One-Dimensional Representations of U (p,q) and the Howe Correspondence

Paul, Annegret; Trapa, Peter E.

doi:10.1006/jfan.2002.3974

Cited by 8 publications

(1 citation statement)

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“…In a previous paper [32], we investigated the case where π is a unitary character (see also [35]) or a holomorphic discrete series with a scalar minimal K -type. For these representations, the multiplicity in the associated cycle is 1, and the situation is much simpler than the present case of (almost all) unitary lowest weight representations.…”

Section: Introductionmentioning

confidence: 99%

Theta lifting of unitary lowest weight modules and their associated cycles

Nishiyama¹,

Zhu²

2004

Duke Math. J.

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Abstract. We consider a reductive dual pair (G, G ) in the stable range with G the smaller member and of Hermitian symmetric type. We study the theta lifting of (holomorphic) nilpotent K C -orbits in relation to the theta lifting of unitary lowest weight representations of G . We determine the associated cycles of all such representations. In particular, we prove that the multiplicity in the associated cycle is preserved under the theta lifting. We also develop a theory for the lifting of covariants arising from double fibrations by affine quotient maps.

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Section: Introductionmentioning

confidence: 99%

Theta lifting of unitary lowest weight modules and their associated cycles

Nishiyama¹,

Zhu²

2004

Duke Math. J.

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Representations with scalarK-types and applications

Zhu

2003

Isr. J. Math.

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

2017

Invent. math.

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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 ...

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One-Dimensional Representations of U (p,q) and the Howe Correspondence

Cited by 8 publications

References 31 publications

Theta lifting of unitary lowest weight modules and their associated cycles

Theta lifting of unitary lowest weight modules and their associated cycles

Representations with scalarK-types and applications

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

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