Matrix-valued Berezin kernels

Dijk, Gerrit van; Pevzner, M.

doi:10.4064/bc55-0-13

Cited by 2 publications

(1 citation statement)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

View full text Add to dashboard Cite

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 .

show abstract

Section: Introductionmentioning

confidence: 99%

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

Nakahama¹

2023

SIGMA

View full text Add to dashboard Cite

show abstract

Covariant quantization: spectral analysis versus deformation theory

Pevzner

2008

Jpn. J. Math.

View full text Add to dashboard Cite

Formal deformation or rather symbolic calculus? To which extent do these approaches complete each other in the study of symmetry-preserving quantization procedures for homogeneous spaces? The representation theory of underlying Lie groups shows that the answer is much more delicate than initially thought and that it cannot be always reduced to asymptotic expansions with respect to some Planck's constant. The main goal of this survey is to give hints regarding the aims of each approach and, on the domain where these intersect, to compare the answers they lead to.

show abstract

Matrix-valued Berezin kernels

Cited by 2 publications

References 9 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

Covariant quantization: spectral analysis versus deformation theory

Contact Info

Product

Resources

About