On twisted Gelfand pairs through commutativity of a Hecke algebra

For a locally compact, totally disconnected group $G$, a subgroup $H$ and a character $\chi :H\to {\mathbb{C}}^{\times }$ we define a Hecke algebra ${\mathcal{H}}_{\chi }$ and explore the connection between commutativity of ${\mathcal{H}}_{\chi }$ and the $\chi$-Gelfand property of $(G,H)$, i.e. the property ${\mathrm{d}\mathrm{i}\mathrm{m}}_{\mathbb{C}}({\rho }^{\ast }{)}^{(H,{\chi }^{-1})}\le 1$ for every $\rho \in \mathrm{I}\mathrm{r}\mathrm{r}(G)$, the irreducible representations of $G$. We show that the conditions of the Gelfand-Kazhdan criterion imply commutativity of ${\mathcal{H}}_{\chi }$, and verify in several simple cases that commutativity of ${\mathcal{H}}_{\chi }$ is equivalent to the $\chi$-Gelfand property of $(G,H)$. We then show that if $G$ is a connected reductive group over a $p$-adic field $F$, and $G\mathrm{/}H$ is $F$-spherical, then the cuspidal part of ${\mathcal{H}}_{\chi }$ is commutative if and only if $(G,H)$ satisfies the $\chi$-Gelfand property with respect to all cuspidal representations $\rho \in \mathrm{I}\mathrm{r}\mathrm{r}(G)$. We conclude by showing that if $(G,H)$ satisfies the $\chi$-Gelfand property with respect to all irreducible $(H,{\chi }^{-1})$-tempered representations of $G$ then ${\mathcal{H}}_{\chi }$ is commutative.