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**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].

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)

### 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 |