### 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 language | English (US) |
---|---|

Journal | Memoirs of the American Mathematical Society |

Volume | 146 |

Issue number | 693 |

State | Published - Jul 2000 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Invariant measures for unitary groups associated to Kac-Moody Lie algebras.** / Pickrell, Douglas M.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Invariant measures for unitary groups associated to Kac-Moody Lie algebras

AU - Pickrell, Douglas M

PY - 2000/7

Y1 - 2000/7

N2 - 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.

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

UR - http://www.scopus.com/inward/record.url?scp=33749144329&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749144329&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:33749144329

VL - 146

JO - Memoirs of the American Mathematical Society

JF - Memoirs of the American Mathematical Society

SN - 0065-9266

IS - 693

ER -