Modular Representation Theory of Finite and p-adic Groups - IMS

Modular Representation Theory of Finite and p-adic Groups
(1 - 26 Apr 2013)

~ Abstracts ~

On a modulo p representation of pro-p Iwahori-Hecke algebra
Noriyuki Abe, Hokkaido University, Japan

For a split reductive p-adic group and its pro-p Iwahori subgroup, one can define its Hecke algebra, called pro-p Iwahori-Hecke algebra. We will discuss modules of this algebra over characteristic p field.

Iwahori-Hecke model for supersingular representations of GL(2,Q_p)
U. K. Anandavardhanan, Indian Institute Of Technology, India

In this talk, we first describe a regular supersingular representation of GL(2,Q_p) as a quotient of a representation induced from the Iwahori subgroup of GL(2,Q_p). This setting provides a natural way to realize all the self-extensions of a supersingular representation that admit a four dimensional space of invariants under the pro-p-Iwahori subgroup. We also investigate the structure of invariants under the principal congruence subgroups and give an alternative proof of a recent result of S. Morra. This work is joint with Gautam H. Borisagar.

Modules of constant Jordan type for elementary abelian p-groups
Shawn Baland, Bielefeld University, Germany

Let E be an elementary abelian p-group of rank r and k an algebraically closed field of characteristic p. In this talk we discuss kE-modules of constant Jordan type, which were defined by Carlson, Friedlander and Pevtsova. In particular, we focus on a construction of Benson and Pevtsova that gives a relationship between kE-modules of constant Jordan type and algebraic vector bundles on projective (r-1)-space. This will allow us to use the theory of Chern classes to place some restrictions on such modules. We close with a discussion of the limitations of our technique and avenues of future investigation.

Lie powers of group modules and the Lie module for the symmetric group
Roger Bryant, University of Manchester, UK

If V is a module for a group G over a field F then the homogeneous components of the free Lie algebra over F freely generated by (a basis of) V are also modules for G over F, called the Lie powers of V (by analogy with the tensor powers of V). I shall survey some of the results on the structure of these Lie powers with emphasis on the case where F is a field of prime characteristic p. The case where G is the general linear group on V is of particular interest and this is linked with the study of the Lie module Lie(r) for the symmetric group of degree r.

Global/local conjectures in the representation theory of finite groups
Marc Cabanes, Université Denis Diderot - Paris 7, France

An underlying idea of the so-called global/local conjectures - notably the conjectures from Alperin, Brauer and McKay - is that certain aspects of the representation theory of a finite group should be determined "locally", that is, by the representation theory of normalisers of certain p-subgroups. Much of the recent work in the representation theory of finite groups is centered around theorems reducing those conjectures to the checking of (usually stronger) ad hoc statements on quasi-simple groups. The classification of finite (quasi-)simple groups is then used. I will report on the reduction theorems and the cases already checked.

Brauer algebras associated to non-exceptional complex reflection groups
Anton Cox, City University London, UK

This talk will describe how to associate a Brauer-type algebra to a complex group of type G(n,p,m). We will review the classical Brauer algebra theory, and the construction of unoriented cyclotomic Brauer algebras (which corresponds to G(m,1,n)). We will also explain how the decomposition numbers in the general case can be deduced from those in the cyclotomic case by a combination of explicit diagram algebra combinatorics and Clifford theory.

Twisted category algebras and quasi-heredity
Susanne Danz, University of Kaiserslautern, Germany

In this talk we shall consider twisted category algebras over fields of characteristic 0. The underlying category will always be finite and will have an additional property, which is called 'split'. The multiplication in such an algebra is essentially induced by the composition of morphisms in the category. Prominent examples of twisted category algebras are various classes of diagram algebras (for suitable parameters) such as Brauer algebras, Temperley--Lieb algebras, or partition algebras. Twisted category algebras also arise in connection with double Burnside rings and biset functors.

We shall show that a twisted split category algebra in characteristic 0 is quasi-hereditary, that is, the corresponding module category is a highest weight category. Moreover, we shall give an explicit description of its standard modules with respect to a particular partial order on the set of isomorphism classes of simple modules. This provides, in particular, a unified proof of the known fact that the aforementioned diagram algebras are quasi-hereditary.

This is joint work with Robert Boltje.

On local Langlands correspondence for mod l representations
Jean-Francois Dat, Institut de Mathématiques de Jussieu, France

Vigneras has established a bijection between irreps (mod l) of a p-adic GL_n and (mod l) Weil-Deligne representations of dimension n. Despite its similarity with the usual l-adic correspondence, this one has a different nature since the "Deligne part" does not seem to have an arithmetic origin. We will explain a geometric interpretation of this Deligne part inspired by Arthur's "second SL_2 factor".

