Geometry and Representation Theory around the Langlands programs

Liste des Contributions

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

Maarten Solleveld

07/07/2025 09:00

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...

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

Yujie Xu

07/07/2025 10:30

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".

4. Categorical diagonalization and Drinfeld centers

Ben Elias

07/07/2025 14:00

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...

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

Olivier Dudas

07/07/2025 15:30

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...

6. Positivity properties for the lowest summand of the asymptotic Hecke algebra

Stefan Dawydiak

08/07/2025 09:00

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...

7. Towards a categorical Künneth formula for motives

Jakob Scholbach

08/07/2025 10:30

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...

8. Disconnected reductive groups

Dmitriy Rumynin

08/07/2025 14:00

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...

9. Representations of shifted affine quantum groups and Coulomb branches

Michela Varagnolo

08/07/2025 15:30

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...

10. Spectral actions and central functors

Arnaud Etève

09/07/2025 09:00

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...

11. Positive-depth Deligne-Lusztig theory

Charlotte Chan

09/07/2025 10:30

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...

17. Conference Dinner

09/07/2025 19:30

Hôtel Oceania
82 Boulevard François Mitterrand
63000 Clermont-Ferrand

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

Benjamin Schraen

10/07/2025 09:00

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...

13. Applications of localization theorems of locally analytic representations

Christine Huyghe

10/07/2025 10:30

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...

1. Symmetries of geometric Eisenstein series

Andreas Hayash

10/07/2025 14:00

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...

14. On modular representations of affine Lie algebras

Gurbir Dhillon

10/07/2025 15:30

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...

15. The motivic Satake equivalence and central motives

Thibaud van den Hove

11/07/2025 09:00

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...

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

Lucien Hennecart

11/07/2025 10:30

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...