R-groups, elliptic representations, and parameters for GSpin groups

Abstract

Abstract We study parabolically induced representations for GSpin m ( F ) ${\mathrm {GSpin}_m(F)}$ with F a p-adic field of characteristic zero. The Knapp–Stein R-groups are described and shown to be elementary two groups. We show the associated cocycle is trivial proving multiplicity one for induced representations. We classify the elliptic tempered spectrum. For GSpin 2 n + 1 ( F ) ${\mathrm {GSpin}_{2n+1}(F)}$ , we describe the Arthur (Endoscopic) R-group attached to Langlands parameters, and show these are isomorphic to the corresponding Knapp–Stein R-groups. This relies on a non-trivial equality of certain L-functions, and we give a direct proof of this equality. We also reduce the isomorphism, in the general case of split groups, to the maximal case.