Transfer of R-groups between p-adic inner forms of SL_n

2012/11/21 by Choiy, Kwangho, Goldberg, David
#11F70 (Primary) 22E35 #22E50 (Secondary) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1211.5054

Abstract

We study the Knapp-Stein R--groups for inner forms of the split group SL_n(F), with F a p--adic field of characteristic zero. Thus, we consider the groups SL_m(D), with D a central division algebra over F of dimension d², and m=n/d. We use the generalized Jacquet-Langlands correspondence and results of the first named author to describe the zeros of Plancherel measures. Combined with a study of the behavior of the stabilizer of representations by elements of the Weyl group we are able to determine the Knapp-Stein R--groups in terms of those for SL_n(F). We show the R--group for the inner form embeds as a subgroup of the R--group for the split form, and we characterize the quotient. We are further able to show the Knapp-Stein R--group is isomorphic to the Arthur, or Endoscopic R--group as predicted by Arthur. Finally, we give some results on multiplicities and actions of Weyl groups on L--packets.

Related

• Behavior of R-groups for p-adic inner forms of quasi-split special unitary groups
• Transfer of Plancherel Measures for Unitary Supercuspidal Representations between p-adic Inner Forms
• Rigid inner forms of real and p-adic groups
• R-groups and geometric structure in the representation theory of SL(N)
• Invariance of R-groups between p-adic inner forms of quasi-split classical groups