Base change and theta-correspondences for supercuspidal representations of \(SL(2)\) (Q372651)

In the paper under review the author considers the quadratic base change for supercuspidal representations of the group \(SL(2, F)\) over a \(p\)-adic field \(F\), with \(p\) odd, as a theta-correspondence.NEWLINENEWLINELet \(E/F\) denote a quadratic extension and let \(G = SL(2, F)\). Further, let \(O\) denote the orthogonal group attended to a quadratic form on a four-dimensional vector space \(V\) over \(F\) and of Witt rank one over \(F\). Then \(O(V, F) = O(V)\) contains \(PSL(2, F)\) as a normal subgroup. The author considers the reductive dual pair \((G,O(V ))\) and the smooth Weil representation \(\omega_{\chi}^{\infty}\) attached to a nontrivial additive character \(\chi\) of \(F\). The restriction of \(\omega_{\chi}^{\infty}\) to \(G \times O(V)\) gives rise to a map \(\theta_{\chi} : R_{\chi}(G) \rightarrow R_{\chi}(O(V))\), where \(R_{\chi}(G)\) stands for the set of irreducible smooth representations of \(G\) that occur as quotients of \(\omega_{\chi}^{\infty} |_{G}\) and \(R_{\chi}(O(V))\) is defined similarly. For a representation \(\pi\) of \(G\), let \(L(\pi)\) denote its associated \(L\)-packet and let \(bc(L(\pi))\) denote the \(L\)-packet of \(PSL(2,E)\) obtained from the one of \(L(\pi)\) by base change with respect to \(E\).NEWLINENEWLINEThe author proves that if \(\pi \in R_{\chi}(G)\) is a supercuspidal representation such that \(bc(L(\pi))\) consists of supercuspidal representations, then \(\theta_{\chi}(\pi) |_{PSL(2,E)}\) consists of representations in \(bc(L(\pi))\).