Construction of relative discrete series representations for p-adic GLn

Let F be a nonarchimedean local field of characteristic zero and odd residual characteristic. Let G be the F-points of a connected reductive group defined over F and let θ be an F-rational involution of G. Define H to be the closed subgroup of θ-fixed points in G. The quotient variety H\G is a p-adic symmetric space. A fundamental problem in the harmonic analysis on H\G is to understand the irreducible subrepresentations of the right-regular representation of G acting on the space L2(H\G) of complex-valued square integrable functions on H\G. The irreducible subrepresentations of L2(H\G) are called relative discrete series representations. In this thesis, we give an explicit construction of relative discrete series representations for three p-adic symmetric spaces, all of which are quotients of the general linear group. We consider GLn (F) × GLn(F)\GL2n(F), GLn(F)\GLn(E), and UE/F(F)\GL2n(E), where E is a quadratic Galois extension of F and UE/F is a quasi-split unitary group. All of the representations that we construct are parabolically induced from θ-stable parabolic subgroups admitting a certain type of Levi subgroup. In particular, we give a sufficient condition for the relative discrete series representations that we construct to be non-relatively supercuspidal. Finally, in an appendix, we describe the relative discrete series for the space GLn−1(F) × GL1(F)\GLn(F).Ph.D