Seminar Darstellungstheorie

Das Seminar Darstellungstheorie findet normalerweise dienstags, 14.15 - 15.15 Uhr in G.13.18 statt.

Beteiligte Dozenten: Sascha Orlik, Kay Rülling, Tobias Schmidt, Britta Späth.

Nicolas Dupré: Singularity categories of pro-p Iwahori--Hecke modules

Let G be the group of rational points of a split p-adic reductive group. The smooth representation theory of G over fields of positive characteristic is of particular interest in the mod-p Langlands program, however it is still to this day poorly understood. Associated to the group G, there is a so-called pro-p Iwahori--Hecke algebra H and a natural adjunction relating smooth representations over G and H-modules. In earlier joint work with Kohlhaase, we considered Hovey's so-called Gorenstein projective model structure on Mod(H) and used it to study this adjunction. Later, the structure of the associated homotopy category Ho(H) and in particular the homotopy classes of simple H-modules was also investigated. In this talk, I will report on this work. Time permitting, I also hope to explain how, when G is of rank one, the category Ho(H) can be described in terms of the singularity categories of some relatively straightforward schemes.

Georg Linden: Classification of Equivariant Line Bundles on the Drinfeld Upper Half Plane

We explicitly determine the group of isomorphism classes of equivariant line bundles on the Drinfeld upper half plane associated to a non-archimedean local field F for GL2(F), for its subgroup of matrices whose determinant has trivial valuation, and for GL2(O_F ). Our results extend a recent classification of torsion equivariant line bundles with connection due to Ardakov and Wadsley, but we use a different approach. A crucial ingredient is a construction due to Van der Put which relates invertible analytic functions on the Drinfeld upper half plane to currents on the Bruhat–-Tits tree. Another tool we use is condensed group cohomology.

Franziska Jahnke: Perfectoid fields and almost purity - a model theoretic perspective

In this talk, I present a model-theoretic approach to understanding perfectoid fields, which in particular gives rise to a new proof of the almost purity theorem for valuation rings. I then discuss applications and potential future directions, including: which arithmetic properties (e.g. Lang's property C_i) of a field (un)tilt? How can we see the almost purity theorem for perfectoid K-algebras in this setting?
 

Gaëtan Mancini: On the Alperin-McKay Conjecture for Groups of Exceptional Type

The Alperin-McKay conjecture relates the number of height 0 characters in a block and in its Brauer correspondent. The proof of the conjecture has been reduced to the verification of certain conditions on (quasi-)simple groups. In this talk we will discuss the techniques used to verify these conditions for finite reductive groups of exceptional type.

Julian Reichardt: On D-cap-Modules of Finite Length

Andrés Sarrazola Alzate: Local–Global Diagrams for G–Equivariant D_Ω-cap–Modules on the Drinfeld Plane

• Andrés Sarrazola Alzate: Local–Global Diagrams for G–Equivariant D_Ω-cap–Modules on the Drinfeld Plane