Titles and abstracts

Titles and abstracts

(In alphabetical order)

Anne-Marie AUBERT (Institut Mathématique de Jussieu – Paris)

Title : Non-singular representations of reductive $p$-adic groups

Abstract:We call non-singular the smooth irreducible representations of a reductive $p$-adic group $G$, whose supercuspidal supports are non-singular in the sense of Kaletha.  We will explain how non-singular supercuspidal representations are constructed, and provide examples.

In previous works in collaboration with Moussaoui, Solleveld, and Xu, we have used the generalized Springer correspondence for disconnected complex reductive groups to construct a Langlands correspondence for $G$, under certain hypotheses.

In this talk, we will focus on the part of our Langlands correspondence, which is correlated with the ordinary (i.e. not generalized) Springer correspondence: we will characterize the cuspidal supports of the occurring enhanced $L$-parameters (we will called them ``ordinary") and show that the corresponding representations of $G$  are non-singular.

When $G$ is an inner form of a general linear group, we will characterize the non-singular representations of $G$  which correspond to ordinary enhanced $L$-parameters.

Raphaël BEUZART-PLESSIS (Institut Mathématique de Luminy - Marseille)

Corinne BLONDEL (Institut Mathématique de Jussieu – Paris)

Title : On the road to functoriality

Abstract : Guy Henniart has 109 publications since 1977, according to Zentralblatt. This is far too much to even show the list, and the problem when diving into this list of works is that you would like to read it all… impossible.So we willjustfollow the guy on his journey and make a few stops with him,mostly describing the landscapeandthe good types encountered on the way.

Peiyi CUI (University of East Anglia - Norwich, Angleterre)

Title: Blocks of the category of l-modular representations of SL_n(F)

Abstract: Let F be a p-adic field and k an algebraically closed field of characteristic l different from p. In this talk, we introduce the blocks decomposition of Rep_k(SL_n(F)), which is the category of l-modular representations of SL_n(F).

Jessica FINTZEN (Universität Bonn - Bonn, Allemagne)

Title: Representations of p-adic groups and Hecke algebras

Abstract: We show that one can reduce a lot of problems about the (category of) (smooth, complex) representations of p-adic groups to problems about representations of finite groups of Lie type, where answers might already be known or are easier to achieve. More precisely, the category of representations of p-adic groups decomposes into subcategories, called Bernstein blocks. Some of these blocks, called depth-zero blocks, correspond essentially to blocks in the category of representations of finite groups of Lie type and are much better understood than arbitrary Bernstein blocks. I will discuss a joint project in progress with Jeffrey Adler, Manish Mishra and Kazuma Ohara in which we show that general Bernstein blocks are equivalent to the much better understood depth-zero Bernstein blocks. This is achieved via an isomorphism of Hecke algebras.
To put everything into context, I will also recall what is known about the explicit construction of (supercuspidal) representations of p-adic groups (and types).

Jan KOHLHAASE (Universität Duisburg-Essen - Essen, Allemagne)

Title: Homotopy theory of smooth representations in characteristic p

Abstract: I will report on ongoing work with Nicolas Dupré concerning homotopical methods in the theory of smooth mod-p representations of a p-adic reductive group G. I will recall how to match up the category of Gorenstein projective modules over the corresponding pro-p Iwahori-Hecke algebra H with a full subcategory of smooth G-representations. I will also explain how to interpret this result from the point of view of model categories. In the remaining time I will introduce singularity categories of pro-p Iwahori-Hecke modules and present first ideas how to analyze them.

Robert KURINCZUK (University of Sheffield - Sheffield, Angleterre)

Title: Blocks for p-adic groups via type theory

Abstract: I will explain a method for approaching the R-block decomposition of the category of smooth representations of a p-adic reductive group, where R is a coefficient ring where p is invertible, using the theory of semisimple characters and types. In the special case of inner forms of classical p-adic groups, when the residue characteristic is not 2, I will explain this decomposition explicitly when R is the ring of algebraic integers with p inverted.  This is joint work with Helm, Skodlerack, and Stevens.

