$Sp(d,{\mathbb R})$的三角子群与再生公式
Triangular Subgroups of $Sp(d,{\mathbb R})$ and Reproducing Formulae
摘要：
本文研究Heisenberg群与辛群的半直积$\mathcal{G}={\mathbb H}^d\rtimes Sp(d,{\mathbb R})$的（扩展）亚辛表示。子群$H=Σ\rtimes D$，$Σ$是一个$d\times d$对称矩阵和$D$的封闭子群$GL(d,{\mathbb R})$，是我们的主要关注。我们将给出一个一般设置的再现性，这些群体包括和组装的单一例子中处理的[5]。作为一个副产品，限制在某些类这样的子群的扩展元代数表示是${\mathbb R}^{2d}$的薛定谔表示或${\mathbb R}^d\rtimes D$的小波表示，其中$D$是$GL(d,{\mathbb R})$的闭子群。最后，我们将提供在维度$d=2$中再生$H=Σ\rtimes D$类型群的新例子。
英文摘要：
We consider the (extended) metaplectic representation of the semidirect product $\mathcal{G}={\mathbb H}^d\rtimes Sp(d,{\mathbb R})$ between the Heisenberg group and the symplectic group. Subgroups $H=Σ\rtimes D$, with $Σ$ being a $d\times d$ symmetric matrix and $D$ a closed subgroup of $GL(d,{\mathbb R})$, are our main concern. We shall give a general setting for the reproducibility of such groups which include and assemble the ones for the single examples treated in [5]. As a byproduct, the extended metaplectic representation restricted to some classes of such subgroups is either the Schrödinger representation of ${\mathbb R}^{2d}$ or the wavelet representation of ${\mathbb R}^d\rtimes D$, with $D$ closed subgroup of $GL(d,{\mathbb R})$. Finally, we shall provide new examples of reproducing groups of the type $H=Σ\rtimes D$, in dimension $d=2$.
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