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Zbl 0960.43001
Pickrell, Doug
Invariant measures for unitary groups associated to Kac-Moody Lie algebras.
(English)
[J] Mem. Am. Math. Soc. 693, 125 p. (2000). ISSN 0065-9266

The author discusses the problems of existence and uniqueness of invariant measures in the case of groups associated to infinite Kac-Moody algebras, loop groups, and $\text{Diff} (S^1)$. Let $G(A)$ denote the complex algebraic group associated to a Cartan matrix A. The author considers the action of the unitary real form $K(A)$ on the formal completion space ${\cal G}$ of $G(A)$; the space ${\cal G}$ or $G(A)_{\text {formal}}$ is the maximal space on which the matrix coefficients of integrable highest weight representations of $G(A)$ can be defined. Let $C_0$ denote the identity component of the center of $G(A)$. The author proves his conjecture of existence of a $K(A)$ bi-invariant measure on ${\cal G}/C_0$. The considerations in this paper are by no means uniform. Each case has to be discussed and proved separately; even in the class of affine groups the group $SL(\infty)$ plays a special role. The following table of contents gives an impression of what is going on: General introduction: Some conjectures on the existence of invariant measures -- Basic intuition and compactness -- Comments on contents and organization. Part I. General Theory. Chapter 1. The formal completions of $G(A)$ and $G(A)/B$. Chapter 2. Measures on the formal flag space. Part II. Infinite classical groups. Chapter 0. Introduction to Part II. Chapter 1. Measures on the formal flag space. Chapter 2. The case ${\germ g}=sl (\infty, \bbfC)$. Chapter 3. The case ${\germ g}=sl(2\infty, \bbfC)$. Chapter 4. The cases ${\germ g}=0(2 \infty,\bbfC)$, $0(2\infty+ 1,\bbfC)$, $sp(\infty,\bbfC)$. Part III. Loop groups. Chapter 0. Introduction to Part III. Chapter 1. Extensions of loop groups. Chapter 2. Completions of loop groups. Chapter 3. Existence of the measure $\nu_{\beta,k}$, $\beta>0$. Chapter 4. Existence of invariant measures. Part IV. Diffeomorphisms of $S^1$. Chapter 0. Introduction to Part IV. Chapter 1. Completions and classical analysis. Chapter 2. The extension $\widehat D$ and determinant formulas. Chapter 3. The measures $\nu_{\beta,e,h}$, $\beta>0,e,h\ge 0$. Chapter 4. On the existence of invariant measures. Concluding comments. Acknowledgements. References.
[H.Boseck (Greifswald)]
MSC 2000:
*43A05 Measures on groups, etc.
28C10 Set functions and measures on topological groups
58D05 Groups of diffeomorphisms and homeomorphisms as manifolds
58B25 Group structures and generalizations on infinite-dim. manifolds
17B67 Kac-Moody algebras
22E67 Loop groups and related constructions

Keywords: invariant measures; Kac-Moody algebra; loop groups

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