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). Time permitting, we will also discuss some results about Ho(H) in higher rank.