The c-function for affine Kac-Moody algebras

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).