Compatibility between Satake and Bernstein isomorphisms in characteristic $p$

Abstract

We study the center of the pro-$p$ Iwahori–Hecke ring ${\tilde{H}}_{\mathbb{Z}}$ of a connected split $p$-adic reductive group $G$. For $k$ an algebraically closed field of characteristic $p$, we prove that the center of the $k$-algebra ${\tilde{H}}_{\mathbb{Z}}{\otimes }_{\mathbb{Z}}k$ contains an affine semigroup algebra which is naturally isomorphic to the Hecke $k$-algebra $\mathcal{\mathscr{H}}\left(G,\rho \right)$ attached to an irreducible smooth $k$-representation $\rho$ of a given hyperspecial maximal compact subgroup of $G$. This isomorphism is obtained using the inverse Satake isomorphism defined in our previous work.

We apply this to classify the simple supersingular ${\tilde{H}}_{\mathbb{Z}}{\otimes }_{\mathbb{Z}}k$-modules, study the supersingular block in the category of finite-length ${\tilde{H}}_{\mathbb{Z}}{\otimes }_{\mathbb{Z}}k$-modules, and relate the latter to supersingular representations of $G$.