Gan, Wee Teck; Gross, Benedict H.; Prasad, Dipendra Restrictions of representations of classical groups: examples. (English) Zbl 1279.22023 Gan, Wee Teck et al., Sur les conjectures de Gross et Prasad. I. Paris: Société Mathématique de France (SMF) (ISBN 978-2-85629-348-5/pbk). Astérisque 346, 111-170 (2012). In an earlier paper [Astérisque 346, 1–109 (2012; Zbl 1280.22019)], the authors of the paper under review proposed a very influential conjecture on representations of classical groups, which is now known as Gan-Gross-Prasad conjecture. The conjecture has a local part and a global part. In the paper under review, the authors provide strong evidences for the local part of the conjecture, in the case of unitary groups.To be more precise, let \(k/k_0\) be a quadratic field extension of local fields. Let \(V\) be a finite-dimensional Hermitian space over \(k\), and let \(V_0\) be a non-degenerate subspace of it. First assume that the orthogonal complement \(V_0^\perp\) is odd dimensional and contains a split non-degenerate subspace of codimension 1. Write \(G=U(V)\times U(V_0)\) for the product of the unitary groups. Attached to a maximal flag of totally isotropic subspaces of \(V_0^\perp\), there is a certain algebraic subgroup \(H\) of \(G\) whose Levi component equals the group \(U(V_0)\) diagonally embedded in \(G\). Fix a generic character \(\nu\) of \(H\). When \(k\) is non-Archimedean, let \(\pi\) be an irreducible admissible smooth representation of \(G\); when \(k\) is archimedean, let \(\pi\) be an irreducible admissible smooth Fréchet representation of \(G\) of moderate growth, as considered by Casselman-Wallach. At least when \(k\) has characteristic zero, it is now known that the space \(\text{Hom}_H(\pi, \nu)\) is at most one-dimensional. Assume that \(\pi\) belongs to a generic L-packet. The local Gan-Gross-Prasad conjecture predicts that the space \(\text{Hom}_H(\pi, \nu)\) is nonzero if and only if under Vogan-Langlands correspondence, \(\pi\) corresponds to the “distinguished character” of the component group of the L-parameter. A similar conjecture is made when \(V_0^\perp\) is even-dimensional and split. One of the main results of the paper is that if \(\pi\) is a depth zero supercuspidal representation of \(G\) which corresponds to the distinguished character in the Gan-Gross-Prasad conjecture, then \(\text{Hom}_H(\pi, \nu)\) is nonzero. Using theta correspondences and some global arguments, the authors also verify the conjecture in all cases when \(G=U(1)\times U(1)\), \(U(2)\times U(1)\) or \(U(2)\times U(2)\); and in many cases when \(G=U(3)\times U(2)\).For the entire collection see [Zbl 1257.22001]. Reviewer: Binyong Sun (Beijing) Cited in 27 Documents MSC: 22E50 Representations of Lie and linear algebraic groups over local fields 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11R39 Langlands-Weil conjectures, nonabelian class field theory Keywords:Gross-Prasad conjectures; unitary groups; depth-zero supercuspidals; theta correspondence Citations:Zbl 1280.22019 × Cite Format Result Cite Review PDF Full Text: arXiv