2019/03/26 by Broussous, Paul, Matringe, Nadir
#11F70 #22E50 #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.1903.11051
Let E/F be a quadratic extension of non-archimedean local fields of characteristic different from 2. Let A be an F-central simple algebra of even dimension so that it contains E as a subfield, set G=A^× and H for the centralizer of E^× in G. Using a Galois descent argument, we prove that all double cosets H g H⊂ G are stable under the anti-involution g↦ g-1, reducing to Guo's result for F-split G which we extend to fields of positive characteristic different from 2. We then show, combining global and local results, that H-distinguished irreducible representations of G are self-dual and this implies that (G,H) is a Gelfand pair: dimℂ(HomH(π,ℂ))≤ 1 for all smooth irreducible representations π of G. Finally we explain how to obtain the the multiplicity one statement in the archimedean case using the criteria of Aizenbud and Gourevitch, and we then show self-duality of irreducible distinguished representations in the archimedean case too.