{"title":"Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups","authors":"Ryosuke Nakahama","doi":"10.3842/SIGMA.2023.049","DOIUrl":null,"url":null,"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 $D_1=G_1/K_1\\subset D=G/K$, realized as bounded symmetric domains in complex vector spaces ${\\mathfrak p}^+_1:=({\\mathfrak p}^+)^\\sigma\\subset{\\mathfrak p}^+$ 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 $\\Vert f\\Vert_\\lambda$ 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 $f(x_2)\\in{\\mathcal P}({\\mathfrak p}^+_2)$, $x=(x_1,x_2)$, $z\\in{\\mathfrak p}^+={\\mathfrak p}^+_1\\oplus{\\mathfrak p}^+_2$, 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 ${\\mathcal P}({\\mathfrak p}^+_2)$.","PeriodicalId":49453,"journal":{"name":"Symmetry Integrability and Geometry-Methods and Applications","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2022-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Symmetry Integrability and Geometry-Methods and Applications","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.3842/SIGMA.2023.049","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
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 $D_1=G_1/K_1\subset D=G/K$, realized as bounded symmetric domains in complex vector spaces ${\mathfrak p}^+_1:=({\mathfrak p}^+)^\sigma\subset{\mathfrak p}^+$ 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 $\Vert f\Vert_\lambda$ 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 $f(x_2)\in{\mathcal P}({\mathfrak p}^+_2)$, $x=(x_1,x_2)$, $z\in{\mathfrak p}^+={\mathfrak p}^+_1\oplus{\mathfrak p}^+_2$, 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 ${\mathcal P}({\mathfrak p}^+_2)$.
期刊介绍:
Scope
Geometrical methods in mathematical physics
Lie theory and differential equations
Classical and quantum integrable systems
Algebraic methods in dynamical systems and chaos
Exactly and quasi-exactly solvable models
Lie groups and algebras, representation theory
Orthogonal polynomials and special functions
Integrable probability and stochastic processes
Quantum algebras, quantum groups and their representations
Symplectic, Poisson and noncommutative geometry
Algebraic geometry and its applications
Quantum field theories and string/gauge theories
Statistical physics and condensed matter physics
Quantum gravity and cosmology.