwitz, Harmonic analysis on reductive p-adic groups and Lie algebras
- Beuzart-Plessis, The local Gan-Gross-Prasad conjecture, https://ggp-2014.sciencesconf.org/resource/page/id/1
- Beuzart-Plessis, A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the Archimedean case (including the -adic case)

Let be a quadratic extension of local fields of characteristic 0 (i.e., or -adic). Let be hermitian spaces over , with odd dimensional and is quasi-split (automatic in the -adic case). Equivalently, there exists a basis of such that if and only if . Gan-Gross-Prasad construct a triple (unique up to -conjugacy), consisting of

- ,
- ,
- .

Here

- is the unipotent radical of the parabolic subgroup Notice and we have the embedding
- The character , where we choose the non-degenerate character Notice that is -invariant and it is generic with that property.

Example 5

- (codimension 1 case) When . We have , .
- (Whittaker case) When , we have is a maximal unipotent subgroup and 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 is even (the *Fourier-Jacobi* case), where 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 be the set of isomorphism classes of irreducible smooth admissible representations of -representations of . Recall that in the archimedean case, smooth admissible means the Casselman-Wallach globalization of a Harish-Chandra module.

Definition 49
For , we define its *multiplicity* the dimension of the space of continuous -equivariant linear forms on .

Theorem 24 (Aizenbud-Gourevitch-Rallis-Schiffman (-adic, codimension 1), Gan-Gross-Prasad (-adic, general), Jiang-Sun-Zhu (archimedean)) We have .

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 for a Harish-Chandra module . But in general we do not have (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.

Recall that is bijection with the set of isomorphism classes of hermitian spaces with . We denote by the hermitian space associated to . Similarly, is bijection with the set of isomorphism classes of pairs of hermitian spaces with and . We denote by the pair of hermitian spaces associated to . The natural embedding is then given by explicitly

Definition 50

- A
*pure inner form*of is a group of the form for . - A
*pure inner form*of is a GGP triple associated to the pair for some .

Remark 27
In the -adic case there are only 2 pure inner forms of . In the archimedean case, there are different pure inner forms of , determined by the signature of .

04/08/2019

Recall that in the local Langlands correspondence (Theorem 13), the bijection depends on the choice of a Whittaker datum.

Definition 51
A *Whittaker datum* (for the family of pure inner forms ) is a triple (up to -conjugacy), where

- such that is quasi-split.
- is a maximal unipotent subgroup.
- is generic (i.e. nondegenerate).

Remark 28

- A unitary group of even rank has unique quasi-split pure inner form ( in the archimedean case), but there are two conjugacy classes of Whittaker data on this pure inner form.
- A unitary group of odd rank has two quasi-split pure inner forms (, in the archimedean case), and each of these support a unique conjugacy class of Whittaker data.
- Therefore there are 4 different choices of Whittaker data for . Among them there is a unique quasi-split inner form , 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 and the -orbits of nontrivial characters . So fixing allows us to fix the local Langlands correspondence for and its pure inner forms.

Definition 52
An -parameter is *generic* if is regular at . It is known that is generic if and only if contains generic representations (i.e. admitting a Whittaker model). If is generic, then is the unique representation in the packet admitting a Whittaker model of type (i.e., ).

Let be an -parameter. By base change (Proposition 7), can be identified with a pair where

- is a -conjugate-dual representation of of dimension .
- is a -conjugate-dual representation of of dimension .

Let be the nondegenerate -conjugate-dual form on the space of (and similarly for ). Then More precisely, if is the decomposition into -conjugate-dual, -conjugate-dual and non-conjugate-dual irreducible representations. Then (and similarly ).

Let be nontrivial. Then the standard properties of local root numbers implies that if is a conjugate-dual representation of , then .

Definition 53
We define characters and similarly for . We define the *GGP character* of to be Notice that it depends on the choice of .

Now we can state more precisely the local GGP conjecture. Fix a nontrivial , and use it to normalize the local Langlands correspondence and the GGP character.

