UBC Number Theory Seminar: Nicolas Dupré

Let G be a p-adic reductive group and k a field of characteristic p. The category Rep(G) of smooth k-linear representations is at the heart of the mod-p Langlands program. If we let I be a pro-p Iwahori subgroup of G, there is an associated convolution algebra H=k[I\G/I], called the pro-p Iwahori-Hecke algebra of G, that is well-understood. Taking I-invariants then defines a functor U from Rep(G) to the category Mod(H) of H-modules, which is expected to provide a strong relationship between the two categories. However, aside from small cases, the functor U fails to be exact and its behaviour remains quite mysterious. In earlier joint work with J. Kohlhaase, we used the language of model categories to study this situation. In that approach, one instead studies the interplay between certain associated homotopy categories of representations and H-modules. In this talk, I will give an overview of this new approach and describe when distinct simple (supersingular) H-modules can become homotopy equivalent to one another.