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 $(G,H)$, that is, the property $\dim _{\mathbb{C}} (\rho ^*)^{(H,\chi ^{-1})} \leq 1$ for every $\rho \in \textrm{Irr}(G)$, 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 ${\rho \in \textrm{Irr}(G)}$. We conclude by showing that if $(G,H)$ satisfies the $\chi$-Gelfand property with respect to all irreducible $(H\backslash G,\chi ^{-1})$-tempered representations of $G$ then ${\mathcal{H}}_\chi$ is commutative.