Infinite Dimensional Kähler Manifolds pp 179-229 | Cite as

# Borel-Weil Theory for Loop Groups

## Abstract

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 \(L\mathfrak{k} : = {C^\infty }({\mathbb{S}^1},\mathfrak{k})\)
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].

## Keywords

Central Extension Holomorphic Section Loop Group Smooth Action Frechet Space## Preview

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