Spherical functions on affine Lie groups
Abstract
We describe vector valued conjugacy equivariant functions on a group K in two cases  K is a compact simple Lie group, and K is an affine Lie group. We construct such functions as weighted traces of certain intertwining operators between representations of K. For a compact group $K$, PeterWeyl theorem implies that all equivariant functions can be written as linear combinations of such traces. Next, we compute the radial parts of the Laplace operators of $K$ acting on conjugacy equivariant functions and obtain a comple tely integrable quantum system with matrix coefficients, which in a special case coincides with the trigonometric CalogeroSutherlandMoser multiparticle system. In the affine Lie group case, we prove that the space of equivariant functions having a fixed homogeneity degree with respect to the action of the center of the group is finitedimensional and spanned by weighted traces of intertwining operators. This space coincides with the space of WessZuminoWitten conformal blocks on an elliptic curve. We compute the radial part of the second order Laplace operator on the affine Lie group acting on equivariant functions, and find that it is a certain parabolic partial differential operator, which degenerates to the elliptic CalogeroSutherlandMoser hamiltonian as the central charge tends to minus the dual Coxeter number (the critical level). Quantum integrals of this hamiltonian are obtained as radial part of the higher Sugawara operators which are central at the critical level.
 Publication:

arXiv eprints
 Pub Date:
 July 1994
 arXiv:
 arXiv:hepth/9407047
 Bibcode:
 1994hep.th....7047E
 Keywords:

 High Energy Physics  Theory;
 Mathematics  Quantum Algebra
 EPrint:
 30 pages. Minor misprints and errors in Def. 1.3, 2.2, formulas after (2.8), and Lemma 3.9 were corrected