Iwahori component of the Gelfand--Graev representation for reductive groups

Let $G$ be a connected reductive group over a $p$-adic field $F$, $U$ the unipotent radical of a minimal parabolic subgroup, $\psi$ a depth-zero non-degenerate character of $U(F)$, and $I$ an Iwahori subgroup of $G(F)$.

We show that, as a module over the Iwahori-Hecke algebra ${H}$, the space of $I$-fixed vectors in the Gelfand-Graev representation $\mathrm{ind}_U^G\psi$ is isomorphic to ${H} \otimes_{{H}_{W_0}} \mathrm{sgn}$.

Here $\mathrm{sgn}$ is the sign representation of the finite Hecke subalgebra ${H}_{W_0}$ attached to the relative Weyl group.

This extends the theorem of Chan-Savin from split groups to all connected reductive groups.