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 $\mathrm{dim}_\mathbb{C}(\rho^*)^{(H,\chi^{-1})} \leq 1$ for every $\rho \in \mathrm{Irr}(G)$, 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 $G/H$ 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 ${\rho \in \mathrm{Irr}(G)}$. 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.