dc.contributor.advisor Pickrell, Doug en_US dc.contributor.author Dang, Son Xuan, 1964- dc.creator Dang, Son Xuan, 1964- en_US dc.date.accessioned 2013-05-09T11:37:04Z dc.date.available 2013-05-09T11:37:04Z dc.date.issued 1996 en_US dc.identifier.uri http://hdl.handle.net/10150/290700 dc.description.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). dc.language.iso en_US en_US dc.publisher The University of Arizona. en_US dc.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. en_US dc.subject Mathematics. en_US dc.title The c-function for affine Kac-Moody algebras en_US dc.type text en_US dc.type Dissertation-Reproduction (electronic) en_US thesis.degree.grantor University of Arizona en_US thesis.degree.level doctoral en_US dc.identifier.proquest 9720679 en_US thesis.degree.discipline Graduate College en_US thesis.degree.discipline Mathematics en_US thesis.degree.name Ph.D. en_US dc.identifier.bibrecord .b34585229 en_US refterms.dateFOA 2018-08-29T21:31:40Z html.description.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).
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