Schedule for: 21w5228 - Basic Functions, Orbital Integrals, and Beyond Endoscopy (Online)

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In my talk I describe the ideas behind the proof of analytic continuation of Eisenstein series by myself and E. Lapid (see arXiv:1911.02342). The proof is based on some general properties of automorphic forms that are of independent interest.

We establish the analog for real spherical varieties of the Scattering Theorem of Sakellaridis and Venkatesh for p-adic wavefront spherical varieties. We use properties of the Harish-Chandra homomorphism of Knop for invariant differential operators of the variety, special coverings of the variety and spectral projections. We have to make an analog of the Discrete Series Conjecture of Sakellaridis and Venkatesh.

Most of the spectrum of locally homogeneous arithmetic manifolds is presumably transcendental. We discuss what is expected, what can be proven, and the role of these transcendental objects in the theory of automorphic forms.

Relative trace formulas are central tools in the study of relative functoriality. In many cases of interest, basic stability problems have not been addressed. In this talk, I will discuss a theory of endoscopy in the context of symmetric varieties with the global goal of stabilizing the associated relative trace formula. I outline how, using the dual group of the symmetric variety, one can give a good notion of endoscopic symmetric variety and conjecture a matching of relative orbital integrals in order to stabilize the relative trace formula. In the case of unitary Friedberg-Jacquet periods, I explain my proof stabilizing the elliptic terms of the relative trace formula.

I am going to introduce several new types of harmonic analysis on reductive groups arising from the proposal of Braverman and Kazhdan. This is based on my joint work with D. Jiang and L. Zhang, D. Jiang, and B. C. Ngô.

We will discuss a variant of the multiplicity one theorem for automorphic forms on GL(n), and consider the question of whether the set of Hecke eigenvalues of a cusp form on GL(n) is contained in the set of Hecke eigenvalues of a cusp form on GL(m) for n≤m, and try to understand the question in some cases. We will also discuss an analogous question about group representations which seems not to have been considered before, and seems to be of independent interest. Joint work with R. Raghunathan.

Kim and Murnaghan developed a theory of asymptotic expansions of characters, which describe their behaviour near the identity in terms of Fourier transforms of semisimple orbital integrals. In 2016, I showed that, like Harish-Chandra's local character expansion, these asymptotic expansions could be centred everywhere, thus effectively providing an inductive formula for characters of tame supercuspidal representations of p-adic groups G in terms of the analogous representations of tame, twisted Levi subgroups G'. However, unrolling the induction presented technical difficulties. In this talk, I will describe how those difficulties were overcome by a refined understanding of the Fourier transforms appearing in the asymptotic expansions. This work provides a pleasant simultaneous justification of the local character expansion, Kim–Murnaghan asymptotic expansions, the Shalika germ expansion, and an asymptotic result of Waldspurger on Fourier transforms of semisimple orbital integrals.

The beyond endoscopy proposal hinges on obtaining geometric expressions for residues of L-functions using trace formulae. We explain how this can be accomplished for the Rankin-Selberg L-function of a pair cuspidal automorphic representations of $GL_2$. In contrast to previous methods, I work with the whole reductive monoid as opposed taking traces, thus the output is a sum over a ``boundary term'' for a reductive monoid. This makes explicit the connection between ideas of Braverman-Kazhdan-L. Lafforgue-Ngo-Sakellaridis and the beyond endoscopy proposal. Crucially, this boundary term can be integrated over subgroups of $GL_2^4$. In particular, it can be integrated to produce a geometric formula for a weighted sum over cuspidal automorphic representations of $GL_2$ that are invariant under a nonabelian Galois group. Making this precise is work in progress, but I will discuss the key idea.

This is a semi-expository talk. After a quick review of Godement-Jacquet's generalization of Tate's thesis to GL(n) and the starting point of Braverman-Kazhdan/Ngo program, I will discuss Renner's construction of reductive monoids attached to representations of the L-group and conclude with the construction for the cases of symmetric powers of GL(2). Next, I discuss corresponding Schwartz spaces and Fourier transforms, selecting a natural subspace of the conjectured Schwartz space whose functions are uniformly smooth which I will prove to contain the basic function. This space seems to be adequate in proving some of the basic results in the program. These results are joint work with my student William Sokurski.

