Branching laws for discrete Wallach points

Merigon, Stéphane; Seppänen, Henrik

doi:10.1016/j.jfa.2010.01.025

Cited by 4 publications

(4 citation statements)

References 31 publications

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“…[2,3,4,15,16,32] under different realizations of H λ (D), H ε 1 λ (D 1 , P k(p + 2 )), and those for anti-holomorphic type cases are studied by e.g. [37,42,43,44,46,53,54,55,57,58,59,60]. In our setting, for (G, G 1 ) of holomorphic type, each H ε 1 λ (D 1 , P k(p +…”

Section: ++mentioning

confidence: 99%

Computation of weighted Bergman inner products on bounded symmetric domains and Parseval--Plancherel-type formulas under subgroups

Nakahama¹

2022

Preprint

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Let (G, G 1 ) = (G, (G σ ) 0 ) be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces D 1 = G 1 /K 1 ⊂ D = G/K, realized as bounded symmetric domains in complex vector spaces p + 1 := (p + ) σ ⊂ p + respectively. Then the universal covering group G of G acts unitarily on the weighted Bergman spaceon D for sufficiently large λ. Its restriction to the subgroup G 1 decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua-Kostant-Schmid-Kobayashi's formula in terms of the K 1 -decomposition of the space P(p + 2 ) of polynomials on p + 2 := (p + ) −σ ⊂ p + . The object of this article is to understand the decomposition of the restriction H λ (D)| G1 by studying the weighted Bergman inner product on each K 1 -type in P(p + 2 ) ⊂ H λ (D). For example, by computing explicitly the norm f λ for f = f (x 2 ) ∈ P(p + 2 ), we can determine the Parseval-Plancherel-type formula for the decomposition of H λ (D)| G1 . Also, by computing the poles of f (x 2 ), e (x|z) p + λ,x for f (x 2 ) ∈ P(p + 2 ), x = (x 1 , x 2 ), z ∈ p + = p + 1 ⊕ p + 2 , we can get some information on branching of O λ (D)| G1 also for λ in non-unitary range.In this article we consider these problems for all K 1 -types in P(p + 2 ).

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Section: ++mentioning

confidence: 99%

Computation of weighted Bergman inner products on bounded symmetric domains and Parseval--Plancherel-type formulas under subgroups

Nakahama¹

2022

Preprint

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“…This formula is regarded as an analogue of the Parseval or the Plancherel theorems for the Fourier analysis. Such Parseval-Plancherel-type formulas for symmetric pairs of holomorphic type are studied, e.g., by [3,4,5,16,17,34] under different realizations of H λ (D), H ε 1 λ D 1 , P k p + 2 , and those for antiholomorphic type cases are studied, e.g., by [40,45,46,47,48,56,57,58,60,61,62,63]. In our setting, for (G, G 1 ) of holomorphic type, each H ε 1 λ D 1 , P k p + 2 -isotypic component in H λ (D) is generated by the minimal K 1 -type P k p + 2 ⊂ P(p + ) = H λ (D) K , and we assume that…”

Section: Introductionmentioning

confidence: 99%

Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups

Nakahama¹

2023

SIGMA

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Let (G, G 1 ) = (G, (G σ ) 0 ) be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces D 1 = G 1 /K 1 ⊂ D = G/K, realized as bounded symmetric domains in complex vector spaces p + 1 := (p + ) σ ⊂ p + respectively. Then the universal covering group G of G acts unitarily on the weighted Bergman spaceon D for sufficiently large λ. Its restriction to the subgroup G 1 decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua-Kostant-Schmid-Kobayashi's formula in terms of the K 1 -decomposition of the space P p + 2 of polynomials on p + 2 := (p + ) −σ ⊂ p + . The object of this article is to understand the decomposition of the restriction H λ (D)| G1 by studying the weighted Bergman inner product on each K 1 -type in P p + 2 ⊂ H λ (D). For example, by computing explicitly the norm ∥f ∥ λ for f = f (x 2 ) ∈ P p + 2 , we can determine the Parseval-Plancherel-type formula for the decomposition of H λ (D)| G1 . Also, by computing the poles of f (x 2 ), e (x|z) p + λ,x for f (x 2 ) ∈ P p + 2 ,x = (x 1 , x 2 ), z ∈ p + = p + 1 ⊕ p + 2 , we can get some information on branching of O λ (D)| G1 also for λ in non-unitary range. In this article we consider these problems for all K 1 -types in P p + 2 .

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“…[10,13]. More examples of branching laws for small representations are [5,6,17,20,21,23,25,28,29,30,36].…”

Section: Introductionmentioning

confidence: 99%

Branching laws for small unitary representations ofGL(n, ℂ)

Möllers

Schwarz

2014

Int. J. Math.

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Branching laws for discrete Wallach points

Cited by 4 publications

References 31 publications

Computation of weighted Bergman inner products on bounded symmetric domains and Parseval--Plancherel-type formulas under subgroups

Computation of weighted Bergman inner products on bounded symmetric domains and Parseval--Plancherel-type formulas under subgroups

Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups

Branching laws for small unitary representations ofGL(n, ℂ)

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