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.

Wang: Representations of SL_2(F_q) via crystalline cohomology

Let C be the projective plane curve given by xy^q-x^qy-z^{q+1}. Drinfeld proved that the discrete series representations of SL_2(F_q) arise in the l-adic cohomology of C. Analogously, the crystalline cohomology group H^1(C) is a Z_p-module, which carries actions of SL_2(F_q), a non-split torus, and the Frobenius endomorphism. Haastert and Jantzen constructed a filtration of H^1(C) by taking the preimage under the Frobenius endomorphism of the natural filtration of H^1(C) by pH^1(C), p^2H^1(C),.... In this talk, we explain the explicit computations of this filtration and the actions on H^1(C) following the papers of Haastert and Jantzen.

Dupré: Pro-p Iwahori--Hecke modules and singularity categories

Let G be the group of rational points of a split reductive group over a nonarchimedean local field F of residue characteristic p, and let H be the associated pro-p Iwahori--Hecke algebra over a field k of characteristic p. The mod-p Langlands program aims to relate the representation theory of G over k to that of the absolute Galois group of F. The representations of G in this context are however still very poorly understood. On the other hand, the H-modules are much better understood and there even are results relating them to Galois representations. In earlier work, we investigated the so-called Gorenstein projective model structure on the category of H-modules and its associated homotopy category Ho(H). Assuming G has semisimple rank 1, we will explain in this talk how this category Ho(H) identifies with the singularity category of a suitable scheme parametrising Galois representations. This scheme appeared previously in work of Dotto--Emerton--Gee and of Pépin--Schmidt. After taking a suitable notion of support, this recovers (most of) the semisimple mod-p Langlands correspondence for GL_2(Q_p).

Meier: The Right Derived Functors of Ordinary Parts

Emerton's Ordinary Parts functor Ord plays an important role in the theory of mod p representations of p-adic reductive groups. The right derived functors of Ord are conjectured by Emerton to be given in terms of group cohomology. In this talk I will discuss parts of the proof of a variant of this conjecture. A key step in this proof is a comparison between certain compact and parabolic inductions which I will explain for an example. This is based on a paper joint with Manuel Hoff and Michael Spieß and appendix joint with Claudius Heyer.

Gilles: G_m-cohomology of p-adic Stein spaces

In this talk I will explain a computation that we did with Damien Junger, in which we determined the étale cohomology of G_m coefficients of some p-adic analytic spaces. One difficulty is that rigid analytic spaces do not have enough points and it is not possible to deduce the global behavior of G_m from its stalks. Instead we consider the quotient G_m/U of G_m by the group of principal units U. The cohomology of this quotient (in some cases) can then be computed using Kummer exact sequence, whereas the cohomology of U can be computed by passing to the pro-étale site, via p-adic methods. As an application, we compute the G_m-cohomology of Drinfeld's symmetric space.

Gautier: On the decomposition of principal series of GL_3(F)

In order to get information on a group, one can look at its representations. Here, we are looking at principal series of GL(3,F), F non-archimedean local field, that is to say representations obtained by inducing the trivial character of the subgroup of upper triangular matrices. When reducing to the maximal compact subgroup of GL(3,F) one can get a decomposition of such representations to a direct sum of irreducible subrepresentations. In this talk, I will explain how to (partly) get this decomposition depending on the character that is being induced.

• Seminar Darstellungstheorie vergangener Semester