R-groups for unitary principal series of Spin groups

Abstract

We study the unitary principal series of the split group Spinm(F), where F is a p-adic field. Let χ˜ be a unitary character of a maximal F-split torus T˜ of GSpinm(F), and let χ be its restriction to T=T˜∩Spinm(F). The R-groups Rχ˜ and Rχ of the corresponding principal series representations fit in the exact sequence 0→Rχ˜→Rχ→Rχ/Rχ˜→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 GSpin(F) to Spin(F). Further, based on Keys' result, we prove Arthur's conjecture on R-groups for unitary principal series of Spin.