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

IF 0.9 3区 物理与天体物理 Q2 MATHEMATICS Symmetry Integrability and Geometry-Methods and Applications Pub Date : 2022-07-24 DOI:10.3842/SIGMA.2023.049
Ryosuke Nakahama
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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 ${\mathcal P}({\mathfrak p}^+_2)$.
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有界对称域上加权Bergman内积的计算及子群下的Parseval Plancherel型公式
设$(G,G_1)=(G,(G^\sigma)_0)$是全纯型对称对,我们考虑了复向量空间${\mathfrak p}^+_1:=({\mathfrak p}^+)^\sigma\subset{\mathfrak p}^+$中的一对Hermitian对称空间$D_1=G_1/K_1\subset D=G/K$。则$G$的泛覆盖群$\widetilde{G}$对$D$上的加权Bergman空间${\mathcal H}_\lambda(D)\subet{\mathcalO}。它对子群$\widetilde的限制{G}_1$是离散的和多重的自由分解,它的分支律是由Hua Kostant-Schmid Kobayashi公式以$\宽分形式明确给出的{K}_1$-空间${\mathcalP}({\mathfrak P}^+_2)$在${\math frak P}^+_2上的分解:=({\math Frak P}^+)^{-\sigma}\subet{\mathfrak P}^+$。本文的目的是理解限制${\mathcal H}_\lambda(D)| _{\widetilde的分解{G}_1}通过研究每个$\宽颚化符上的加权Bergman内积{K}_1$-在${\mathcal P}({\mathfrak P}^+_2)\subet{\mathical H}_\lambda(D)$中键入。例如,通过显式计算$f=f(x_2)\in{\mathcal P}({\math frak P}^+_2)$的范数$\Vert f\Vert_\lambda$,我们可以确定用于分解${\mathcal H}_\lambda(D)|_{\widetilde的Parseval Plancherel型公式{G}_1}$。此外,通过计算$f(x_2)\in{\mathcal p}({\math frak p}^+_2)$,$x=(x_1,x_2)$,$z\in{\mathfrak p}^+={\math Frak p}^+_1\oplus{\mashfrak p}^+_2$的极点,我们可以得到关于${数学O}_\lambda(D)|_{\wide波浪号{G}_1}$也适用于非酉范围中的$\lambda$。在本文中,我们考虑所有$\widetilde的这些问题{K}_1$-在${\mathcal P}({\math frak P}^+_2)$中键入。
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来源期刊
CiteScore
1.80
自引率
0.00%
发文量
87
审稿时长
4-8 weeks
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.
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