Representations of $p$-adic groups via graded Hecke algebras and perverse sheaves

• Maarten Solleveld

Graded Hecke algebras can arise in several ways. On the one hand they describe categories of representations of reductive $p$-adic groups. On the other hand they admit a geometric construction, in terms of equivariant constructible sheaves on complex algebraic varieties. We will discuss how this can be applied to solve some problems in the representation theory of $p$-adic groups. We will focus on the $p$-adic Kazhdan-Lusztig conjecture, which expresses the multiplicity of an irreducible representation in a standard module as the multiplicity of a local system in a perverse sheaf.

Hecke algebras for $p$-adic groups and applications to the Langlands correspondence

• Yujie Xu

I will talk about several results on Hecke algebras attached to Bernstein blocks of arbitrary reductive $p$-adic groups, and their applications to the local Langlands program. One such application is an explicit understanding of the (classical, arithmetic) Local Langlands correspondence with explicit L-packets. If time permits, I will talk about certain categorical "upgrades".

Categorical diagonalization and Drinfeld centers

• Ben Elias

What does it mean to diagonalize a functor? In linear algebra, given an operator f with a multiplicity-free minimal polynomial, Lagrange interpolation tells you how to construct idempotents projecting to eigenspaces as polynomials in f. We categorify this construction (only for invertible functors) with a healthy dose of homological algebra. A very common tool in representation theory is to simultaneously diagonalize the center of an algebra A (or the centralizer of a subalgebra), or to simultaneously diagonalize large commutative subalgebras. The representation theory of the Hecke algebra in type A can be understood by examining the Young-Jucys-Murphy (YJM) operators, which form a large commutative subalgebra containing the center. One proves that these operators are (simultaneously) diagonalizable, and classifies their spectrum via standard tableaux. We categorify this story, though there are a few "twists." What it means for a functor to be in the center (or to centralize a subcategory) is more complicated than its decategorified analogue (e.g. the Drinfeld center). We have recently defined a stronger notion of categorical center (the A_infinity Drinfeld center), and explain how the categorical YJM operators can be equipped with this additional structure. Everything is joint work with Matt Hogancamp. Most of the talk will be based on work from 2017 and 2018 (but you're a new crowd for me), while the Drinfeld center work is from 2024.

SL(2,Z)-action on unipotent characters for finite reductive groups

• Olivier Dudas

The space of class functions on a reductive group over a finite field (such as GL(n,q), Sp(2n,q), etc.) admits two particularly interesting bases: - an algebraic basis, given by the characters of irreducible representations, - a geometric basis, given by the characteristic functions of character sheaves. These two bases are related through a transformation that generalizes the classical Fourier transform on finite abelian groups. In ongoing joint work with Bonnafé, Broué, Malle, Michel, and Rouquier, we aim to provide a geometric interpretation of this transformation in the case of unipotent class functions, using traces of braid group operators acting on Deligne–Lusztig varieties. This transformation is part of a broader SL(2,Z)-action that encapsulates the Fourier transform, the Frobenius eigenvalues on the cohomology of Deligne–Lusztig varieties, and Shintani’s twisting operator (which interchanges the Frobenius with its inverse). One of the key advantages of this approach is that it extends naturally to settings where the underlying root datum is governed by complex reflection groups rather than just Weyl groups. This provides natural candidates for Fourier matrices and unipotent character sheaves in the context of "Spetses".

Positivity properties for the lowest summand of the asymptotic Hecke algebra

• Stefan Dawydiak

Affine Hecke algebras play a prominent role in the representation theory of a $p$-adic group $G$, with the principal Bernstein block being equivalent to modules over the Iwahori-Hecke algebra. Braverman-Kazhdan proposed Lusztig's asymptotic Hecke algebra $J$ as means to an algebraic version of tempered representations for the principal block. In particular, elements of $J$ are certain rapidly-decaying functions on $G(F)$. I will explain positivity properties, as conjectured by Braverman-Kazhdan, for two bases of the best understood part of the ring $J$, in terms of K-theory of the flag variety of $\hat{G}$ and spherical Kazhdan-Lusztig polynomials for $G$, and report on joint work in progress with Bezrukavnikov to give an asymptotic characterization of these functions in terms of the wonderful compactification of $G$. Time permitting, I will explain partial results on positivity properties for two-sided cells corresponding to Levi subgroups of the general linear group with one block size.

Towards a categorical Künneth formula for motives

• Jakob Scholbach

In various geometric situations, one can describe the category of sheaves on $X \times Y$ (a product of varieties over a field $k$) in terms of sheaves on $X$ and sheaves on $Y$. Results of this form are referred to as categorial Künneth formulas. Eying applications in the function field Langlands program, we have established, in joint work with Hemo and Richarz (https://arxiv.org/abs/2012.02853) such a categorical Künneth formula for étale Weil sheaves. More recently (https://arxiv.org/abs/2503.14416), Richarz and I have proposed a conjecture concerning a categorical Künneth formula for motivic sheaves. In this talk I will present this circle of ideas, give evidence for the conjecture and report on recent results using the interplay between motives and non-commutative motives in order to further extend our knowledge regarding this conjecture.

Disconnected reductive groups

• Dmitriy Rumynin

A disconnected reductive group is a linear algebraic group whose connected component of the identity is a reductive group. Even those, only interested in connected reductive groups, encounter disconnected ones as various subgroups, e.g., centralisers, normalisers, intersections. In this talk I will spell out how to classify disconnected reductive groups up to an isomorphism. Later on, I will examine their representations and several challenges that they render. The talk is based on joint work with Dylan Johnston and Diego Martin Duro.