In this talk we propose an adaptation of Shahidi's enhanced genericity conjecture to ABV-packets: for every Langlands parameter for a p-adic group, the associated ABV-packet contains a generic representation if and only if the orbit of the parameter in the moduli space is open. We relate this genericity conjecture for ABV-packets to other standard conjectures and verify its validity in some special cases. Joint work with Andrew Fiori, Ahmed Moussaoui and Qing Zhang.

The Gan-Gross-Prasad (GGP) conjecture relates the non-vanishing of some periods of cuspidal automorphic forms to that of the central value of some related L-functions. In the talk, we will focus on the case of the (regularized) period of some Eisenstein series in the case of the diagonal subgroup U(n) of U(n)xU(n+1). We will discuss an extension of the usual GGP conjecture in this situation and an application to the Bessel periods of unitary groups. (Based on an ongoing work with Raphaël Beuzart-Plessis).

In the program to generalize Tate-Godement-Jacquet approach of establishing directly the functional equation of general automorphic L-function $L(s,\pi,\rho)$, a main ingredient would be a formula for the $\rho$-Fourier transform where rho is a finite-dimensional representation of the Langlands dual group of $G$. Such a formula is well understood in the case of tori. By reduction to maximal tori we get a stably invariant function depending on $\rho$ from which we hope to produce the correct kernel by means of a transform which is independent of $\rho$. Such a transform has been proposed by L. Lafforgue in the case $GL(2)$. We propose a transform for $GL(n)$ using some intricate invariant theory. This is a joint work with Z. Luo.

Orbital L-functions are geometric analogues of automorphic L-functions. For GL(n), they should be attached to the regular elliptic terms on the geometric side of the trace formula, as opposed to the cuspical automorphic terms on the spectral side. They were introduced for GL(2) by Zagier in 1976, and played an important role in the Poisson summation formula of Ali Altug for GL(2) that allowed him to isolate the nontempered one-dimensional representations. They are also closely related to the zeta functions defined for GL(n) by Z. Yun. We shall introduce orbital L-functions for GL(3), in a form suitable for application. It turns out that they have surprisingly simple formulas, which specialize to even simpler formulas for the elliptic orbital integrals. If time permits, we shall add some remarks on their possible analogues for higher rank, and their future role in Beyond Endoscopy.

A fundamental problem in the representation theory of p-adic groups is the construction of the buildings blocks of all (irreducible, smooth, complex or mod-$\ell$) representations of p-adic groups: the supercuspidal representations. I will provide an overview of our current understanding of the construction of these supercuspidal representations focusing on recent developments including joint work with Kaletha and Spice on a twist of Yu's construction of supercuspidal representations by a quadratic character. This twist is crucial for applications towards character formulae and an explicit local Langlands correspondence.

Abstract: Arthur (1989) conjectured that the discrete spectrum of automorphic representations of a connected reductive group over a number field can be decomposed into A-packets, in terms of which he also conjectured a multiplicity formula. In this talk I will give an introduction to these conjectures and report on the progress for symplectic and orthogonal similitude groups based on the works of Arthur and Moeglin for classical groups.

In this talk, I will explain how to construct convolution operators that isolate certain cuspidal representations from the rest of the automorphic spectrum. For this, we combine the action of spherical Hecke algebras at unramified places with that of an algebra of "multipliers" at Archimedean places. In particular, it is crucial that the multiplier algebra we use be sufficiently large. Time permitting, I might also explain an application of this construction to the global Gan-Gross-Prasad conjecture for unitary groups. This is based on joint work with Yifeng Liu, Wei Zhang and Xinwen Zhu.

