Computation of weighted Bergman inner products on bounded symmetric domains and Parseval-Plancherel-type formulas for ($Sp$($r$, $mathbb{R}$), $Sp$($r'$, $mathbb{R}$)$times$$Sp$($r''$, $mathbb{R}$)) (Various Issues on Representation Theory and Related Topics)

Abstract

Let (G, G') = (G, (G[δ]⁻)₀ ) be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces D' = G'/K' ⊂ D = G/K, realized as bounded symmetric domains in complex vector spaces P₁⁺ := (p⁺)[δ] ⊂ p⁺ respectively. Then the universal covering group G~ of G acts unitarily on the weighted Bergman space H[λ](D) ⊂ O(D) = O[λ](D) on D for sufficiently large λ. Its restriction to the subgroup G~' decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua-Kostant-SchmidKobayashi's formula in terms of the K~'-decomposition of the space P(p₂⁺) of polynomials onp₂⁺ := (p⁺)⁻[δ] ⊂ p⁺. Our goal is to understand the decomposition of the restriction H[λ](D)|[G~'] by studying the weighted Bergman inner product on each K~'-type in P(p₂⁺) ⊂ H[λ](D). In this article we mainly deal with the symmetric pair (G, G') = (Sp(r, ℝ), Sp(r', ℝ) x Sp(r'', ℝ)).