Nicolas Dupré

Title: Homotopy classes of simple pro-p Iwahori-Hecke modules

Abstract: 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.