半直积与不变联络
Semidirect products and invariant connections
摘要：
设$S$是一个复约化群，它通过全纯自同构全纯作用在复李群$N$上.设$K(S)\subset S$是极大紧子群.半直积$G := N\rtimes K(S)$通过双全纯作用于$N$。给出了$N$上$G$ -等变几乎全纯埃尔米特主丛的同构类的一个明确描述.在假定$K(S)$的中心子群$Z= \text{U}(1)$作用于$\text{Lie}(N)$上作为通过单个非平凡特征标的乘法的条件下，我们给出了$N$上$G$ -等变全纯埃尔米特主丛的同构类的一个明确描述.
英文摘要：
Let $S$ be a complex reductive group acting holomorphically on a complex Lie group $N$ via holomorphic automorphisms. Let $K(S)\subset S$ be a maximal compact subgroup. The semidirect product $G := N\rtimes K(S)$ acts on $N$ via biholomorphisms. We give an explicit description of the isomorphism classes of $G$-equivariant almost holomorphic hermitian principal bundles on $N$. Under the assumption that there is a central subgroup $Z= \text{U}(1)$ of $K(S)$ that acts on $\text{Lie}(N)$ as multiplication through a single nontrivial character, we give an explicit description of the isomorphism classes of $G$-equivariant holomorphic hermitian principal bundles on $N$.
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