Types in reductive $p$-adic groups: The Hecke algebra of a cover

In this paper, $F$ is a non-Archimedean local field and $G$ is the group of $F$-points of a connected reductive algebraic group defined over $F$. Also, $\tau$ is an irreducible representation of a compact open subgroup $J$ of $G$, the pair $(J,\tau )$ being a type in $G$. The pair $(J,\tau )$ is assumed to be a cover of a type $(J_{L},\tau _{L})$ in a Levi subgroup $L$ of $G$. We give conditions, generalizing those of earlier work, under which the Hecke algebra $\mathcal H(G,\tau )$ is the tensor product of a canonical image of $\mathcal H(L,\tau _{L})$ and a sub-algebra $\mathcal H(K,\tau )$, for a compact open subgroup $K$ of $G$ containing $J$.