论文标题
在子组下的有限对称域和parseval-plancherel-type公式上的加权伯格曼内部产品计算
Computation of Weighted Bergman Inner Products on Bounded Symmetric Domains and Parseval-Plancherel-Type Formulas under Subgroups
论文作者
论文摘要
令$（g，g_1）=（g，（g^σ）_0）$为对称的圆形类型，我们考虑一对遗传学对称空间$ d_1 = g_1/k_1/k_1 \ subset d = g/k $ p}^+）^σ\ subset {\ mathfrak p}^+$。然后，$ g $的通用覆盖组$ \ widetilde {g} $在加权的Bergman空间$ {\ Mathcal H}_λ（d）\ subset {\ Mathcal O}（d）= {\ Mathcal O}_λ（D）$ d $ for $ d $ for for $ $ $λ上。它限制了子组$ \ wideTilde {g} _1 $ $分解和多样性，其分支定律由hua-kostant-schmid-kobayashi的公式明确地给出，以$ \ wideTilde {k} _1 $ -deccol $ -deccort of the Mathcosect of $ \ widectrive of。 $ {\ mathfrak p}^+_ 2：=（{{\ Mathfrak p}^+）^{ - σ} \ subset {\ mathfrak p}^+$ plyenmials的p}^+_ 2）$多项式$。本文的目的是了解限制$ {\ Mathcal H}_λ（d）| _ {\ widetilde {g} _1} $的分解，通过研究每种$ \ widetilde {k} _1 $ -type in $ { p}^+_ 2）\ subset {\ Mathcal H}_λ（d）$。例如，通过明确计算$ f = f = f（x_2）\ in {\ mathcal p} in {\ mathfrak p}^+_ 2）$ in n narm $ \ vert f \ vert f \vert_λ$，我们可以确定parseval-plancherel-type type form $ {\ mather-type typemula $ { h}_λ（d）| _ {\ widetilde {g} _1} $。另外，通过计算$ \ langle f（x_2）的杆子，{\ rm e}^{（x | \ overline {z}）_ {{\ Mathfrak p}^+}} \ rangle_ {λ} $ for $ f（x_2） p}^+_ 2）$，$ x =（x_1，x_2）$，$ z \ in {\ mathfrak p}^+= {\ mathfrak p}^+_ 1 \ oplus {\ mathfrak p}^+_ 2 $ o}_λ（d）| _ {\ widetilde {g} _1} $也适用于非独立范围的$λ$。在本文中，我们考虑了所有$ \ widetilde {k} _1 $ -Types in $ {\ Mathcal P}（{\ Mathfrak P}^+_ 2）$。
Let $(G,G_1)=(G,(G^σ)_0)$ be a symmetric pair of holomorphic type, and we consider a pair of Hermitian symmetric spaces $D_1=G_1/K_1\subset D=G/K$, realized as bounded symmetric domains in complex vector spaces ${\mathfrak p}^+_1:=({\mathfrak p}^+)^σ\subset{\mathfrak p}^+$ respectively. Then the universal covering group $\widetilde{G}$ of $G$ acts unitarily on the weighted Bergman space ${\mathcal H}_λ(D)\subset{\mathcal O}(D)={\mathcal O}_λ(D)$ on $D$ for sufficiently large $λ$. 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 ${\mathfrak p}^+_2:=({\mathfrak p}^+)^{-σ}\subset{\mathfrak p}^+$. The object of this article is to understand the decomposition of the restriction ${\mathcal H}_λ(D)|_{\widetilde{G}_1}$ by studying the weighted Bergman inner product on each $\widetilde{K}_1$-type in ${\mathcal P}({\mathfrak p}^+_2)\subset{\mathcal H}_λ(D)$. For example, by computing explicitly the norm $\Vert f\Vert_λ$ for $f=f(x_2)\in{\mathcal P}({\mathfrak p}^+_2)$, we can determine the Parseval-Plancherel-type formula for the decomposition of ${\mathcal H}_λ(D)|_{\widetilde{G}_1}$. Also, by computing the poles of $\langle f(x_2),{\rm e}^{(x|\overline{z})_{{\mathfrak p}^+}}\rangle_{λ,x}$ 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 ${\mathcal O}_λ(D)|_{\widetilde{G}_1}$ also for $λ$ in non-unitary range. In this article we consider these problems for all $\widetilde{K}_1$-types in ${\mathcal P}({\mathfrak p}^+_2)$.
