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).

• Seminar Darstellungstheorie vergangener Semester