辛向量空间上的矩映射与振子表示
The moment map on symplectic vector space and oscillator representation
摘要：
本文的目的是表明，正则量子化的矩映射的辛向量空间自然产生的振子表示。更精确地说，令$(W,ω)$表示一个实辛向量空间，在该空间上李群$G$在左辛作用，其中$G$表示本文中的实约化李群$\mathrm{Sp}(n,\mathbb R), \mathrm U(p,q)$或$\mathrm O^*(2n)$。然后我们对矩映射$μ: W \to \mathfrak g_0^*$进行映射，其中$\mathfrak g_0^*$表示$G$的李代数$\mathfrak g_0$的对偶空间.也就是说，在取$W$的复化的复拉格朗日子空间$V$之后，对于每个$X \in \mathfrak g_0$，我们将$V$的Weyl代数的元素分配给$< μ, X >$，我们将其表示为$< \hatμ, X >$。它表明，映射$X \mapsto \mathrm i <\hatμ, X >$给出了一个表示的$\mathfrak g_0$，它扩展到一个$\mathfrak g$，复合$\mathfrak g_0$，通过线性。在每种情况下，通过适当选择复拉格朗日子空间$V$，该表示与$\mathfrak g$的振子表示一致。
英文摘要：
The aim of this paper is to show that the canonical quantization of the moment maps on symplectic vector spaces naturally gives rise to the oscillator representations. More precisely, let $(W,ω)$ denote a real symplectic vector space, on which a Lie group $G$ acts symplectically on the left, where $G$ denotes a real reductive Lie group $\mathrm{Sp}(n,\mathbb R), \mathrm U(p,q)$ or $\mathrm O^*(2n)$ in this paper. Then we quantize the moment map $μ: W \to \mathfrak g_0^*$, where $\mathfrak g_0^*$ denotes the dual space of the Lie algebra $\mathfrak g_0$ of $G$. Namely, after taking a complex Lagrangian subspace $V$ of the complexification of $W$, we assign an element of the Weyl algebra for $V$ to $< μ, X >$, which we denote by $< \hatμ, X >$, for each $X \in \mathfrak g_0$. It is shown that the map $X \mapsto \mathrm i <\hatμ, X >$ gives a representation of $\mathfrak g_0$ which extends to the one of $\mathfrak g$, the complexification of $\mathfrak g_0$, by linearity. With a suitable choice of the complex Lagrangian subspace $V$ in each case, the representation coincides with the oscillator representation of $\mathfrak g$.
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