  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 $K$, Peter-Weyl
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 Calogero-Sutherland-Moser multi-particle
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 finite-dimensional and spanned by weighted traces of
intertwining operators. This space coincides with the space of
Wess-Zumino-Witten 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
Calogero-Sutherland-Moser 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.
