## Workshop on modular Iwahori-Hecke algebras

17th to 21st March 2014

Abstracts:

Marie-France Vigneras (Université Paris Diderot - Paris 7)
Title (1): Pro-p-Iwahori Hecke algebras of reductive p-adic groups

Iwahori-Matsumoto presentations, Alcove walks, Bernstein bases, Bernstein relations, Center.

Title (2): Modular pro-p-Iwahori Hecke algebras.

Satake isomorphisms, supersingular modules.

Guy Henniart (Université Paris-Sud)
Title: A classification of mod. p irreducible admissible representations of reductive p-adic groups.

This talk, presenting joint work with N. Abe, F. Herzig and M.-F. Vigneras, follows the talks by Vigneras and is closely related to the talk by Abe.
Let F be a locally compact non-Archimedean field, of residue characteristic p, and G the group of F-points of a connected reductive group over F. Fix an algebraically closed field C of characteristic p. We classify irreducible admissible C-representations of G in terms of supercuspidal representations of the Levi subgroups of G. The main tool is to investigate the " weights" of those representations, that is the irreducible K-representations they contain, where K is a special parahoric subgroup of G. A crucial step is provided by a "change of weight " which is established using the structure of the pro-p Iwahori-Hecke algebras of Vigneras' talks.

Noriyuki Abe (Hokkaido University)
Title: "Change of weight" theorem.

This is a continuation of the talk by Henniart. We have the classification theorem of modulo p representations of a p-adic group.One crucial step of the proof is "change of weight" theorem. It is proved using the pro-p-Iwahori Hecke algebra. In thiks talk, how to use the structure theory of the pro-p-Iwahori Hecke algebra is explained. This is a joint work with G. Henniart, F. Herzig and M.-F. Vigneras.

Steffen König (Universität Stuttgart)
Title: An introduction to cellular structures.

Semisimple finite-dimensional algebras are direct sums of their Wedderburn components, which are just matrix rings over fields. Finite-dimensional cellular algebras can be defined by the existence of two-sided ideal filtrations, where the subquotients are matrix rings with a slightly deformed multiplication: A · B := A Φ B for a fixed matrix Φ. Group algebras of symmetric groups in any characteristic as well as most Hecke algebras of finite type and many other classes of algebras arising in algebraic Lie theory are cellular. Affine cellular algebras may have infinite dimension. But by definition, they still have a finite filtration with subquotients being slightly deformed matrix rings, now over affine commutative rings. Extended affine Hecke algebras of type A are affine cellular, and so are finite type Khovanov-Lauda-Rouquier-algebras, and other classes of algebras arising in algebraic Lie theory. Cell structures can be used to parametrise simple representations and to introduce linear algebra methods to describe them. When trying to put cohomological conditions into the theory, one is naturally led to the subclass of quasi-hereditary algebras, which have finite global dimension. Affine quasi-hereditary algebras are currently being defined.
References: Cellular algebras have been defined by Graham and Lehrer. Much of their structure theory has been developed jointly with Changchang Xi. The generalisation to affine cellular algebras also is joint with Xi, while the current work on affine quasi-hereditary algebras is joint with Kleshchev.

Tobias Schmidt (Humboldt Universität zu Berlin)
Titel: Affine flag varieties and Hecke algebras in characteristic p.

In this talk I will explain an extension of a result of Kostant-Kumar on complex analytic affine flag varieties to positive chacteristic. In particular, the T-equivariant K-theory of any affine flag variety in positive chacteristic admits an action of the corresponding modular Iwahori-Hecke algebra. This geometric Hecke module can be computed explicitly in terms of the affine Weyl group and the torus.

Karol Koziol (Columbia University, Vancuver)
Title: Towards a Langlands correspondence for Hecke modules of SLn in characteristic p.

In this talk, we show how to realize the pro-p-Iwahori-Hecke algebra of SLn as a subalgebra of the pro-p-Iwahori-Hecke algebra of GLn. Using the interplay between these two algebras, we deduce two main results: one on a numerical Langlands correspondence between "packets" of Hecke modules and mod-p projective Galois representations, and another on an equivalence of categories between Hecke modules and mod-p representations of SLn.

Elmar Große-Klönne (Humboldt Universtität zu Berlin)
Title: Pro-p-Iwahori Hecke modules and Gal(\overline{\mathbb Q}p/ {\mathbb Q}p)-representations.

In Colmez' work on the mod-p local Langlands correspondence for GL}2({\mathbb Q}p), an important ingredient is a certain functor from smooth admissible mod p representations of GL2({\mathbb Q}p) to mod p representations of Gal(\overline{\mathbb Q}p/ {\mathbb Q}p). I want to discuss a partial generalization of this functor for more general split reductive groups G over {\mathbb Q}p. This functor is defined for smooth admissible mod p representations V of G({\mathbb Q}p), but (at least in its present state) it is sensitive only to the space of invariants of V under a pro-p-Iwahori subgroup I0. More precisely, it factors through a functor from finite dimensional ${\mathcal H}$-modules to $(\varphi,\Gamma)$-modules, and hence (by Fontaine's functor) to Gal(\overline{\mathbb Q}p/ {\mathbb Q}p)-representations. Here ${\mathcal H}$ denotes the mod p pro-p-Iwahori Hecke algebra for I0G({\mathbb Q}p). I want to discuss how this functor behaves on simple supersingular ${\mathcal H}$-modules.

