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Invariant Measures for Unitary Groups Associated to Kac-Moody Lie Algebras
Doug Pickrell, University Arizona, Tucson, AZ
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Memoirs of the American Mathematical Society
2000; 125 pp; softcover
Volume: 146
ISBN-10: 0-8218-2068-0
ISBN-13: 978-0-8218-2068-1
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 Kac-Moody 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 bundle-valued 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.

Researchers in Lie groups, representation theory, stochastic analysis and geometry, and conformal field theory.

• General introduction
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