Supercuspidal Gelfand pairs (Q1394922)

Let \(G\) be a totally disconnected unimodular group, and let \(H\) be a closed subgroup of \(G\). Then \((G,H)\) is called a Gelfand pair if the dimension of \(\text{Hom}_H (\pi,1)\) is at most one for every irreducible admissible representation \(\pi\) of \(G\). According to the Gelfand-Kazhdan lemma, \((G,H)\) is a Gelfand pair if the double coset space \(H \backslash G/H\) is preserved by an antiautomorphism of \(G\) of order two. If \(\dim \text{Hom}_H (\pi,1) \leq 1\) for every irreducible admissible supercuspidal representation \(\pi\) of \(G\), then \((G,H)\) is called a supercuspidal Gelfand pair. There exist supercuspidal Gelfand pairs that are not Gelfand pairs. In this paper the author proves, under certain general restrictions, that \((G,H)\) is a supercuspidal Gelfand pair if the symmetry property holds only for almost all double cosets.