Invariant measures for unitary groups associated to Kac-Moody Lie algebras

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8 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
JournalMemoirs of the American Mathematical Society
Volume146
Issue number693
StatePublished - Jul 2000
Externally publishedYes

Fingerprint

Lie groups
Kac-Moody Algebras
Unitary group
Invariant Measure
Algebra
Lie Algebra
Line Bundle
Affine Kac-Moody Algebra
Haar Measure
Compact Lie Group
Homogeneous Space
Completion
Bundle
Uniqueness
Projection
Invariant

Keywords

  • Infinite classical groups
  • Invariant measures
  • Kac-Moody algebras
  • Loop groups
  • Virasoro algebra
  • Wiener measure

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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abstract = "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.",
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AB - 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.

KW - Infinite classical groups

KW - Invariant measures

KW - Kac-Moody algebras

KW - Loop groups

KW - Virasoro algebra

KW - Wiener measure

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