SOME RESULTS ON THE ADMISSIBLE REPRESENTATIONS OF NON-CONNECTED REDUCTIVE 0ADIC GROUPS

DAVID GOLDBERG REBECCA HERB

Abstract. We examine the theory of induced representations for non-connected reductive p-adic groups for which G/G^{0} is abelian. We first examine the structure of those representations of the form Ind_{P^{0}}^{G}(σ), where P^{0} is a parabolic subgroup of G^{0} and σ is a discrete series representation of the Levi component of P^{0}. Here we develop a theory of R–groups, extending the theory in the connected case. We then prove some general results in the theory of representations of non-connected p-adic groups whose component group is abelian. We define the notion of cuspidal parabolic for G in order to give 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 a cuspidal parabolic subgroup of G. Here we also develop a theory of R–groups, and show that these groups parameterize the induced representations in a manner that is consistent with the connected case and with the first set of results as well.