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Invariant measures for unitary groups associated to Kac-Moody Lie algebras. (English) Zbl 0960.43001
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.

43A05Measures on groups and semigroups, etc.
28C10Set functions and measures on topological groups or semigroups, etc.
58D05Groups of diffeomorphisms and homeomorphisms as manifolds
58B25Group structures and generalizations on infinite-dimensional manifolds
17B67Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
22E67Loop groups and related constructions, group-theoretic treatment