Spherical functions on affine Lie groups
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 affine Jack polynomials. One considers also higher order Laplace operators. In particular, for the affine Lie group \(\widetilde {SL}_2\), one obtains the classical Lamé operator.
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