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

Abstract

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 ).