On twisted Gelfand pairs through commutativity of a Hecke algebra

For a locally compact, totally disconnected group $G$, a subgroup $H$ and a character $χ:H \to \mathbb{C}^{\times}$ we define a Hecke algebra $\mathcal{H}_χ$ and explore the connection between commutativity of $\mathcal{H}_χ$ and the $χ$-Gelfand property of $(G,H)$, i.e. the property $\mathrm{dim}_\mathbb{C}(ρ^*)^{(H,χ^{-1})} \leq 1$ for every $ρ\in \mathrm{Irr}(G)$, the irreducible representations of $G$. We show that the conditions of the Gelfand-Kazhdan criterion imply commutativity of $\mathcal{H}_χ$, and verify in several simple cases that commutativity of $\mathcal{H}_χ$ is equivalent to the $χ$-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 $\mathcal{H}_χ$ is commutative if and only if $(G,H)$ satisfies the $χ$-Gelfand property with respect to all cuspidal representations ${ρ\in \mathrm{Irr}(G)}$. We conclude by showing that if $(G,H)$ satisfies the $χ$-Gelfand property with respect to all irreducible $(H,χ^{-1})$-tempered representations of $G$ then $\mathcal{H}_χ$ is commutative.

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