# The c-function for affine Kac-Moody algebras

http://hdl.handle.net/10150/290700
Title:
The c-function for affine Kac-Moody algebras
Author:
Dang, Son Xuan, 1964-
Issue Date:
1996
Publisher:
The University of Arizona.
Rights:
Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
Abstract:
In this paper, we will first study the Harish-Chandra transform and the c-function for finite type Kac-Moody groups. The Harish-Chandra transform is essentially the Gelfand transform on L¹( K\G/K). From our point of view, the c-function arises as the Fourier transform of the diagonal distribution for Haar measures of K. A brief account of Kac-Moody algebras, especially affine Kac-Moody algebras, is also presented. Then we use a formula of Harish-Chandra for the c-function for finite type Kac-Moody groups to discuss the definition of the c-function for affine Kac-Moody algebras, especially the twisted affine Kac-Moody algebras. It turns out the c-function for an affine Kac-Moody algebra can be written as a product of trigonometric functions over the positive roots of the corresponding finite type Kac-Moody algebra. This finite type Kac-Moody algebra is the Lie algebra of a finite type Kac-Moody group G. Then the c-function can be thought as the Fourier transform of the diagonal distribution for a Haar type measure of G. For the affine Kac-Moody algebra of type A⁽¹⁾₁ , G is SL(2, C) and the measure is (trace(g* g))⁻³. This leads to the question of whether (trace( g* g))⁻ᵐ on SL(n, C) is that measure for the affine Kac-Moody algebra of type A⁽¹⁾(n-1). In the last part, for any positive integer l, the Harish-Chandra transform of (trace(g* g) ⁻⁽⁽ⁿ⁽ⁿ⁻¹⁾⁾/²⁾⁻¹ on SL(n, C) is calculated to check if the Fourier transform of the diagonal distribution for (trace(g* g) ⁻⁽⁽ⁿ⁽ⁿ⁻¹⁾⁾/²⁾⁻¹ is the c-function for the affine Kac-Moody algebra of type A⁽¹⁾(n-1).
Type:
text; Dissertation-Reproduction (electronic)
Keywords:
Mathematics.
Degree Name:
Ph.D.
Degree Level:
doctoral
Degree Program:
Degree Grantor:
University of Arizona
Pickrell, Doug

DC FieldValue Language
dc.language.isoen_USen_US
dc.titleThe c-function for affine Kac-Moody algebrasen_US
dc.creatorDang, Son Xuan, 1964-en_US
dc.contributor.authorDang, Son Xuan, 1964-en_US
dc.date.issued1996en_US
dc.publisherThe University of Arizona.en_US
dc.rightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.en_US
dc.description.abstractIn this paper, we will first study the Harish-Chandra transform and the c-function for finite type Kac-Moody groups. The Harish-Chandra transform is essentially the Gelfand transform on L¹( K\G/K). From our point of view, the c-function arises as the Fourier transform of the diagonal distribution for Haar measures of K. A brief account of Kac-Moody algebras, especially affine Kac-Moody algebras, is also presented. Then we use a formula of Harish-Chandra for the c-function for finite type Kac-Moody groups to discuss the definition of the c-function for affine Kac-Moody algebras, especially the twisted affine Kac-Moody algebras. It turns out the c-function for an affine Kac-Moody algebra can be written as a product of trigonometric functions over the positive roots of the corresponding finite type Kac-Moody algebra. This finite type Kac-Moody algebra is the Lie algebra of a finite type Kac-Moody group G. Then the c-function can be thought as the Fourier transform of the diagonal distribution for a Haar type measure of G. For the affine Kac-Moody algebra of type A⁽¹⁾₁ , G is SL(2, C) and the measure is (trace(g* g))⁻³. This leads to the question of whether (trace( g* g))⁻ᵐ on SL(n, C) is that measure for the affine Kac-Moody algebra of type A⁽¹⁾(n-1). In the last part, for any positive integer l, the Harish-Chandra transform of (trace(g* g) ⁻⁽⁽ⁿ⁽ⁿ⁻¹⁾⁾/²⁾⁻¹ on SL(n, C) is calculated to check if the Fourier transform of the diagonal distribution for (trace(g* g) ⁻⁽⁽ⁿ⁽ⁿ⁻¹⁾⁾/²⁾⁻¹ is the c-function for the affine Kac-Moody algebra of type A⁽¹⁾(n-1).en_US
dc.typetexten_US
dc.typeDissertation-Reproduction (electronic)en_US
dc.subjectMathematics.en_US
thesis.degree.namePh.D.en_US
thesis.degree.leveldoctoralen_US