论文标题
实际还原性谎言组和梯度图的射影表示
Projective representations of real reductive Lie groups and the gradient map
论文作者
论文摘要
令$ g $为连接的半胶合noncampact真实谎言组，让$ρ：g \ longrightArrow \ mathrm {sl}（v）$是有限维矢量空间$ v $ over $ \ mathbb r $的代表，$ \ mathbb r $，$ \ mathrm中的$ \ \ m m mathrm cleash $ \ m m iallm {sl} $ {v）$。用$ρ（g）$识别$ g $，我们假设存在$ k $ invariant scalar产品$ \ mathtt g $，以至于$ g = k = k \ exp（\ mathfrak p）$，其中$ k = \ mathrm {so}（so}（v，v，\ mathtt g） g）\ cap \ mathfrak g $和$ \ mathfrak g $表示$ g $的lie代数。这里$ \ mathrm {sym} _o（v，\ mathtt g）$表示具有跟踪零的对称内态。使用$ g $ - 级别的地图技术，我们分析了$ \ Mathbb p（v）$上的$ g $的天然投影表示形式。
Let $G$ be a connected semisimple noncompact real Lie group and let $ρ: G \longrightarrow \mathrm{SL}(V)$ be a representation on a finite dimensional vector space $V$ over $\mathbb R$, with $ρ(G)$ closed in $\mathrm{SL}(V)$. Identifying $G$ with $ρ(G)$, we assume there exists a $K$-invariant scalar product $\mathtt g$ such that $G=K\exp(\mathfrak p)$, where $K=\mathrm{SO}(V,\mathtt g)\cap G$, $\mathfrak p=\mathrm{Sym}_o (V,\mathtt g)\cap \mathfrak g$ and $\mathfrak g$ denotes the Lie algebra of $G$. Here $\mathrm{Sym}_o (V,\mathtt g)$ denotes the set of symmetric endomorphisms with trace zero. Using the $G$-gradient map techniques we analyze the natural projective representation of $G$ on $\mathbb P(V)$.
