Abstract :
If G is a totally disconnected group and H is a closed subgroup then, according to the Gelfand–Kazhdan Lemma, if the double coset space H-45 degree ruleG/H is preserved by an antiautomorphism of G of order two then (G,H) must be a Gelfand pair in the sense that HomH(π,1) has dimension at most one for each irreducible, admissible representation π of G. Under certain rather general restrictions, we show that if the symmetry property holds only for almost all double cosets, then (G,H) is a supercuspidal Gelfand pair in the sense that dimHomH(π,1) for all irreducible, supercuspidal representations π of G. There exist examples of supercuspidal Gelfand pairs which are not Gelfand pairs.
