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Dual \(R\)-groups of the inner form of \(\mathrm{SL}(N)\). (English) Zbl 1295.22020
This paper studies the \(R\)-groups – both the Knapp-Stein \(R\)-groups defined using intertwining operators and the Arthur \(R\)-groups defined in terms of \(L\)-parameters – of the inner forms of \(\text{SL}(N)\) over a \(p\)-adic field \(F\). The main approach is the restriction from the inner forms of \(\text{GL}(N)\). The authors use the Langlands correspondence for the inner forms of \(\text{SL}(N)\) established by K. Hiraga and H. Saito [Mem. Am. Math. Soc. 1013, iii–v, 97 p. (2012; Zbl 1242.22023)].
Let \(G^\sharp\) be an inner form of \(\text{SL}(N)\). Let \(\sigma^\sharp\) be a square integrable representation of a Levi subgroup of \(G^\sharp(F)\). On the representation-theoretic side, we have the Knapp-Stein \(R\)-group \(R_{\sigma^\sharp}\) and the corresponding central extension \(\widetilde{R}_{\sigma^\sharp}\) defined using the normalized intertwining operators.
On the dual side, \(\sigma^\sharp\) belongs to an \(L\)-packet \(\Pi_{\phi^\sharp}\). We have the \(R\)-group \(R_{\phi^\sharp}\) corresponding to the packet \(\Pi_{\phi^\sharp}\) and the subgroup \(R_{\phi^\sharp, \sigma^\sharp} \subseteq R_{\phi^\sharp}\) attached to \(\sigma^\sharp \in \Pi_{\phi^\sharp}\).
The authors prove that there is a canonical isomorphism \(R_{\sigma^\sharp} \simeq R_{\phi^\sharp, \sigma^\sharp}\), as conjectured by Arthur. In addition, the give a “concrete” description of \(R_{\phi^\sharp}\) and \(R_{\phi^\sharp, \sigma^\sharp}\), and a description of the 2-cocycles attached to \(R\)-groups. Finally, they give examples such that
(i) \(\widetilde{R}_{\sigma^\sharp} \twoheadrightarrow R_{\sigma^\sharp} \) is not split, or
(ii) \(R_{\phi^\sharp, \sigma^\sharp} \neq R_{\phi^\sharp}\).

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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