A derived local Langlands correspondence for GL_n
David Helm, University of Texas at Austin, USA

We describe joint work (with David Ben-Zvi and David Nadler) that constructs an equivalence between the derived category of smooth representations of GL_n(Q_p) and a certain category of coherent sheaves on the moduli stack of Langlands parameters for GL_n. The proof of this equivalence is essentially a reinterpretation of K-theoretic results of Kazhdan and Lusztig via derived algebraic geometry. We will also discuss (conjectural) extensions of this work to other quasi-split groups, and to the modular representation theory of GL_n.

The Breuil-Mézard conjecture for non-scalar split residual representations
Yongquan Hu, Institut de Recherche Mathématiques de Rennes, France

Following the approach of Paskunas, we prove the Breuil-Mézard conjecture for split (non-scalar) residual representations of the absolute Galois group of Qp (when p>3). Combined with the cases previously proved by Kisin and by Paskunas, this completes the proof of the conjecture. This is a joint work with Fucheng Tan.

On Donovan's conjecture
Radha Kessar, City University London, UK

Many questions in the modular representation theory of finite groups revolve around the extent to which the structure of the p-modular algebra of a finite group G is controlled by the G-poset of p-subgroups of G. The Donovan conjecture asserts that there only are finitely many Morita equivalence classes of blocks of modular group algebras with a given defect. In my talk, I will give an introduction to the conjecture and report on recent joint work with C. Eaton, B. Kulshammer and B. Sambale.

Supercuspidal representations of U(2,1)
Karol Koziol, Columbia University, USA

Classifying the mod-p supercuspidal representations of a connected, reductive, p-adic group is an important question in the context of the mod-p Langlands program, and has only been achieved in the case of GL_2(Q_p). In this talk, we will show how to construct mod-p supercuspidal representations of the unramified unitary group U(2,1) in three variables by adapting a method of Paskunas. We will also mention some complications that arise with this method. This work is joint with P. Xu.

l-modular representations of classical p-adic groups (p not equal to l)
Robert Kurinczuk, University of East Anglia, UK

The l-modular representation theory of classical p-adic groups has striking differences to the relatively well known complex theory. We will examine the situation in detail for unramified p-adic U(2,1), where many interesting phenomena already appear, and remark on current work (joint with Shaun Stevens) to extend these results to all classical p-adic groups (p not equal to 2).

On simple modules over twisted category algebras
Markus Linckelmann, City University London, UK

We show that Alperin's weight conjecture admits a formulation for twisted category algebras. The main ingredient is joint work with Michal Stolorz, where we give a description of a parametrisation of the isomorphism classes of simple modules over twisted category algebras. For semigroup algebras, this parametrisation goes back to work of Clifford, Munn, and Ponizovskii; a recent proof, due to Ganyushkin, Mazorcuk, and Steinberg in terms of Schur functors and Green relations in semigroups has been a key ingredient for the extension of this parametrisation to category algebras.

Graded homomorphisms between Specht modules for KLR algebras of type A
Sinéad Lyle, University of East Anglia, UK

The Khovanov-Lauda-Rouquier algebras, or cyclotomic quiver Hecke algebras, are certain Z-graded algebras which depend on an oriented quiver. Remarkably, it has been shown by Brundan and Kleshchev that the cyclotomic quiver Hecke algebras of type A are isomorphic to the cyclotomic Hecke algebras of type G(r,1,n), also known as the Ariki-Koike algebras. These algebras include as special cases the Hecke algebras of type A and type B and hence also the symmetric group algebra. Thus one application of Brundan and Kleshchev's result is that it defines a Z-grading on the symmetric group algebra. A further result of Brundan, Kleshchev and Wang shows that the Specht modules are graded. It therefore makes sense to talk about graded decomposition numbers and graded homomorphisms between Specht modules.

This talk will discuss some recent work on KLR algebras.

Zelevinsky involution and Langlands classification of modulo l irreducible representations of GL(n,F)
Alberto Minguez, Institut de Mathématiques de Jussieu, France

In the l-adic representation theory of GL(n) (and its inner forms) over a p-adic field there are two classification schemes due to Zelevinsky and Langlands in which the building blocks are certain segment representations Z(\Delta) and L(\Delta).

When one considers modulo l representations, l different from p, there exists a Zelevinsky classification of the irreducible representations of GL(n) (due to Vignéras) and its inner forms (due to Mínguez-Sécherre). In this talk we will present how to construct a Langlands classification using the Zelevinsky involution. This is a joint work with V. Sécherre.

Iwahori-Hecke algebras are Gorenstein, parts 1 and 2
Rachel Ollivier, Columbia University, USA

Let G be a split connected reductive group over a nonarchimedean local field of residual characteristic p, and let H be the (pro-p) Iwahori-Hecke algebra of G with coefficients in an arbitrary field k. In the classical case, where k has characteristic zero, H is known, by Bernstein, to be a regular ring. This means that any H-module has a finite projective resolution. This is no longer the case if k has characteristic p. However, we prove that H is always a Gorenstein ring.