Conjecture 7 (Gan-Gross-Prasad)
Let be a generic -parameter. Then there exists a unique such that . More precisely, we have

Remark 29

- In a series of 5 papers, Waldspurger and Moeglin-Waldspurger have established the analogous conjecture for -adic orthogonal groups.
- This conjecture was proved by similar techniques for -adic unitary groups (Beuzart-Plessis, the tempered case; Gan-Ichino, the general case).
- 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).

Assume for the moment that is compact (in the -adic case this implies that ), then by the orthogonality relations of characters we have here , is the character of , and we normalize the Haar measure so that . A striking discovery of Waldspurger is that there is a similar formula in the noncompact case as well.

Let . For , we have an operator on defined by It is of finite rank in the -adic case and is of trace-class in the archimedean case. We define a distribution

Theorem 25 (Harish-Chandra, Barbasch—Vogan)

- There exists , called the
*character*of , which is smooth on the regular semisimple locus such that for any , - Let be a semisimple element. Then there is a
*local character expansion*in a small neighborhood of 0, where- is the set of regular nilpotent orbits in (which is empty when is not quasi-split).
- is the Fourier transform of the orbital integral over , i.e., the unique locally integral function on (smooth on the regular semisimple locus) such that where the Fourier transform is defined using a -invariant form (e.g. the Killing form)
- .
- .

04/10/2019

Definition 55
Let be the set of semisimple -conjugacy classes. We define Here is *regular elliptic* if is a compact torus. Concretely, let be the set of representatives of the isomorphism classes (i.e., -orbits by Witt's theorem) of nondegenerate subspaces of , then Here is a set of representatives of conjugacy classes of elliptic maximal torus in , and is the Weyl group relative to .

Definition 56
We equip with the unique Borel measure such that for any , Here we normalize the Haar measure on by , and is the Weyl discriminant. Notice that with measure 1, so the contribution of 1 to the multiplicity formula will *not* be negligible.

Theorem 26 (Beuzart-Plessis)

- This expression makes sense, i.e., the integral converges for and the limit exists.
- If is tempered, then .

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

Remark 30
An -parameter is *tempered* if and only if is bounded, if and only if contains only tempered representations. Moreover, we have a parametrization of the tempered dual of by

Proof
Without loss of generality we may assume that is quasi-split. We will use the following three proprieties of the -packets:

- (stability) For any , the sum is
*stable*, i.e., constant on -conjugacy classes). - (transfer) For any , is the transfer of , where is the Kottwitz sign (in the -adic case, there are only two pure inner forms with and ). Namely, for every and which are conjugate inside , we have .
- (whittaker) For every Whittaker datum of , there exists a unique such that .

There is a natural bijection Moreover we have (F. Rodier, -adic case; H. Matumoto, archimedean case) It follows from (whittaker) and the uniqueness of Whittaker model that

By Theorem 26, we have Let be the space of stable conjugacy classes, and let be the natural projection. Then by (stability) we obtain Since every is stably conjugate to an , by (transfer) we obtain

Write for short. One can show that there exists an isotropic torus such that such that under the natural map . It follows that Moreover, the composition is a group homomorphism which is nontrivial when . Thus unless , and so This completes the proof. ¡õ

04/15/2019

For , we would like to show that where is a space of conjugacy classes, and is the regularization of the Harish-Chandra character of . Our goal is to compute using local trace formula.

Definition 58
We say is *strongly cuspidal* if for all proper parabolic subgroup , we have for all . We denote the space of strongly cuspidal functions by .

Proof
It suffices to show that 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 , constructed a *boundary degeneration* (where is a -orbit at ; or a "spherical root"). Concretely, where is a limit of conjugates of in the Grassmannian of (a certain point in the closure of the conjugates of in the Grassmannian). Then one can show that nary (after identifying a neighborhood of in and ) is equal to the kernel function of acting on . (In the archimedean case, the difference between and is rapidly decreasing).

Now the crucial point is that is *parabolic induced*, i.e., there exists and such that (this is true for most spherical varieties). So if is strongly cuspidal, then identically. Hence near the boundary.
¡õ

Our next goal is to expand both geometrically and spectrally.

