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Transfer of R-groups between p-adic inner forms of SL n

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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 d 2, 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.

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Authors and Affiliations

  1. Department of Mathematics, Oklahoma State University, Stillwater, OK, 74078-1058, USA

    Kwangho Choiy

  2. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    David Goldberg

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  1. Kwangho Choiy
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  2. David Goldberg
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Correspondence to Kwangho Choiy.

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Choiy, K., Goldberg, D. Transfer of R-groups between p-adic inner forms of SL n . manuscripta math. 146, 125–152 (2015). https://doi.org/10.1007/s00229-014-0689-3

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