Traces of intertwining operators and Macdonald's polynomials

Abstract

Let $\Phi:V\to V\otimes U$ be an intertwining operator between representations of a simple Lie algebra (quantum group, affine Lie algebra). We define its generalized character to be the following function on the Cartan subalgebra with values in $U$: $\chi_\Phi(h)=\Tr_V (\Phi e^h)$. These generalized characters are a rich source of special functions, possessing many interesting properties; for example, they are common eigenfunctions of a family of commuting differential (difference) operators. We show that the special functions that can be obtained this way include Macdonald's polynomials of type $A$, and this technique allows to prove inner product and symmetry identities for these polynomials (though proved earlier by other methods). Generalized characters for affine Lie algebras are closely related with so-called correlation functions on the torus in the Wess-Zumino-Witten (WZW) model of conformal field theory. We derive differential equations satisfied by these correlation functions (elliptic Knizhnik-Zamolodchikov, or Knizhnik-Zamolodchikov-Bernard equations) and study their monodromies.