We replace the GGP triple by , where is any connected reductive group over . Then the action of on is given by right and left translations. Let . The action of on is given by the kernel function We assume that is compact.

Proof
The proof is similar to Theorem 28. Here the boundary degenerations are of the form where is a parabolic subgroup, and is the opposite parabolic subgroup. In particular, integrating along is already 0 since is strongly cuspidal.
¡õ

Definition 60
Let be a Levi subgroup. Let . Let be a special maximal compact subgroup (all we need is satisfies the Iwasawa decomposition for any parabolic ). Associated to these data, we define the *weighted orbital integral* where is *Arthur's weight function*.

Remark 31

- The weight depends on the choice of .
- When , the weight and .
- Let . We have a morphism For any parabolic subgroup , we define Then is the volume of the convex hull of for all .

Definition 61
Let be the set of regular semisimple conjugacy classes in . We equip with the unique measure such that where is a set of representatives of conjugacy classes of maximal tori in .

Remark 32
Since is strongly cuspidal, the weighted orbital integral only depends on the orbit of and it does not depend on the choice of . Moreover, for all other Levi subgroups .

Theorem 31 (Harish-Chandra)
Let , , a compact open subgroup. Then the function is compactly supported, and

Definition 63
Let be a Levis subgroup. Let be a tempered representation of . Let be a special maximal compact subgroup. Let . We define the *weighted character* to be where is *Arthur's weight*, defined using standard intertwining operators.

Definition 64
Let be the category of finite length tempered representations. For , we have a normalized parabolic induction functor It has a left adjoint 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 (i.e., generalized eigenspaces corresponding to unitary characters of ).

Definition 65
Let be the Grothendieck group of . Then by the exactness, the above two functors descends to functors on the Grothendieck groups (only depends on the Levi ) and . We define and Then it is a fact that Moreover, Arthur constructed a basis of , whose elements are called *elliptic representations*.

Remark 34

- contains , the isomorphism classes of square-integral representations of , but it is usually bigger (unless ). For example, if , is nontrivial, and , then .
- is invariant under unramified unitary twists. So the group of unramified unitary characters acts on .

04/17/2019

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

Definition 66
Let Then we have an embedding We equip with a measure such that Here is a set of representatives of conjugacy classes of Levi subgroups, and comes from a Haar measure on (of which is a quotient by a finite subgroup).

Remark 35
A priori the distribution depends on the the choice of and but as is strongly cuspidal, it actually does not depend on these choices.

Let .

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

By Weyl's integration formula, the geometric side is equal to By the definition of Harish-Chandra's character , the spectral side is equal to Therefore we obtain a equivalent form of Theorem 33, justing the notation as a *Fourier transform*. This implies that the weighted orbital integral behaves *like a character*. In particular, it has the same kind of local germ expansion as characters. More precisely, for , we have in a neighborhood of 0,

Define using the same formula as Definition 54 (replacing by ). Then by comparing the local expansion of both sides of Arthur's local trace formula, we obtain

Theorem 34 (local trace formula for GGP triples)
For , we have Here 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 -distinguished representations, via local periods).

Now we deduce Theorem 26 from Theorem 34. Recall that By Arthur's local trace formula, the geometric side of is equal to So Theorem 34 is equivalent to One can show that for every , there exists such that . Then using standard techniques (action of the Bernstein center/center of universal enveloping algebra + Stone-Weierstrass), we obtain for , As forms a basis of , 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 be an increasing sequence of compact subsets of such that . Let . Then Using Weyl's integration formula on , we have Here is a maximal torus in and can be thought of as a weight. Arthur proves that the limit of these -weighted orbital integrals is equal to the desired weighted orbital integral. However, the error term *explodes* near in the GGP case and does not vanish sufficiently fast to control it (i.e., we cannot apply dominant convergence theorem).

The strategy to overcome this difficulty:

- Use the semisimple descent to localize near . If we are left with a similar expression for the triple , which turns out to be a product of a
*smaller*GGP triple*and*an Arthur triple), and we may use induction to conclude. - Near , 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.