Modular representations of p-adic groups

I will survey some results in the theory of modular representations of a reductive $p$-adic group, in positive characteristic $\ell \neq p$ and $\ell=p$

The pro-p-Iwahori–Hecke algebra of a reductive p-adic group, II

For any commutative ring R and any reductive p-adic group G, we describe the center of the pro-p-Iwahori–Hecke R-algebra of G. We show that the pro-p-Iwahori–Hecke algebra is a finitely generated module over its center and is a finitely generated R-algebra. When the ring R is noetherian, the center is a finitely generated R-algebra and the pro-p-Iwahori–Hecke R-algebra is noetherian. This generalizes results known only for split groups

Modulo pp representations of reductive pp-adic groups: functorial properties

Let $F$ be a local field with residue characteristic $p$, let $C$ be an algebraically closed field of characteristic $p$, and let $\mathbf{G}$ be a connected reductive $F$-group. In a previous paper, Florian Herzig and the authors classified irreducible admissible $C$-representations of $G=\mathbf{G}(F)$ in terms of supercuspidal representations of Levi subgroups of $G$. Here, for a parabolic subgroup $P$ of $G$ with Levi subgroup $M$ and an irreducible admissible $C$-representation $\tau$ of $M$, we determine the lattice of subrepresentations of $\mathrm{Ind}_P^G \tau$ and we show that $\mathrm{Ind}_P^G \chi \tau$ is irreducible for a general unramified character $\chi$ of $M$. In the reverse direction, we compute the image by the two adjoints of $\mathrm{Ind}_P^G$ of an irreducible admissible representation $\pi$ of $G$. On the way, we prove that the right adjoint of $\mathrm{Ind}_P^G$ respects admissibility, hence coincides with Emerton's ordinary part functor $\mathrm{Ord}_{\overline{P}}^G$ on admissible representations.Comment: 39 page

Representations of a reductive pp-adic group in characteristic distinct from pp

We investigate the irreducible cuspidal $C$-representations of a reductive $p$-adic group $G$ over a field $C$ of characteristic different from $p$. When $C$ is algebraically closed, for many groups $G$, a list of cuspidal $C$-types $(J,\lambda)$ has been produced satisfying exhaustion, sometimes for a restricted kind of cuspidal representations, and often unicity. We verify that those lists verify Aut($C$)-stability and we produce similar lists when $C$ is no longer assumed algebraically closed. Our other main results concern supercuspidality. This notion makes sense for the representations $\lambda$ in the cuspidal $C$-types $(J,\lambda)$ as above, which involve finite reductive groups. We check that an irreducible cuspidal representation of $G$ induced from $\lambda$ is supercuspidal if and only $\lambda$ is supercuspidal.Comment: 45 pages This is the final versio