Ottawa, 7 - 11 juin 2021
I intend to provide an overview of our understanding of the complex and mod-$\ell$ representations of p-adic groups and outline recent developments and applications.
In characteristic p things are still poorly understood and the role of the pro-p Iwahori-Hecke algebra H is played by a differential graded Hecke algebra. In particular, by work of Peter Schneider, the module category over the d.g. Hecke algebra is equivalent to the derived category of smooth representations of G. Unlike in the case of H, we know little about the structure of this d.g. Hecke algebra.
In this talk I will report on joint work with Peter Schneider where we study the cohomology of the d.g. Hecke algebra. When $G= SL(2,\mathbb Q_p)$ we now understand its structure well enough to deduce some properties of mod p representations of $SL(2,\mathbb Q_p)$. We also have results for certain more general groups.