Abstracts

Rachel Ollivier

Title: TBA.

Abstract: TBA.

Peter Schneider

Title: TBA.

Abstract: TBA.

Konstantin Ardakov

Title: TBA.

Abstract: TBA.

Sabin Cautis

Title: Some remarks on Coulomb branches.

Abstract: One can view the Coulomb branch mathematically as a generalization of the affine Grassmannian and the Satake category. We will explain what this means, sketch some general structural results from joint work with Harold Williams and try to speculate a little on its connections to derived Hecke algebras.

Nicolas Dupré

Title: Model categories and pro-p Iwahori-Hecke modules.

Abstract: Let k be a field and let G be the group of rational points of a split connected reductive p-adic group. There is a natural adjunction between the category of smooth k-linear G-representations and the category of modules over the so-called pro-p Iwahori-Hecke algebra. This adjunction is known to have reasonably good properties when the characteristic of k is different than p, and this has proved useful in order to understand the smooth representations of G over such fields. However, in characteristic p, the behaviour of this adjunction is not sufficiently understood beyond G=GL_2(Q_p) and a few related cases. In this talk I will present a new approach, in the case where char(k)=p, to study this adjunction which uses the language of model categories. The Hecke algebra is known by work of Ollivier-Schneider to be Gorenstein, and from work of Hovey it follows that its category of modules has a so-called Gorenstein projective model structure. We use this to define a model structure on smooth G-representations. Our adjunction then naturally induces a derived adjunction between homotopy categories whose behaviour can be better understood. Along the way, we generalise a result of Cabanes on representations of finite reductive groups. This is joint work with Jan Kohlhaase.

Claudius Heyer

1. Title: The left adjoint of derived parabolic induction.

Abstract: Let G be a p-adic reductive group and P=MU a parabolic subgroup. By exactness, parabolic induction extends to a derived functor RInd_P^G between the derived categories D(M) and D(G) of unbounded complexes of smooth mod p representations. We sketch the argument for why RInd_P^G commutes with arbitrary derived products and hence admits a left adjoint L(U,-). In the remaining time, we investigate some properties of L(U,-) and compute it on the irreducible smooth representations of GL_2(Q_p).

2. Title: Parabolic induction via the parabolic pro-p Iwahori--Hecke algebra.

Abstract: Let G be a p-adic reductive group and P a parabolic subgroup with Levi M. On pro-p Iwahori--Hecke modules over any coefficient ring, there is defined a parabolic induction functor given by extension of scalars H(G) \otimes_{H(M^+)} -, where H(M^+) is the positive subalgebra of the Levi algebra H(M). In this talk, we present an extended construction of parabolic induction, replacing H(M^+) with the parabolic pro-p Iwahori--Hecke algebra H(P). This amounts to describing two algebra morphisms \Theta^P_M: H(P) \rightarrow H(M) and \Xi^P_G : H(P) \rightarrow H(G). Time permitting, we explain an alternative proof of the transitivity property of parabolic induction.

James Timmins

Title: The augmented Iwasawa algebra.

Abstract: The Iwasawa algebra of a compact p-adic Lie group was first defined by Lazard in 1965. Iwasawa algebras have many interesting and useful properties, and play an important role in the mod p representation theory of (compact) p-adic Lie groups. In this talk we'll introduce the augmented Iwasawa algebra, a more recent generalisation to any p-adic Lie group. We'll explain why it plays a crucial role in studying smooth representations. On the other hand, we present a result showing that augmented Iwasawa algebras have poorer properties than in the compact case, and ask some related questions.