R-groups for unitary principal series of Spin groups

Abstract

Abstract We study the unitary principal series of the split group S p i n m ( F ) , where F is a p-adic field. Let χ ˜ be a unitary character of a maximal F-split torus T ˜ of G S p i n m ( F ) , and let χ be its restriction to T = T ˜ ∩ S p i n m ( F ) . The R-groups R χ ˜ and R χ of the corresponding principal series representations fit in the exact sequence 0 → R χ ˜ → R χ → R χ / R χ ˜ → 0 . We give a complete answer to the question of splitting of this exact sequence. We also prove that the multiplicity is one when irreducible constituents in unitary principal series are restricted from G S p i n ( F ) to S p i n ( F ) . Further, based on Keys result, we prove Arthur s conjecture on R-groups for unitary principal series of S p i n .