Representations of shifted affine quantum groups and Coulomb branches

• Michela Varagnolo

I will present an equivalence between the category O for shifted quantum loop groups (associated with arbitrary Cartan matrices, including non-symmetric ones) and a module category over a new type of quiver Hecke algebra. This equivalence is based on the computation of the K-theoretic analogue of Coulomb branches with symmetrizers introduced by Nakajima and Weekes. At the decategorified level, this yields a connection between the Grothendieck group of O and a finite-dimensional module over a simple Lie algebra of unfolded symmetric type. In some cases, this module can be computed explicitly; more generally, one can describe its crystal structure via a combinatorial rule. Joint with Eric Vasserot.

Spectral actions and central functors

• Arnaud Etève

Let $G$ be a reductive group over an algebraic closure of $\mathbb{F}_q$. In a joint work in progress with D. Gaitsgory, A. Genestier and V. Lafforgue, we construct a spectral action on the stack of $G$-isocrystals over $\mathbb{F}_q((t))$. The construction that we propose works in greater generality and not just for the stack of isocrystal, as an example, it also recovers the construction of the central functor of Gaitsgory. The goal of this talk will be to discuss the machinery at play, some examples and consequences.

Positive-depth Deligne-Lusztig theory

• Charlotte Chan

Representation theory and the geometry of flag varieties are deeply intertwined. For finite groups of Lie type, Deligne and Lusztig's breakthrough work in 1976 defined Frobenius-twisted versions of flag varieties whose cohomology realizes all representations of these groups. In the last quarter-century, generalizations of Deligne-Lusztig varieties have allowed us to study representations of p-adic groups explicitly. I will describe recent advances in this subject and their relationship to the Langlands program.

Conference Dinner Hôtel Oceania 82 Boulevard François Mitterrand 63000 Clermont-Ferrand

On the finite slope part in the $p$-adic Langlands correspondence

• Benjamin Schraen

In the hypothetical $p$-adic Langlands correspondence beyond the case of $GL_2$, the study of the finite slope locally analytic representations is, so far, the most accessible aspect. We can study locally analytic representations of $GL_n(\mathbb{Q}_p)$ appearing in the completed cohomology of Shimura varieties (or spaces of $p$-adic automorphic forms) and study their finite slope part. The description of this finite slope part can be done via the study of certain coherent sheaves on spaces of trianguline $p$-adic parameters. These coherent sheaves can be conjecturally described via a functor constructed by Bezrukavnikov. In this talk I will describe the context of this conjecture and explain the proof when $n=3$.

Applications of localization theorems of locally analytic representations

• Christine Huyghe

In this talk I will recall the localization result of locally analytic representations of a p-adic Lie group with p-adic coefficients and with central character. The localization theorem involves equivariant coadmissible D-modules over the p-adic rigid analytic flag variety and is due to several authors. As an application, I will explain how Ardakov-Schmidt used this result to recover Orlik-Strauch irreducibility results of certain locally analytic representations.

Symmetries of geometric Eisenstein series

• Andreas Hayash

I will report on work in progress, joint with Justin Campbell. For a reductive group $G$, we prove a factorizable Koszul duality result for the spherical category associated to the affine Grassmannian of $G$. As a consequence, we construct an action of the Langlands dual Lie algebra on the intersection cohomology of the moduli of quasimaps from a curve $X$ into the flag variety $G/B$. If time allows, I will explain how we develop along the way some infinite-dimensional deformation theory using techniques inspired by recent developments in "solid" geometry

On modular representations of affine Lie algebras

• Gurbir Dhillon

While the highest weight representation theory of affine Lie algebras in characteristic zero is by now fairly well understood, far less has been known in characteristic $p > 0$. We will report on some basic results (partly in progress) in this direction, e.g. the Harish-Chandra center of the enveloping algebra at all levels, the linkage principle, an analogue of the Kac-Kazhdan conjecture, and a local de Rham geometric Langlands correspondence. This is based on joint work with Ivan Losev.

The motivic Satake equivalence and central motives

• Thibaud van den Hove

The spherical and Iwahori-Hecke algebras of a reductive group are of great importance in the Langlands program. They are categorified by equivariant sheaves on the affine Grassmannian and affine flag variety respectively. Similarly, the Satake and Bernstein isomorphisms are categorified by the geometric Satake equivalence and Gaitsgory's central functor, for which one needs a choice of cohomology theory. In this talk, I will explain how to construct motivic refinements of these functors, so that the choice of cohomology theory is irrelevant. In particular, this yields "mixed" versions of both constructions. This is joint work with Robert Cass and Jakob Scholbach.

Purity of the moduli stack of G-Higgs bundles over a smooth projective curve

• Lucien Hennecart

The moduli spaces of Higgs bundles over a smooth projective curve, and more generally of G-Higgs bundles for a reductive group G, play a central role in the geometric Langlands program. It has been observed that working with the moduli stacks themselves is equally (if not more) natural: they exhibit richer structures that facilitate their study. For instance, these stacks admit derived enhancements that carry a symplectic structure. From an enumerative viewpoint, taking the stack into account is crucial, as it reflects the presence of strictly semistable objects. In this talk, I will explain how to prove that a certain class of stacks, which includes the stacks of semistable G-Higgs bundles over a smooth projective curve, has pure Borel-Moore homology. To establish this, we employ tools from cohomological Donaldson-Thomas theory. A key ingredient is the construction of a cohomological integrality isomorphism.