Borel-Weil Theory for Loop Groups
Borel-Weil Theory for Loop Groups
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环群的 Borel-Weil 理论
DOI：
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发表时间：
2001
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影响因子：
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通讯作者：
K. Neeb
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文献类型：
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作者：
K. Neeb
设K为紧李群，（LK: = {C^ inty}({mathbb{S}^1}，K)）为值在K中的光滑环路群。这是一个点乘法下的群，它具有在Frechet空间(Lmathfrak{K}: = {C^ inty}（{mathbb{S}^1},mathfrak{K}）上建模的具有值在（mathfrak}的李代数）中的光滑环路的李群的结构。这些注释来自于对Borel-Weil理论的环群证明的重写，正如Pressley和Segal在书中提出的那样（[PS86]）。我们的主要目标是从解析的角度出发，在光滑环的Frechet群的背景下，发展与这一理论相关的技术。我们将描述不可约正能量表示的内在构造，它不涉及环路群嵌入到无限维经典巴纳赫李群中，如[GW84]和[Ner83]。对于一般KacMoody群的Borel-Weil定理的代数版本，我们参考[Ka85b， p. 192]。在[NRW99]中讨论了将bot - borel - weil理论推广到李群的直接极限。在全纯截面的Frechet空间中，群O（∞，C）的自旋表示由Neretin在[Ner87]中构造。
Let K be a compact Lie group and (LK : = {C^infty }({mathbb{S}^1},K)) the group of smooth loops with values in K. This is a group under pointwise multiplication and it carries the structure of a Lie group modeled over the Frechet space (Lmathfrak{k} : = {C^infty }({mathbb{S}^1},mathfrak{k})) of smooth loops with values in the Lie algebra of (mathfrak{k}). These notes grew out of a reworking of the proof of the Borel-Weil theory for loop groups as it is presented in the book of Pressley and Segal ([PS86]). Our main objective is to develop the techniques which are relevant for this theory in the setting of the Frechet groups of smooth loops from an analytic point of view. We will describe an intrinsic construction of the irreducible positive energy representations which does not refer to embeddings of loop groups into infinite-dimensional classical Banach Lie groups as in [GW84] and in [Ner83]. For an algebraic version of a Borel-Weil Theorem for general KacMoody groups, considered as algebraic groups of infinite type, we refer to [Ka85b, p. 192]. Generalizations of Bott-Borel-Weil theory to direct limits of Lie groups are discussed in [NRW99]. A realization of the spin representation of the group O(∞, C) in a Frechet space of holomorphic sections is constructed by Neretin in [Ner87].