Transfer of \(R\)-groups between \(p\)-adic inner forms of \(\mathrm{SL}_n\)
The study of R-groups is crucial to the understanding of the elliptic tempered spectrum of reductive \(p\)-adic groups. The article focuses on the interesting case of the inner forms of the special linear group. Let \(G=\mathrm{SL}_n(F)\), with \(F\) a \(p\)-adic field of characteristic zero, and let \(G'=\mathrm{SL}_m(D)\), where \(D\) is a central division algebra of degree \(d^2\) over \(F\), and \(md=n\). Let \(\mathrm{tG}=\mathrm{GL}_n(F)\) and \(\mathrm{tG}'=\mathrm{GL}_m(D)\). Let \(\sigma\) be an irreducible discrete series representation of a Levi subgroup \(M\) of \(G\), and let \(W(\sigma)\) denote its stabilizer in the Weyl group of \(M\). We denote by \(W_\sigma'\) the subgroup of \(W(\sigma)\) generated by the root reflections in the zeros of the rank-one Plancherel measures. The Knapp-Stein (or analytic) R-group of \(\sigma\) can be realized as the quotient of \(W(\sigma)\) by \(W'_\sigma\). Let \(\sigma\) be an irreducible discrete series representation of a Levi subgroup \(tM\) of \(\mathrm{tG}\) such that \(\sigma\) occurs in the restriction of \(\sigma\) to \(M=tM\cap G\). We will use analogous notation for the corresponding data attached to \(G'\) instead of \(G\). If \(\sigma\) and \(\sigma'\) correspond under the Jacquet-Langlands correspondence then we have \(W'_\sigma=W'_{\sigma'}\). The inclusion \(W(\sigma')\hookrightarrow W(\sigma)\) induces an inclusion \(R_\sigma\hookrightarrow R_{\sigma'}\). The authors provide a characterization of the quotient \(R_{\sigma'}/R_{\sigma'}\) in Remark 4.21 of the article. Important results of the article (Theorem 4.10 and Theorem 4.16, respectively) show that \(R_{\sigma}\) and \(R_{\sigma'}\) are isomorphic to the Arthur (or geometric) R-group of \(\sigma\) and \(\sigma'\), respectively. Moreover, Theorem 4.19 proves that all the irreducible discrete series representations belonging to a given L-packet of \(G'\) admit the same R-group. The similar result for \(G\) was proved previously by the second author. Some results on multiplicity which arise for the inner forms of \(G\) are discussed in the last section of the article. Close results have been proved independently and almost simultaneously by K. F. Chao and W. W. Li.
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