Memoirs of the American Mathematical Society 2000; 125 pp; softcover Volume: 146 ISBN10: 0821820680 ISBN13: 9780821820681 List Price: US$50 Individual Members: US$30 Institutional Members: US$40 Order Code: MEMO/146/693
 The main purpose of this paper is to prove the existence, and in some cases the uniqueness, of unitarily invariant measures on formal completions of groups associated to affine KacMoody algebras, and associated homogeneous spaces. The basic invariant measure is a natural generalization of Haar measure for a simply connected compact Lie group, and its projection to flag spaces is a generalization of the normalized invariant volume element. The other "invariant measures" are actually measures having values in line bundles over these spaces; these bundlevalued measures heuristically arise from coupling the basic invariant measure to Hermitian structures on associated line bundles, but in this infinite dimensional setting they are generally singular with respect to the basic invariant measure. Readership Researchers in Lie groups, representation theory, stochastic analysis and geometry, and conformal field theory. Table of Contents Part I. General Theory  The formal completions of \(G(A)\) and \(G(A)/B\)
 Measures on the formal flag space
Part II. Infinite Classical Groups  Introduction for Part II
 Measures on the formal flag space
 The case \({\mathfrak g}=\)sl\((\infty,\mathbb{C})\)
 The case \({\mathfrak g}=\)sl\((2\infty,\mathbb{C})\)
 The cases \({\mathfrak g}=\) o\((2\infty,\mathbb{C}),o(2\infty+1,\mathbb{C}),\) sp\((\infty,\mathbb)\)
Part III. Loop Groups  Introduction for Part III
 Extensions of loop groups
 Completions of loop groups
 Existence of the measures \(\nu_{\beta,k},\beta>0\)
 Existence of invariant measures
Part IV. Diffeomorphisms of \(S^1\)  Introduction for Part IV
 Completions and classical analysis
 The extension \(\hat{\mathcal{D}}\) and determinant formulas
 The measures \(\nu_{\beta,c,h}, \beta>0,c,h\geq0\)
 On existence of invariant measures
 Concluding comments; acknowledgements
 References
