#### Vol. 8, No. 5, 2014

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Compatibility between Satake and Bernstein isomorphisms in characteristic $p$

### Rachel Ollivier

Vol. 8 (2014), No. 5, 1071–1111
##### Abstract

We study the center of the pro-$p$ Iwahori–Hecke ring ${\stackrel{̃}{H}}_{ℤ}$ 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 ${\stackrel{̃}{H}}_{ℤ}{\otimes }_{ℤ}k$ contains an affine semigroup algebra which is naturally isomorphic to the Hecke $k$-algebra $\mathsc{ℋ}\left(G,\rho \right)$ 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 ${\stackrel{̃}{H}}_{ℤ}{\otimes }_{ℤ}k$-modules, study the supersingular block in the category of finite-length ${\stackrel{̃}{H}}_{ℤ}{\otimes }_{ℤ}k$-modules, and relate the latter to supersingular representations of $G$.

##### Keywords
Hecke algebras, characteristic $p$, Satake isomorphism, supersingularity
Primary: 20C08
Secondary: 22E50