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

Kwangho Choiy; David Goldberg

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

Representation Theory 2012-11-27 v2

Authors: Kwangho Choiy , David Goldberg

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.

Keywords

knapp-stein r-groups inner forms p-adic fields jacquet-langlands correspondence plancherel measures weyl groups l-packets

Cite

@article{arxiv.1211.5054,
  title  = {Transfer of R-groups between p-adic inner forms of SL_n},
  author = {Kwangho Choiy and David Goldberg},
  journal= {arXiv preprint arXiv:1211.5054},
  year   = {2012}
}

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

Abstract

Keywords

Cite

Related papers