## Number Theory and Branching Laws

### Titles and Abstracts

1. Jeffrey Adams, University of Maryland.

2. Title: Unipotent Representations of Real Groups

Abstract: Jim Arthur has conjectured the existence of certain representations of a reductive group over a local field, which he calls unipotent. In general there is no single well defined definition of this term. Even in the case of real groups there is more than one definition of the term unipotent. Furthermore, even in cases when there is a precise definition there is typically no known algorithm to explicitly compute these representations, say in terms of their Langlands parameters. I will discuss some progress on computing unipotent representations in the real case.

This is part of the Atlas of Lie Groups and Representations project.

3. Raphaël Beuzart-Plessis, Aix-Marseille Université.

4. Title: Isolation of the cuspidal spectrum and application to the Gan-Gross-Prasad conjecture for unitary groups

Abstract: In this talk, I will explain a new technique for isolating components on the spectral side of trace formulas which takes its roots in a beautiful paper of Lindenstrauss and Venkatesh from 2005. Applying it to the Jacquet-Rallis trace formula, this leads to a proof of the Gan-Gross-Prasad conjecture and its refinement by Ichino-Ikeda for U(n)xU(n+1) in the stable case. This is joint work with Yifeng Liu, Wei Zhang and Xinwen Zhu.

5. Kei Yuen Chan, Fudan University.

6. Title: Bernstein-Zelevinsky theory in branching laws of (GL(n+1),GL(n))

Abstract: In the Duke paper of D. Prasad in 1993, he applied the Bernstein-Zelevinsky filtration, among other things, to study branching laws for general linear groups. This talk will be on some extensions of this method. In particular, I shall explain the notion of left and right Bernstein-Zeleinsky filtrations, which can be used to study the projectivity and indecomposability of representations under restriction. If time permits, I also discuss the Hecke algebra structure appearing in Bernstein-Zelevinsky filtrations and branching laws. Part of the work is joint with Gordan Savin.

7. Sarah Dijols, Yau Center, Tsinghua University.

8. Title: Representations of G2 distinguished by SO4

Abstract: Much has been studied in the context of p-adic symmetric spaces pertaining to the classification of representations of linear and classical groups, in the recent years (for instance Prasad on Galois distinction), but none has been known for exceptional groups. In a joint work with A. Mitra, in progress, we are considering the p-adic symmetric space G2/SO4 and applying the orbit method and Mackey theory to obtain distinction results for representations of G2. I will recall the method, and present some preliminary results on this work. I will also recall some (recent) contributions of D. Prasad to distinction problems.

9. Radhika Ganapathy, Indian Institute of Science.

10. Title: Some Hecke algebra isomorphisms over close local fields

Abstract: Two non-archimedean local fields F and F' are m-close if the rings OF/pFm and OF'/pF'm are isomorphic. For a split connected, reductive group G over Z, Kazhdan proved that the Hecke algebras H(G(F), Km) and H(G(F'), K'm) are isomorphic if the fields F and F' are sufficiently close, where Km := Ker(G(OF) → G(OF/pFm)) and K'm is the corresponding object over F'. In this talk we will discuss the generalization of this isomorphism (and a variant of this isomorphism with respect to the Iwahori filtration) to non-split groups.

11. Michael Harris, Columbia University.

12. Title: A local Langlands parametrization for G2

Abstract: This is a report on joint work with C. Khare and J. Thorne. We construct a bijection between generic supercuspidal representations of the semisimple group G2 over a p-adic field and local Langlands parameters for G2. The construction of the map is simply a matter of assembling arguments that are already in the literature, together with a previously unpublished theorem of G. Savin on exceptional theta correspondences, included as an appendix. The proof that the map is a bijection is arithmetic in nature, and specifically uses automorphy lifting theorems. These can be applied thanks to a recent result of Hundley and Liu on automorphic descent.

13. Volker Heiermann, Aix-Marseille Université.

14. Title: Representations of p-adic groups and affine Hecke algebras

Abstract: The aim of my talk is to give examples of how results on representations of affine Hecke algebras can help to prove results for the category of smooth representations of a p-adic group.

15. Colette Moeglin, Institut de Mathématiques de Jussieu - Paris Rive Gauche.

16. Title: Examples of the conjecture of Sakellaridis and Venkatesh about discrete spectrum

Abstract: The context is the real field and homogeneous spaces for reductive groups. The main case solved with D. Renard is the case of symmetric spaces and I will explain a case where the conjecture of S-V is subtle, that is the case of U(p,q)/U(r,s)xU(p-r,q-s) with p+q odd. At the beginning I will explain the much simpler case of U(p,q)/U(r,s) and SO(p,q)/SO(r,s) where a variant of the conjecture of S-V also holds and which is, in fact, already in the literature.