Thomas LANARD (Laboratoire de Mathématiques de Versailles - Versailles)

Title: Modulo l distinction problems

Abstract: Let G be a p-adic group. A representation of G is said to be distinguished with respect to a subgroup H if it admits a non-trivial H-invariant linear form. Distinguished representations are central objects in the study of the relative Langlands program. On $\mathbb{C}$ they have been studied intensively, but much remains to be done for modular representations, i.e. with coefficients in $\overline{\mathbb{F}}_l$, where l is a prime other than p. In this talk, I will discuss distinguished modular representations for the pair (G,H)=(GL_n(E),GL_n(F)), where E is a quadratic extension of F. I will also discuss the validity of Prasad's conjecture (a conjecture describing distinction from the Langlands correspondence) for modular representations. This is joint work with Peiyi Cui and Hengfei Lu.

Ahmed MOUSSAOUI (Laboratoire de Mathématiques et Applications - Poitiers)

 
Title: Affine Hecke algebras for Langlands parameters

Abstract: We will explain how to define a Bernstein decomposition for enhanced Langlands parameters and an explicit construction of an affine Hecke algebra whose simple modules are parametrized by elements of each blocks. This approach could be useful to reduce the local Langlands correspondence to the cuspidal case. This is a joint work with A.-M. Aubert and M. Solleveld.

Masao OI (Kyoto Daigaku/University of Kyoto - Kyoto, Japon)

Title: On explicit local Jacquet--Langlands correspondence for regular supercuspidal representations

Abstract: Let F be a non-archimedean local field.
The local Jacquet--Langlands correspondence (LJLC) relates the irreducible discrete series representations of GL_n(F) to those of inner forms of GL_n(F).
In the 1990s, Henniart obtained an explicit description of the LJLC for certain supercuspidal representations by analyzing the character relation, which is the characterization of the LJLC.
The point of his proof is to show that those supercuspidal representations are determined only by the values of the Harish-Chandra characters on very regular elements, a special class of regular semisimple elements on which the character formulas become quite simple.
In this talk, I will discuss a generalization of Henniart's result to general groups.
This is joint work with Charlotte Chan (University of Michigan).

Vytautas PASKUNAS (Universität Duisburg-Essen - Essen, Allemagne)

Title : Local Galois deformation rings : reductive groups

Abstract: I will report on an ongoing joint work with Julian Quast on deformation rings of Galois representations of p-adic fields valued in reductive groups.

Beth ROMANO (King’s College London - Londres, Angleterre)

Title : Epipelagic representations in the local Langlands correspondence

Abstract : Epipelagic representations, which have minimal positive depth, provide a first setting in which to build our understanding of the positive-depth part of the local Langlands correspondence for a p-adic group G. I’ll talk about a construction of epipelagic representations by Reeder—Yu, and in particular the special case of simple supercuspidal representations (which were first constructed by Gross—Reeder, and were studied for classical groups by Henniart and his collaborators). I’ll describe recent work in which I give a complete classification of the simple supercuspidal representations of G, and describe properties of the corresponding Langlands parameters, in the case when G is a simple, split, adjoint group.

Marteen SOLLEVELD (Radboud University - Nijmigen, Pays-Bas)

Title: Generic representations of p-adic groups and of Hecke algebras

Abstract: Generic representations and Whittaker functionals have proven to be very useful
in the representation theory of a quasi-split reductive p-adic group G. The
generalized injectivity conjecture, posed by Casselman and Shahidi, asserts
that any generic standard G-representation has a generic irreducible subrepresentation.
More generally, one may wonder under which conditions an irreducible constituent
of a standard G-representation is actually a subrepresentation.

We will discuss these issues in the context of affine Hecke algebras and of
graded Hecke algebras. We will introduce a good analogue of genericity for
representations of Hecke algebras. With that, we will provide a geometric proof
of the Hecke algebra version of the generalized injectivity conjecture.
In many cases, the results can be transferred to reductive p-adic groups.

Marie-France VIGNERAS (Institut Mathématique de Jussieu - Paris)