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- W4385068314 abstract "Let $(G,G_1)=(G,(G^sigma)_0)$ be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces $D_1=G_1/K_1subset D=G/K$, realized as bounded symmetric domains in complex vector spaces ${mathfrak p}^+_1:=({mathfrak p}^+)^sigmasubset{mathfrak p}^+$ respectively. Then the universal covering group $widetilde{G}$ of $G$ acts unitarily on the weighted Bergman space ${mathcal H}_lambda(D)subset{mathcal O}(D)={mathcal O}_lambda(D)$ on $D$ for sufficiently large $lambda$. Its restriction to the subgroup $widetilde{G}_1$ decomposes discretely and multiplicity-freely, and its branching law is given explicitly by Hua-Kostant-Schmid-Kobayashi's formula in terms of the $widetilde{K}_1$-decomposition of the space ${mathcal P}({mathfrak p}^+_2)$ of polynomials on ${mathfrak p}^+_2:=({mathfrak p}^+)^{-sigma}subset{mathfrak p}^+$. The object of this article is to understand the decomposition of the restriction ${mathcal H}_lambda(D)|_{widetilde{G}_1}$ by studying the weighted Bergman inner product on each $widetilde{K}_1$-type in ${mathcal P}({mathfrak p}^+_2)subset{mathcal H}_lambda(D)$. For example, by computing explicitly the norm $Vert fVert_lambda$ for $f=f(x_2)in{mathcal P}({mathfrak p}^+_2)$, we can determine the Parseval-Plancherel-type formula for the decomposition of ${mathcal H}_lambda(D)|_{widetilde{G}_1}$. Also, by computing the poles of $langle f(x_2),{rm e}^{(x|overline{z})_{{mathfrak p}^+}}rangle_{lambda,x}$ for $f(x_2)in{mathcal P}({mathfrak p}^+_2)$, $x=(x_1,x_2)$, $zin{mathfrak p}^+={mathfrak p}^+_1oplus{mathfrak p}^+_2$, we can get some information on branching of ${mathcal O}_lambda(D)|_{widetilde{G}_1}$ also for $lambda$ in non-unitary range. In this article we consider these problems for all $widetilde{K}_1$-types in ${mathcal P}({mathfrak p}^+_2)$." @default.
- W4385068314 created "2023-07-23" @default.
- W4385068314 creator A5078179589 @default.
- W4385068314 date "2023-07-21" @default.
- W4385068314 modified "2023-09-30" @default.
- W4385068314 title "Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups" @default.
- W4385068314 doi "https://doi.org/10.3842/sigma.2023.049" @default.
- W4385068314 hasPublicationYear "2023" @default.
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