On a conjecture of Jacquet about distinguished representations of \(\text{GL}(n)\)
Let \(k\) be a \(p\)-adic field, \(K\) a quadratic unramified extension of \(k\), \(\pi\) a cuspidal irreducible admissible representation of \(\text{GL}(n,K)\) whose central character is trivial on \(k^\times\), \(\check\pi\) its contragredient. Let \(\sigma\) denote the non trivial element of \(\text{Gal}(K/k)\), its componentwise action on \(\text{GL}(n,K)\), and \(^\sigma\pi:g\mapsto\pi(\sigma(g))\). Let \(\omega\) be the non trivial character on \(k^\times/N_{K/k}K^\times\). \(\pi\) is called distinguished if it admits a nonzero \(\text{GL}(n,K)\)-invariant linear form, and \(\omega\)-distinguished if there is a linear form \(L:\pi\to\mathbb C\) with \(L(\pi(g)\xi)=\omega(\text{det}(g))L(\xi)\) for all \(g\) in \(\text{GL}(n,K)\) and \(\xi\in\pi\). The main theorem (Theorem 4) asserts: \({}^\sigma\pi\simeq\check\pi\) iff \(\pi\) is distinguished, or also -- when \(n\) is even -- \(\omega\)-distinguished. The proof is based on a result of Bushnell and Kutzko which realizes any cuspidal representation of \(\text{GL}(n,K)\) by compact induction from a finite dimensional representation of an open compact-mod-center subgroup. It has the advantage of being purely local, although to extend the result to ramified quadratic extensions and to non cuspidal representations one should use the global trace formula. The result is the exact statement which according to the author was conjectured by Jacquet in [\textit{H. Jacquet} and \textit{Y. Ye}, C. R. Acad. Sci. Paris 311, 671--676 (1990; Zbl 0715.11026)]. This conjecture follows from the conjecture submitted earlier (but appeared later) in \textit{Y. Flicker} [J. Reine Angew. Math. 418, 139-172 (1991; Zbl 0725.11026)] characterizing the \(\text{GL}(n,K)\)-distinguished \(\pi\) on \(\text{GL}(n,K)\) as stable or unstable base change lifts from the quasi-split unitary group \(\text{U}(n,K/k)\), and the conjectural theory of base change for \(\text{U}(n,K/k)\). The author also refers to \textit{Y. Flicker} and \textit{J. Hakim} [Am. J. Math. 116, 683--736 (1991; Zbl 0725.11026)], to which the first named author contributed all (the simple trace formula, algebraic cycles) but section D (pp. 726-730) on the relative character. Finally the author makes an interesting ``distinguished'' conjecture on the Steinberg representation of \(G(K)\).
scientific article; zbMATH DE number 1820833
scientific article