Spherical functions on affine Lie groups
type: Item
instance of: scholarly article
title: Spherical functions on affine Lie groups
zbMATH Open document ID: 0848.43010
DOI: 10.1215/S0012-7094-95-08003-X
published in: Duke Mathematical Journal
publication date: 1996-05-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/hep-th/9407047
review text: A complex valued function \(f\) on a group \(K\) is said to be spherical if the vector space \(V_f\) spanned by the functions \(x \mapsto f(g^{-1} xg)\) is finite dimensional. If \(V\) is a finite dimensional representation of \(K\) the spherical function \(f\) is said to be of type \(V\) if \(V_f\) is isomorphic to \(V\). Further a function \(\Psi : K \to V^*\) (the dual representation of \(V)\) is said to be equivariant if \(g^{-1} \Psi (gxg^{-1}) = \Psi (x)\). Then, for \(v \in V\), the function \(f(x) = \langle v,\Psi (x) \rangle\) is spherical of type \(V\). In the first section one considers the case of a compact Lie group. Let \(V,W\) be finite dimensional representations of \(K\) and \(\Phi : W \to W \otimes V^*\) an intertwining operator, then \(\Psi (x) = Tr |_W (\Phi x)\) is an equivariant function. The restriction of \(\Psi\) to a maximal torus determines \(\Psi\), and, when \(W = N_\lambda\) is a highest weight representation, is the solution of a differential system \(R_V (Y) \Psi = \chi (Y) (\lambda + \rho) \Psi\), where \(R_V (Y)\) is an operator valued radial part associated to an element \(Y \in {\mathcal Z} (g)\), the center of the universal enveloping algebra of \(g = Lie (K)^C\). This theory is a vector valued generalization of the theory of characters (which corresponds to the special case \(V = C)\). It can be applied to the integrability of the Sutherland operator and to the study of the Jack polynomials. In the second part one extends the preceding results to the case of an affine Lie group: \(G\) is a complex simply connected simple Lie group, \(\widehat G\) is the universal extension of the loop group \(LG\), and \(\widetilde G\) is the semi-direct product \(C^* \times \widetilde G\). One proves that the space of equivaruant functions having a fixed homogeneity degree with respect to the action of the center is finite dimensional. One constructs an eigenbasis of the radial part of the second order Laplace operator. This is related to the a…
Mathematics Subject Classification ID: 43A90
Mathematics Subject Classification ID: 22E67
zbMATH DE Number: 850800
zbMATH Keywords: vector space
zbMATH Keywords: finite dimensional representation
zbMATH Keywords: spherical function
zbMATH Keywords: dual representation
zbMATH Keywords: Lie group
zbMATH Keywords: intertwining operator
zbMATH Keywords: universal enveloping algebra
zbMATH Keywords: theory of characters
zbMATH Keywords: Sutherland operator
zbMATH Keywords: Jack polynomials
zbMATH Keywords: loop group
zbMATH Keywords: equivariant functions
zbMATH Keywords: Laplace operator
zbMATH Keywords: Lamé operator
Wikidata QID: Q115240143
author: Pavel Etingof
author: Igor B. Frenkel
author: Alexander jun. Kirillov
reviewed by: Jacques Faraut
MaRDI profile type: Publication
arXiv ID: hep-th/9407047
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