# Conférence - Théorie des représentations et D-modules

## Schedule :

### Monday

09h00-09h30 : Registration

09h30-10h30 : W. Soergel, Koszul Duality for Real groups

10h30-11h00 : Coffee break

11h00-12h00 : D. Rumynin, D-Affine Varieties

12h00-13h30 : Lunch break

13h30-14h30 : P. Cui, Uniqueness of supercuspidal support of irreducible mod l representations of SL_n(F)

14h30-15h15 : Coffee break + Poster session I

15h15-16h15 : R. Crew, Nilpotent arithmetic D-modules

### Tuesday

09h30-10h30 : S. Riche, Support varieties of tilting modules and p-cells in affine Weyl groups

10h30-11h00 : Coffee break

11h00-12h00 : S. Gunningham, Very central D-modules via Koszul duality

12h00-13h30 : Lunch break

13h30-14h30 : J. Campbell, The fundamental local equivalence at integral level

14h30-15h15 : Coffee break + Poster session II

15h15-16h15 : T. Abe, Vanishing cycle functor for arithmetic D-modules

### Wednesday

09h30-10h30 : K. Ardakov, The first Drinfeld covering and equivariant D-modules on rigid spaces

10h30-11h00 : Coffee break

11h00-12h00 : C. Huyghe, Intermediate extensions and crystalline distribution algebras

Excursion to Saint-Malo

### Thursday

09h30-10h30 : C. Pauly, Opers in positive characteristic

10h30-11h00 : Coffee break

11h00-12h00 : C. Dodd, Witt Differential Operators

12h00-13h30 : Lunch break

13h30-14h30 : Y. Wakabayashi, Enumerative geometry of dormant opers

14h30-15h00 : Coffee break

15h00-16h00 : E. Grosse-Klönne, Hecke modules and Galois representations into the dual group

Conference Dinner

### Friday

09h30-10h30 : A. Langer, Nearby-cycles and semipositivity in positive characteristic

10h30-11h00 : Coffee break

11h00-12h00 : P. Schneider, Support theory for the Iwahori-Hecke Ext-algebra

12h00-13h30 : Lunch

End of the conference

## Abstracts :

Wolfgang Soergel (Universität Freiburg); Koszul Duality for Real groups

I will talk on some old conjectures of mine from an article ”Langlands philosophy and Koszul duality” and how they can be written in a much more precise way using recent advances in the theory of motives, written out with Rahbar Virk and Matthias Wendt, and joint work with Joseph Bernstein. However, there is still no proof known beyond the case of complex groups.

Dmitriy Rumynin (University of Warwick); D-Affine Varieties

I am interested in the following problem: ”classify all smooth connected D-affine projective varieties”. As a part of their proof of Kazhdan-Lusztig Conjecture, Beilinson and Bernstein have proved that the partial flag varieties G/P are D-affine. This is the current state of the art: no other examples are known but there is no proof that the G/P-s exhaust all the possible examples. In my talk I will review well-known facts, discuss some new results and formulate several interesting questions.

Peiyi Cui (Université de Rennes 1/IMJ Paris); Uniqueness of supercuspidal support of irreducible modular l representations of SLn(F)

Let F be a non-archimedean locally compact field of residual characteristic p, and k be an algebraically closed field of characteristic l different from p. In the study of the category Rep_C(SLn(F)) of smooth C-representations of SLn(F), the Bernstein decomposition theorem for this category is based on the fact that the supercuspidal support of irreducible C-representations of Levi subgroups M of SLn(F) is unique up to M-conjugation. In this talk, we will consider Rep_k(SLn(F)), and show that the supercuspidal support of irreducible k-representations of Levi subgroups M of SLn(F) is unique up to M-conjugation, when F is either a finite field of characteristic p or a non-archimedean locally compact field of residual characteristic p.

Richard Crew (University of Florida); Nilpotent Arithmetic D-modules

A particular important class of modules over Berthelot’s ring of arithmetic differential operators of level m are the topologically nilpotent modules. The left module structure of a topologically nilpotent module extends naturally to a module structure for various completions of the ring of level m operators. In this talk I will give some basic algebraic results of these completions and discuss possible applications.

Simon Riche (Université Clermont-Auvergne); Support varieties of tilting modules and p-cells in affine Weyl groups

Given a connected reductive group G over an algebraically closed field of characteristic p > 0, the support variety is a closed subvariety of the nilpotent cone attached to every G-module. These subvarieties are particularly interesting in the case of indecomposable tilting modules, and are the subject of a conjecture of Humphreys (from the 1990’s) relating them to cells in affine Weyl groups through work of Lusztig. In this talk we will explain a recent proof of the conjecture (obtained in joint work with P. Achar and W. Hardesty) in the case when p is large, and outline the new features our approach has revealed, in connection with the theory of p-cells.

Sam Gunningham (University of Edinburgh); Very central D-modules via Koszul duality

Very central D-modules are a natural class of conjugation equivariant D-modules on a reductive group G which behave well with respect to convolution - in particular, they form a symmetric monoidal category. I will explain how this category may be understood as a quantization of the universal regular centralizer group scheme, and give a construction in terms of derived geometric Satake and Koszul duality for Hecke categories. This is joint work with David Ben-Zvi.

Justin Campbell (Caltech); The fundamental local equivalence at integral level

The geometric Satake equivalence plays a basic role in the geometric Langlands program. Gaits- gory and Lurie have predicted that this equivalence admits a quantum deformation, which is expected to be similarly fundamental to the quantum geometric Langlands program. In this deformation the spherical Hecke category is replaced by twisted Whittaker sheaves on the affine Grassmannian, and representations of the Langlands dual group are replaced by a certain category of representations of its affine Kac-Moody algebra. In this talk I will explain some recent joint work with G. Dhillon, in which we construct a tamely ramified version of this fundamental local equivalence at integral level by passing through a singular block of affine category O.

