Iwahori–Hecke algebras are Gorenstein

In the local Langlands program the (smooth) representation theory of $p$-adic reductive groups $G$ in characteristic zero plays a key role. For any compact open subgroup $K$ of $G$ there is a so called Hecke algebra ${\mathcal H}(G,K)$. The representation theory of $G$ is equivalent to the module theories over all these algebras ${\mathcal H}(G,K)$. Very important examples of such subgroups $K$ are the Iwahori subgroup $I$ and the pro-$p$ Iwahori subgroup $I_p$. By a theorem of Bernstein, the Hecke algebras of these subgroups (and many others) have finite global dimension.
In recent years the same representation theory of $G$ but over an algebraically closed field of characteristic $p$ has become more and more important. But little is known yet. Again one can define analogous Hecke algebras. Their relation to the representation theory of $G$ is still very mysterious. Moreover they are no longer of finite global dimension. In a joint work with R. Ollivier, we prove that ${\mathcal H}(G,I)$ and ${\mathcal H}(G,I_p)$ over ANY field are Gorenstein.