Let G be a real reductive algebraic group, and let H be an algebraic subgroup of G. It is known that the action of G on the space of functions on G/H is "tame" if this space is spherical. In particular, the multiplicities of the space of Schwartz functions on G/H are finite in this case. I will talk about a recent joint work with A. Aizenbud in which we formulate and analyze a generalization of sphericity that implies finite multiplicities in the Schwartz space of G/H for small enough irreducible smooth representations of G. In more detail, for every G-space X, and every closed G-invariant subset S of the nilpotent cone of the Lie algebra of G, we define when X is S-spherical, by means of a geometric condition involving dimensions of fibers of the moment map. We then show that if X is S-spherical, then every representation with annihilator variety lying in S has (at most) finite multiplicities in the Schwartz space of X. We give applications of our results to branching problems. Our main tool in bounding the multiplicity is the theory of holonomic D-modules. After formulating our main results, I will briefly recall the necessary aspects of this theory and sketch our proofs.

While studying p-adic L-function and p-adic height of arithmetic diagonal cycles, it is natural to study the p-adic limit of certain relative trace formulas (for a suitable family of test functions). This motivates us to study the p-adic limit of (relative) orbital integrals. I'll describe some results and unsolved problems. This is a joint work with Daniel Disegni.

I am reporting on joint work with Yung-sheng Tai. Each locally symmetric space $X$ for the group $SL(2, \mathbb{C})$ is a hyperbolic 3-dimensional manifold that parametrizes principally polarized complex abelian surfaces with appropriate level structure and anti-holomorphic multiplication, meaning: an action by the integers in a quadratic imaginary number field such that imaginary elements act anti-holomorphically. What happens when these abelian varieties are reduced modulo p? I do not know the answer in general, but for ordinary (principally polarized) abelian varieties it is possible to make sense of anti-holomorphic multiplication. One might say that isomorphism classes of such objects represent ``ordinary points'' of ``$X$ mod $p$'' despite the fact that ``$X$ mod $p$'' does not exist as a scheme or stack, and it suggests that perhaps in some larger world it may be possible to make sense of this object.

In this talk, by using the trace formula method, I will prove a multiplicity formula of K-types for all representations of real reductive groups in terms of the Harish-Chandra character.

Let $G$ be a connected reductive group which is split over a number field $F$. On a subspace generated by wave packets of appropriately normalized Eisenstein series, the spectral decomposition of the space of spherical automorphic forms of $G$ supported by the trivial character of a maximal torus can be made completely explicit, using the theory of residue distributions. The remaining challenge is to prove that this subspace is in fact everything. To address this problem we follow a method which is inspired by Moeglin's contour shift considerations in the classical case. We present a progress report of joint work with Marcelo De Martino and Volker Heiermann.

Mok has defined Arthur packets for quasisplit unitary groups. His definition follows Arthur's work on classical groups, and relies on harmonic analysis. For real groups an alternative definition of Arthur packets has been known since the early 90s. This approach, due to Adams-Barbasch-Vogan, relies on sheaf-theoretic techniques instead of harmonic analysis. We will report on work in progress, joint with N. Arancibia, in proving that these two definitions are equivalent for real quasisplit unitary groups.

Arthur's truncation operator plays a crucial role in the theory of automorphic forms, particularly in the derivation of the Trace Formula, but also in the construction of Eisenstein series and the derivation of the Plancherel formula. However, I don't think it is well understood, and there are many puzzling features to it that become even more puzzling upon closer inspection. In this talk I shall point these out, and perhaps resolve a few.

The Shintani–Casselman–Shalika formula for eigenvectors of the spherical Hecke algebra on the space of Whittaker functions, and its generalizations to other spaces made possible by the method of Casselman and Shalika, hold the key to many fundamental connections between harmonic analysis, L-functions, and geometry. In this talk, I will attempt to explain: (1) How the functional equations of the Casselman–Shalika method calculate the scattering operators of harmonic analysis in terms of gamma factors. (2) The motivic meaning of those functional equations (based on joint work with Jonathan Wang).

I will discuss the work of two of my students, Rui Chen and Jialiang Zou, who show how one can use theta correspondence efficiently to propagate the results of Arthur and Mok on the automorphic discrete spectrum of quasi-split classical groups to their pure inner forms and highlight some remaining problems in this direction.