Titles and abstracts

Non-singular representations of p-adic reductive groups and L-packets

Abstract. We will recall, and illustrate in several examples, the notion of non-singular supercuspidal representation of a tamely ramified p-adic reductive group G, which was recently introduced and studied by Kaletha.

We conjecture that, when p is very good for G, every L-packet contains a representation the supercuspidal support of which is non-singular. We will show that the conjecture holds true under some expected properties of the local Langlands correspondence.

Hidden sign in the Langlands' correspondence

Abstract. Let G be a reductive group over a p-adic field F.

We fix some algebraically closed field C of characteristic 0 and consider smooth representations of the group G in vector spaces over C. The Local Langlands' Correspondence (LLC) attempts to describe the set Irr(G;C) of equivalence classes of simple (irreducible) objects in this category in terms of the dual group.

Namely, denote by Ǧ the algebraic group over C that is the dual group of G (I will talk only about the split groups). Consider the set Lan(G) of Langlands parameters, defined by Lan(G)=Mor(W_F, Ǧ)/Ǧ (quotient with respect to the adjoint action).

Langlands suggested that there should be some correspondence Φ: Lan(G) → Irr(G;C).

After modifying both sides this correspondence is supposed to give a bijection.

Notice that both sets in this correspondence are functorial in C. However, the standard Langlands' correspondence is not compatible with automorphisms of C. 

I will explain why this is the case.

I will also describe another set Lan '(G) of "modified" Langlands' parameters and formulate the LLC as a correspondence Ψ: Lan '(G) → Irr(G;C) that is functorial in C.

This modified correspondence does not differ too much from the standard LLC. But I think that it is important to formulate and prove statement in a correct categorical framework, so I think that "correct" version of the Langlands' correspondence should be formulated using modified Langlands' parameters (for p-adic case, for real case and for automorphic case).

Some analytic aspects of automorphic L-functions

Abstract. In two recent papers, one of them with Peter Sarnak, I have studied lower bounds on quadratic integrals of automorphic L-functions on the line Re(s)= 1/2. The proofs rely on the Mellin transform. The proof implies in particular  an estimate on the abscissa of convergence of L(s) when L(s) is a standard, or Rankin L-function, under the Ramanujan conjecture; lower bounds on the quadratic integrals on 'short' intervals in t (s=1/2+it), of length of the order of log C, C being the analytic conductor; and estimates on the first non-trivial Frobenius in an Artin representation. I will  also advertise an interesting problem relating this to subconvexity.

Interwining operators between principal series of GL(n+1,R) and GL(n,R) and branching laws

Abstract. The multiplicities of the restriction of an irreducible representation of a real reductive group G to a reductive subgroup H are always finite if the corresponding homogeneous space (GxH)/diag(H) is real spherical. Under this assumption, we construct explicit families of intertwining operators between principal series representations of G and H. These families depend meromorphically on the induction parameters, and for generic parameters they span the space of all intertwining operators.

For the special case (G,H)=(GL(n+1,R),GL(n,R)) we find an explicit holomorphic renormalization of the meromorphic families of intertwining operators in terms of local L-factors. We will further explain how to use the holomorphic families to obtain the explicit direct integral decompositions of all irreducible unitary representations of GL(3,R) when restricted to GL(2,R).

The most continuous part of the Plancherel decomposition for a real spherical space -- Part II (joint work with Eitan Sayag)

Abstract. Let Z be a unimodular homogeneous space of a real reductive group G. In this talk I will discuss the derivation of the Plancherel decomposition of the most continuous part of L^2(Z). The starting point is the construction and the properties of the H-fixed functionals discussed in the first part by Eitan Sayag. We endow the spaces of H-fixed funcitonals with inner products. For this we use the constant term maps given by Delorme, Krötz, Souaifi and Beuzart-Plessis. The resulting Hilbert spaces occur in the Plancherel decomposition as the multiplicity spaces. This is a consequence of a refinement of the Maass-Selberg relations which were obtained by Delorme, Knop, Krötz and Schlichtkrull.

On the unipotent contribution to the trace formula for functions fields

Abstract. We are interested in the fine expansion of the unipotent contribution to Arthur trace formula for G(F) where G is a connected reductive group defined over a global field of characteristic p>0. For this we develop a Kempf-Rousseau-Hesselink theory of unipotent F-strata in G(F). These F-strata play the role of the unipotent geometric orbits in Arthur's work. Thanks to the previous work of Chaudouard and Laumon, we obtain the Hoffmann variant of the fine expansion of the unipotent contribution to the trace formula. The result should be true for any p even if today the proof is complete only for p>5.

A new application of the residue method

Abstract. The residue method was introduced by Jacquet and Rallis in order to compute period integrals on residual automorphic representations. It has since been applied by many authors, mainly for residual representations with cuspidal data supported on a maximal Levi subgroup. This talk is about an application of the residue method to certain residual representations of Sp(4n+2), with cuspidal data supported on a co-rank two Levi subgroup. Some further complications occur in this case. We will show that the representations we consider admit a non-zero Sp(2(n+1)) x Sp(2n) -invariant linear form. This is a joint work with Solomon Friedberg and David Ginzburg.

Transfer opertors and Hankel transforms via quantization

Abstract. Langlands "Beyond Endoscopy" program, generalized to the setting of the relative trace formula, one should be able to compare the spectra of differential homogeneous spaces by means of "transfer operators" between test measures and characters of those spaces. It is remarkable facts that, in a small number of law-rank cases where such operators are known, they are given by abelian Fourier transforms on the space of orbits, despite the fact that the characters in question are non-abelian.

I will propose a philosophy, whereby the relative trace formula associated to a homogeneous space is a geometric quantization of a certain symplectic "space". More than just a philosophy, in the aforementioned cases it provides a way to describe the transfer operators as a comparison of two different geometric quantizations of (almost) the same symplectic space. Moreover, the same explanation applies to Jacquet's "Hankel transform", that is, the descent of the Kuznetsov formula of the Godement-Jacquet Fourier transform on the space of n x n matrices.

Congruence properties of the local functorial transfer for cuspidal representations of quasi-split classical groups

Abstract. Let G be the group of rational points of a quasi-split p-adic classical group for some odd prime number p. Following Arthur and Mok, there are a positive integer N, a p-adic field E and a local functorial transfer from the isomorphism classes of irreducible smooth complex representations of G to those of GL(N,E). Fixing a prime number l different from p and an isomorphism between the field of complex numbers and an algebraic closure of the field of l-adic numbers, we get a transfer between representations with l-adic coefficients. Now consider a cuspidal irreducible l-adic representation V of G. One can define its reduction mod l, which is a semi-simple smooth representation of G of finite length, with coefficients in a finite field of characterictic l. Let V’ be another cuspidal irreducible l-adic representation of G whose reduction mod l is isomorphic to that of V. We prove that the local functorial transfers W, W’ of V, V’ have reductions mod l which may not be isomorphic, but which share a unique common generic irreducible component. This is a joint work with Alberto Minguez.