Integral formulas for wave functions of quantum many-body problems and representations of gl(n)

We derive explicit integral formulas for eigenfunctions of quantum integrals of the Calogero-Sutherland-Moser operator with trigonometric interaction potential. In particular, we derive explicit formulas for Jack's symmetric functions. To obtain such formulas, we use the representation of these eigenfunctions by means of traces of intertwining operators between certain modules over the Lie algebra $\frak gl_n$, and the realization of these modules on functions of many variables.Comment: 6 pages. One reference ([FF]) has been corrected. New references and an introduction have been adde

Twisted traces of quantum intertwiners and quantum dynamical R-matrices corresponding to generalized Belavin-Drinfeld triples

This paper is a continuation of math.QA/9907181 and math.QA/9908115. We consider traces of intertwiners between certain representations of the quantized enveloping algebra associated to a semisimple complex Lie algebra g, which are twisted by a ``generalized Belavin-Drinfeld triple'', i.e a triple consisting of two subdiagrams of the Dynkin diagram of g together with an isomorphism between them. The generating functions F(lambda,mu) for such traces depend on two weights lambda and mu. We show that F(lambda,mu) satisfy two sets of difference equations in the variable lambda: the Macdonald-Ruijsenaars (MR) equations and the quantum Knizhnik-Zamolodchikov (qKZB) equations. These equations involve as a main ingredient the quantum dynamical R-matrices constructed in math.QA/9912009. When the generalized Belavin-Drinfeld triple is an automorphism, we show that F(lambda,mu) satisfy another two sets of difference equations with respect to the weight mu. These dual MR and dual qKZB equations involve the usual Felder's dynamical R-matrix. These results were first obtained by the first author and A. Varchenko in the special case of the trivial Belavin-Drinfeld triple. However, the symmetry between lambda and mu which exists in that case is destroyed in the twisted setting. At the end, we brielfly treat the (simialr) case of Kac-Moody algebras g and derive the classical limits of all the previous results.Comment: 30 pages, late

On pointed Hopf algebras associated to unmixed conjugacy classes in S_n

Let s in S_n be a product of disjoint cycles of the same length, C the conjugacy class of s and rho an irreducible representation of the isotropy group of s. We prove that either the Nichols algebra B(C, rho) is infinite-dimensional, or the braiding of the Yetter-Drinfeld module is negative

Universal KZB equations I: the elliptic case

We define a universal version of the Knizhnik-Zamolodchikov-Bernard (KZB) connection in genus 1. This is a flat connection over a principal bundle on the moduli space of elliptic curves with marked points. It restricts to a flat connection on configuration spaces of points on elliptic curves, which can be used for proving the formality of the pure braid groups on genus 1 surfaces. We study the monodromy of this connection and show that it gives rise to a relation between the KZ associator and a generating series for iterated integrals of Eisenstein forms. We show that the universal KZB connection realizes as the usual KZB connection for simple Lie algebras, and that in the sl_n case this realization factors through the Cherednik algebras. This leads us to define a functor from the category of equivariant D-modules on sl_n to that of modules over the Cherednik algebra, and to compute the character of irreducible equivariant D-modules over sl_n which are supported on the nilpotent cone.Comment: Correction of reference of Thm. 9.12 stating an equivalence of categories between modules over the rational Cherednik algebra and its spherical subalgebr