Some results on the admissible representations of non-connected reductive p-adic groups

We examine induced representations for non-connected reductive p-adic groups with G/G(0) abelian. We describe the structure of the representations Ind(P0)(G)(sigma), P-0 a parabolic subgroup of G(0) and sigma a discrete series representation of the Levi component of P-0. We develop a theory of R-groups, extending the theory in the connected case. We then prove some general representation theoretic results for non-connected p-adic groups with abelian component group. The notion of cuspidal parabolic for G is defined, giving a context for this discussion. Intertwining operators for the non-connected case are examined and the notions of supercuspidal and discrete series are defined. Finally, we examine parabolic induction from cuspidal parabolic subgroups of G. We develop a theory of X-groups, and show these groups parameterize the induced representations in a manner consistent with the connected case and with the first set of results as well.