On two questions concerning representations distinguished by the Galois involution

Abstract

Abstract Let E / F {E/F} be a quadratic extension of non-archimedean local fields of characteristic 0. In this paper, we investigate two approaches which attempt to describe the irreducible smooth representations of GL n ⁢ ( E ) {\mathrm{GL}_{n}(E)} that are distinguished by its subgroup GL n ⁢ ( F ) {\mathrm{GL}_{n}(F)} . One relates this class to representations which come as base change lifts from a quasi-split unitary group over F, while another deals with a certain symmetry condition. By characterizing the union of images of the base change maps, we show that these two approaches are closely related. Using this observation, we are able to prove a statement relating base change and distinction for ladder representations. We then produce a wide family of examples in which the symmetry condition does not impose GL n ⁢ ( F ) {\mathrm{GL}_{n}(F)} -distinction, and thus exhibit the limitations of these two approaches.