On the Gan–Gross–Prasad conjecture for U(p, q)

Abstract

In this paper, we give a proof of the Gan-Gross-Prasad conjecture for the discrete series of U (p, q). There are three themes in this paper: branching laws of a small A q (λ), branching laws of discrete series and inductive construction of discrete series. These themes are linked together by a reciprocity law and the notion of invariant tensor product. IntroductionIn [GP], Gross and Prasad formulated a number of conjectures regarding the restrictions of generic representations of the special orthogonal groups over a local field. These conjectures related the restriction problem to local root numbers. In [GGP], Gan, Gross and Prasad extended these local conjectures to all classical groups. These local conjectures are known as the local Gan-Gross-Prasad (GGP) conjectures, or Gross-Prasad conjectures. Among the GGP conjectures, there was a very specific and interesting conjecture about the branching law of the discrete series representations for the real groups. Recently, there has been rapid development concerning the local GGP conjectures over the non-Archimedean fields starting with the work of Waldspurger ([W]). As we understand, all cases of the non-Archimedean local conjectures are close to being completely proved, with some standard assumptions. For the Archimedean fields, Gross and Wallach gave a proof of the GrossPrasad conjecture for a class of small discrete series representation of SO(2n + 1). Since then there has not been much progress towards the Gan-Gross-Prasad conjecture over the real numbers. The purpose of this paper is to give a proof of the GGP conjecture for the discrete series representations of U (p, q). We shall also mention the recent work of Zhang that dealt with the global Gan-Gross-Prasad conjecture for the unitary group ([Zhang]).Discrete series of U (p, q) are parametrized by Harish-Chandra parameters. Following [GP], let (χ, z) be a Harish-Chandra parameter for U (p, q) where χ ∈ R p+q is a sequence of distinct integers or half integers and z ∈ {±1}p+q is a sequence of + and − corresponding to each entry in χ. Here the total number of +'s must be p and the total number of −'s must be q. One may also interpret z as a (p, q)-partition of χ. Let D(η, t) be a discrete series representation of U (p − 1, q). The Gan-GrossPrasad conjecture gave a precise description of those D(η, t) that appear as subrepresentations of D(χ, z)| U(p−1,q) . In addition, the multiplicity of these D(η, t) must be all one. Since the discrete spectrum of D(χ, z)| U(p−1,q) only involves the discrete series, GGP conjecture produces a complete * Key word: unitary groups, discrete series, Gross-Prasad conjecture, Howe's correspondence, branching laws, discrete spectrum, Harish-Chandra parameter, representation with non-zero cohomology. To be more precise, GGP conjecture predicts that D(η, t) appears in D(χ, z)| U(p−1,q) if and only if (η, t) and (χ, z) interlace each other in a very specific way. To describe this interlacing relation, let us recall the branching law for the compact group U (p). Let V λ be an ...

Arthur parameters and cuspidal automorphic modules of classical groups

“…We remark that over the real number field R, H. He proves in [26] the local Gan-Gross-Prasad conjecture for discrete representations via a different approach.…”

“…The version of the local Gan-Gross-Prasad conjecture, which will be stated as Conjecture 3.1, was proved for p-adic local fields by Waldspurger and by Moeglin and Waldspurger in a series of papers (see [76] and [63], for instance) for orthogonal groups, while the archimedean case is still in progress. For unitary groups, Beuzart-Plessis ( [8] and [9]) proves the conjecture (Conjecture 3.1) for tempered local L-parameters over all local fields, and in [26], H. He proves the conjecture for discrete representations over R via a different approach. The extension to the generic local L-parameters was obtained by Gan and Ichino ([16]).…”

Deligne's conjecture for automorphic motives over CM-fields

“…The statement in [Har14] has two hypotheses: the first one is the regularity of the highest weight, while the second one (Hypothesis 4.8 of [Har14]) follows as in Remark 4.9 from the assumption that q ‰ n ´1. We remark that the assumption in loc.cit on the Gan-Gross-Prasad multiplicity one conjecture for real unitary groups has been proved by H. He in [He17].…”

“…Hence, for φ P π, there is a canonical decomposition φ | HpA F `q" ř α φ πα and, similarly, φ 2 | H 2 pA F `q" ř β φ π 2 β for φ 2 P π2 . The following theorem is Conjecture 5.2 of [Har13a], which has finally been proved by [Beu-Ple1], [Beu-Ple2] and [He17].…”

Invariant Tensor Product

“…We also study the invariant tensor functor associated with discrete series representations for classical groups. For motivations and applications [2][3][4].…”