Tomoyuki Abe (KAVLI-IMPU Tokyo); Vanishing cycle functor for arithmetic D-modules

We will discuss how to establish the vanishing cycle theory for arithmetic D-modules. As a small application, we will interpret overconvergent F-isocrytals purely in terms of arithmetic D-modules. This allows us to show the compatibility of isocrystals with respect to proper smooth pushforward of D-modules.

Konstantin Ardakov (University of Oxford); The first Drinfeld covering and equivariant D-modules on rigid spaces

Let p be a prime and let F be a p-adic local field. The p-adic upper half plane Ω is obtained from the projective line viewed as a rigid analytic variety by removing the F-rational points. Drinfeld introduced a tower of finite étale Galois coverings of Ω by interpreting Ω as the rigid generic fibre of the moduli space of certain formal one-dimensional commutative groups with quaternionic multiplication, and introducing level structures to define the coverings. This tower is now known to realise both the Jacquet-Langlands and local Langlands correspondences for G = GL_2(F) in l-adic etale cohomology, where l is a prime not equal to p. Coherent cohomology of the tower is expected to produce representations of G which are admissible in the sense of Schneider and Teitelbaum. Using the theory of equivariant D-modules on rigid spaces we can prove that the dual of the global sections of a non-trivial line bundle arising from the first covering of Ω is an irreducible admissible representation of G. Patel, Schmidt and Strauch have also given an argument for the admissibility of these representations using a formal model for the first covering; whilst similar in certain respects, our approach is significantly different to theirs. This is joint work with Simon Wadsley.

Christine Huyghe (Université de Strasbourg); Intermediate extensions and crystalline distribution algebras

This is joint work with Tobias Schmidt and Matthias Strauch. Let G be a complex reductive algebraic group, and g its Lie algebra. There is an equivalence of categories between the category of D-modules over the flag variety of G and g-modules with central character. The D-module corresponding to the simple quotient of a Verma module with trivial central character is obtained as an intermediate extension of the constant sheaf of a Bruhat cell. I will explain a p-adic analogue of this result, using recent results of Abe-Caro on arithmetic D-modules.

Christian Pauly (Université de Nice); Opers in positive characteristic

Opers have been introduced by Beilinson and Drinfeld in their work on the geometric Langlands program. In this talk I will present known results on opers over a smooth projective curve defined over an algebraically closed field of positive characteristic. In particular, I will outline their relation with the action of the Frobenius morphism on vector bundles over the curve and discuss the importance of the so-called Hitchin-Mochizuki map, which associated to an oper the characteristic polynomial of its p-curvature. Finally, I will describe several generalizations. This is joint work with Kirti Joshi.

Christopher Dodd (University of Illinois); Witt-Differential Operators

We describe a new theory of sheaves of rings of differential operators on the Witt-vectors of a variety of characteristic p. These sheaves are to the de Rham-Witt complex as the usual sheaf of differential operators is to the de Rham complex. Further, the module theory of these rings generalizes and extends the category of crystals. Time permitting, we will describe some potential applications to geometry and representation theory.

Yasuhiro Wakabayashi (Tokyo Institute of Technology); Enumerative geometry of dormant opers

An oper is a certain flat bundle on an algebraic curve and plays fundamental roles in the theory of integrable systems and the geometric Langlands program. The aim of this talk is to explain one aspect of opers in characteristic p > 0 from the view point of enumerative geometry. The main characters are those with vanishing p-curvature (called dormant opers), which concern the algebraic-solution problem of linear differential equations. We would like to discuss the 2d TQFT (= 2-dimensional topological quantum field theory) associated to their moduli spaces and related results (e.g., duality, the Verlinde formula, and the Witten conjecture for dormant opers).

Grosse-Klönne (Humboldt Universität zu Berlin); Hecke modules and Galois representations into the dual group

Let F/Qp be a finite extension. Let G be a simply connected semisimple split algebraic group over O_F , let G^v be its dual. Let H be the pro-p Iwahori Hecke algebra of G(F), with coefficients in a finite extension k of the residue field of O_F . Motivated by the search for mod-p local Langlands correspondences we suggest to assign to irreducible H-modules certain homomorphisms of the absolute Galois group of F into G^v(k). We then ask if such an assignment can be upgraded into an exact functor between suitable abelian categories.

Adrian Langer (University of Warsaw); Nearby-cycles and semipositivity in positive characteristic

I will talk about an analogue of Hodge theory in positive characteristic. In particular, I will show analogues of Schmid’s nilpotent orbit theorem and nearby cycles in positive characteristic. As an application I will prove some strong semipositivity theorems for analogs of complex polarized variations of Hodge structures. This implies semipositivity for the relative canonical divisor of a semistable reduction and it also gives some new results over complex numbers.

Peter Schneider (Universität Münster); Support theory for the Iwahori-Hecke Ext-algebra

One technique to investigate abelian or triangulated categories is to establish a kind of geometric notion of support for their objects. This was originally developed by Gabriel for the abelian module category of certain rings. Eventually we want such a theory for the derived category of smooth representations of a p-adic Lie group in characteristic p. In this talk I will describe a first step in this direction, which is a support theory for the abelian category of graded modules over the corresponding cohomological Ext-algebra. The point is that this Ext-algebra does not satisfy Gabriel’s assumptions.