17. Arvind Nair, Tata Institute of Fundamental Research, Mumbai.

18. TBA

19. Ameya Pitale, University of Oklahoma.

20. Title: Critical L-values and congruences of Siegel modular forms

Abstract: In this talk we will consider the degree 2n+1 standard L-function of irreducible, cuspidal automorphic representations of the symplectic group of genus n, with the holomorphic discrete series as the archimedean component. We obtain an integral representation for the L-function using a pullback formula from a Siegel Eisenstein series of genus 2n, in the spirit of Garrett, Rallis and Piatetski-Shapiro. Using the algebraic and integral properties of the Fourier coefficients of the classical version of the Eisenstein series, we obtain algebraicity results for the critical values of the L-function, as well as, congruences between Siegel modular forms. In the genus two case, we get applications to algebraicity results for symmetric fourth power L-function for GL(2). This is joint work with Abhishek Saha and Ralf Schmidt.

21. A. Raghuram, Indian Institute of Science Education and Research, Pune.

22. TBA

23. C. S. Rajan, Tata Institute of Fundamental Research, Mumbai.

24. TBA

25. Dinakar Ramakrishnan, California Institute of Technology.

26. TBA

27. Yiannis Sakellaridis, Johns Hopkins University.

28. Title: A conjectural geometric setup for the relative Langlands program

Abstract: A recurrent theme of Dipendra Prasad's pioneering work has to do with the question of distinction of representations of a group G by a subgroup H. In the context of the geometric Langlands program, this question of distinction appeared in the work of Gaitsgory and Nadler, who attached a dual subgroup GX ⊊︀ G to every spherical homogeneous space X = G/H. In ongoing work with David Ben-Zvi and Akshay Venkatesh, we propose an conjectural extension of the theorem of Gaitsgory and Nadler, that leads to a representation of GX, so that the total space of the induced vector bundle M --> G/GX is Hamiltonian, and the association T*X <--> M is involutive. Although in the context of geometric Langlands, reasonable expectations about Frobenius traces can lead to conjectures about distinction over local fields, and an explanation of the relationship between periods and L-functions.

29. Gordan Savin, University of Utah.

30. TBA

31. Sandeep Varma, Tata Institute of Fundamental Research, Mumbai.

32. Title: Some homogeneity results in relative harmonic analysis

Suppose $H$ is the group of fixed points of an involution on a reductive $p$-adic group $G$. If $\pi$ is an irreducible smooth representation of $G$ that is distinguished by $H$, then one has an associated spherical character $\Theta$, a distribution on the symmetric space $H \backslash G$. From Rader-Rallis, we know that in a small neighborhood of the identity, which one identifies with a neighborhood of 0 in the `Lie algebra' $\mathfrak{h} \backslash \mathfrak{g}$ of the symmetric space $H \backslash G$, $\Theta$ can be described as the Fourier transform of a distribution supported in the nilpotent cone, and thus has an expansion analogous to the Harish-Chandra-Howe local character expansion for characters. In joint work in progress with Adler and Sayag, assuming that the residue characteristic $p$ of $F$ is "large enough", we seek to find an explicit neighborhood where such an expansion is valid, as DeBacker, following Waldspurger, achieved in the case of characters of groups. The symmetric-space situation presents new difficulties. For example, as Rader-Rallis point out, a nilpotent orbit need not have an invariant measure, and even if it does it may not extend to a distribution on $\mathfrak{h} \backslash \mathfrak{g}$. So far, the main example of the cases we can handle, apart from the easy case where G is the restriction of scalars of a quadratic base-change of H, is the case of a symmetric space of rank 1.

33. Jean-Loup Waldspurger, Institut de Mathématiques de Jussieu - Paris Rive Gauche.

34. Title: Character-sheaves, nilpotent orbital integrals and endoscopy

Abstract: Let G be a connected reductive group defined over a p-adic field F. Let g denote its Lie algebra. The nilpotent orbital integrals are invariant distributions on g(F), which are important for harmonic analysis on this space. They generate a finite dimensional space of distributions denoted by I(g(F))nil*. An unsolved problem is to compute the subspace of stable distributions in I(g(F))nil*. If p is big, we can introduce another space of distributions D which is strongly related to I(g(F))nil*. For each vertex s of the Bruhat-Tits building of G, we can define a connected reductive group Gs over the residual field Fq. Let gs denote its Lie algebra. Lusztig has defined the generalized Green’s functions on gs(Fq). They are restrictions to nilpotent elements of characteristic functions of some character-sheaves. The space D is defined using these Green’s functions, for all vertices s. I will give some properties of this space D concerning endoscopy and I will explain some consequences for the initial problem.