In the first talk we describe the construction of a natural resolution of H as a bimodule over itself. It is obtained thanks to coefficient systems on the semisimple Bruhat-Tits building of G. This resolution allows us to prove that H has finite injective dimension as a module over itself.

In the second talk we first prove, in the case where G is semisimple, that the injective dimension of H is equal to the rank of the group G and that there is a duality functor on the finite length modules. Lastly, we consider the case where k has characteristic p and prove that H has a simple module with infinite projective dimension. The latter result is valid for "most" split groups G.

Joint work with P. Schneider.

Iwahori-Hecke algebras are Gorenstein, parts 1 and 2
Peter Schneider, University of Münster, Germany

Let G be a split connected reductive group over a nonarchimedean local field of residual characteristic p, and let H be the (pro-p) Iwahori-Hecke algebra of G with coefficients in an arbitrary field k. In the classical case, where k has characteristic zero, H is known, by Bernstein, to be a regular ring. This means that any H-module has a finite projective resolution. This is no longer the case if k has characteristic p. However, we prove that H is always a Gorenstein ring.

In the first talk we describe the construction of a natural resolution of H as a bimodule over itself. It is obtained thanks to coefficient systems on the semisimple Bruhat-Tits building of G. This resolution allows us to prove that H has finite injective dimension as a module over itself.

In the second talk we first prove, in the case where G is semisimple, that the injective dimension of H is equal to the rank of the group G and that there is a duality functor on the finite length modules. Lastly, we consider the case where k has characteristic p and prove that H has a simple module with infinite projective dimension. The latter result is valid for "most" split groups G.

Joint work with R. Ollivier.

Patching and the Breuil-Schneider conjecture
Sug Woo Shin, Massachusetts Institute of Technology, USA

Breuil and Schneider made a precise conjecture on when an irreducible smooth representation tensored with an irreducible algebraic representation of a p-adic general linear group admits an invariant norm. The conjecture is central in the p-adic Langlands program. We will review recent results by Hu and Sorensen and report on joint work in progress with Caraiani, Emerton, Gee, Geraghty and Paskunas.

The Bernstein relations in the pro-p-Iwahori Hecke algebra of a general reductive p-adic group
Marie-France Vigneras, Institut de Mathématiques de Jussieu, France

Let $G$ be a general reductive group over a p-adic field $F$ of finite residue field, let $I_p$ be the pro-$p$-radical of an Iwahori subgroup of $G$, and let $R$ be a commutative ring. The pro-p-Iwahori Hecke algebra of $G$ is the algebra $H$ of intertwiners of the regular representation $R[I_p\backslash G]$, naturally isomorphic to the compatly supported bi-$I_p$-invariant functions $G\to R$. This algebra is ubiquitous in the theory of representations of $G$ in the natural characteristic.

To any spherical orientation $o$ of an apartment of the Bruhat-Tits building of $(G,F)$, is associated a basis $(E_o(w))$ satisfying certain relations called the Bernstein relations. These basis play an important role in the classification of simple $H$-modules and of smooth $R$-representations of $G$, when $R$ is an algebraically closed field field of characteristic $0$ (Kazhdan-Lusztig, Ginsburg) and $p$ (V., Ollivier, Abe).

For a split group $G$, the Bernstein relations were proved by Lusztig for the affine Hecke algebras with invertible parameters, by V. for the pro-p-Iwahori Hecke algebras. The introduction by Gortz of the orientations, based on the notion of alcove walk by Arun Ram, allows a much better approach. The elegant proof of Gortz of the Bernstein relations for affine Hecke algebras with invertible parameters was extended by N.Schmidt to the pro-p-Iwahori Hecke algebras.

Quantum Frobenius characteristic map for the centers of Hecke algebras
Weiqiang Wang, University of Virginia, USA

We will establish a precise connection between the centers of Hecke algebras associated to the symmetric groups and the ring of symmetric functions, quantizing the classical Frobenius characteristic map. This leads to an answer to a question of Lascoux on identification of several remarkable bases of the centers with bases of symmetric functions.
This is joint work with Jinkui Wan (Beijing).

Generalized Foulkes modules and decomposition numbers of the symmetric group
Mark Wildon, University of London, UK

Finding the decomposition numbers of blocks of the symmetric group in prime characteristic is one of the main open problems in modular representation theory. I will talk about a new result that gives information about decomposition numbers in blocks of arbitrarily high weight. The result is obtained by applying the Brauer correspondence for p-permutation modules, as developed by M. Broué, to various twists of the permutation module given by the action of the symmetric group S_{2n} by conjugacy on its conjugacy class of fixed-point-free involutions. We classify all the vertices of the indecomposable summands of these modules over fields of odd prime characteristic. In characteristic zero these modules appear in the long-standing Foulkes Conjecture: I will end by mentioning some recent computational results on this problem.

This is joint work with Eugenio Giannelli.

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