Jan Kohlhaase (Universität Münster)
Title: Smooth duality in natural characteristic.

Let G be a p-adic Lie group and let E be a field of characteristic p. On the category of E-linear smooth G-representations the smooth duality functor is known to be rather ill-behaved. In contrast to the classical theory it is not exact and has a large kernel. We explain how its derived functors can be interpreted in terms of the Auslander duality of completed group rings. When G is a p-adic reductive group we compute these derived functors in a large class of examples and discuss the dimension theory of admissible smooth representations of G over E.

Gwyn Bellamy (University of Glasgow)
Title: Modular Cherednik algebras.

These two talks will be a gentle introduction to rational Cherednik algebras in positive characteristic. No prior knowledge of these algebras will be assumed. I'll outline their basic properties and explain how little is actually know about their representation theory (especially in the modular situation).

Marten Bornmann (Universität Münster)
Title: Deligne Lusztig characters associated with Weil group representations and their reductions mod p.

Let ρ be a smooth n-dimensional irreducible representation of the Weil group of a finite extension L of Qp with coefficients in \overline{F}p. With such ρ, we will associate a Deligne-Lusztig character of GLn(k), where k is the residue field of L. In special cases, we will establish a connection of such an object with supersingular irreducible modules over the pro-p Iwahori-Hecke algebra of GLn(L) and compare this construction to Große-Klönne's functor, which establishes a bijection between isomorphism classes of smooth irreducible n-dimensional GalQp-representations and simple supersingular n-dimensional pro-p Iwahori-Hecke modules.

Laura Peskin (California Institute of Technology)
Title: Hecke algebras and representations of p-adic SL2-tilde in characterictic p.

The work discussed in this talk is part of a project to study mod p representations of metaplectic groups, starting with the double cover of SL2. We will describe the spherical and Iwahori Hecke algebras of SL2-tilde in characteristic p and note some interesting differences from the characteristic-0 case. In particular, in characteristic 0, the genuine Iwahori Hecke algebra of SL2-tilde is isomorphic to the Iwahori Hecke algebra of the dual group PGL2, giving a correspondence of irreducible representations with Iwahori-fixed vectors. In contrast, we show that these two algebras are not isomorphic in characteristic p. However, the respective spherical mod p Hecke algebras are isomorphic, and after classifying the ordinary mod p representations of SL2-tilde, we show that this isomorphism gives a correspondence of unramified ordinary representations.

Rachel Ollivier (The University of British Columbia)
Titel: Supersingularity for Hecke modules in characteristic p.

Let G be a split p-adic reductive group and H its pro-p Iwahori-Hecke algebra over a field of characteristic p. I will give an overview of some aspects of the mod p representation theory of H and its link to the mod p representation theory of G. In particular, I will explain that there is a "mod p Bernstein-type isomorphism" which is compatible with the mod p Satake isomorphism. The tools developed to prove this result allow us to classify the simple supersingular modules of H, and to relate the notions of supersingularity for the Hecke modules on one side and the representations of G on the other side.

Peter Schneider (Universität Münster)
Titel: The homological algebra of pro-p Iwahori-Hecke algebras

This is a survey talk on the known homological algebra properties of the pro-p Iwahori-Hecke algebra of a split reductive p-adic group: Auslander-Gorenstein property, duality theory, projective dimensions. I will also suggest a homological framework how to understand the relation between Iwahori-Hecke modules and smooth group representations. Finally I will recall a differential graded version of the above.

Sabin Cautis (The University of British Columbia)
Title: Categorification techniques in representation theory

We will survey some categorification techniques in modular representation theory and geometry.

Konstantin Ardakov (University of Oyford)
Title: Kashiwara's Theorem for $\widehat{\mathcal{D}}$-modules

Coadmissible $\widehat{\mathcal{D}}$-modules on a smooth rigid analytic variety X can be thought of as coherent sheaves on a "rigid analytic quantisation" of the cotangent bundle of X. In the case where X is a rigid analytic flag variety, they give a geometric realisation of coadmissible modules over the Arens-Michael envelope of the corresponding Lie algebra with trivial infinitesimal character. Equivariant $\widehat{\mathcal{D}}$-modules are of interest in connection with admissible locally analytic representations of semisimple compact p-adic Lie groups, and may have applications to the p-adic local Langlands programme. I will describe some recent progress in this theory. This is joint work with Simon Wadsley.

Catharina Stroppel (UNiversität Bonn)
Title: Quiver Schur algebras

In this talk I will introduce the Quiver Hecke algebras and their combinatorics. The construction is a generalization of Khovanov-Lauda quiver Hecke algebras. The construction is geometric, but I will focus on the algebraic and combinatorial description. The talk will be mostly a survey talk with some explicit examples.