Linear periods for unitary representations

Abstract

Let F be a local non-Archimedean field of characteristic zero with a finite residue field. Based on Tadić’s classification of the unitary dual of \({\mathrm {GL}}_{2n}(F)\), we classify irreducible unitary representations of \({\mathrm {GL}}_{2n}(F)\) that have nonzero linear periods, in terms of Speh representations that have nonzero periods. We also give a necessary and sufficient condition for the existence of a nonzero linear period for a Speh representation.