Linear periods for unitary representations

Abstract

Let $F$ be a local non-Archimedean field of characteristic zero with a finite residue field. Based on Tadi\'{c}'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.