# zbMATH — the first resource for mathematics

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
Full Text: