Macdonald's Constant Term Conjectures for Exceptional Root Systems

Self-dual Koornwinder-macdonald Polynomials

We prove certain duality properties and present recurrence relations for a four-parameter family of self-dual Koornwinder-Macdonald polynomials. The recurrence relations are used to verify Macdonald's normalization conjectures for these polynomials. 1. Introduction In a to date unpublished but well-known manuscript, Macdonald introduced certain families of multivariable orthogonal polynomials a...

Macdonald's Evaluation Conjectures and Diierence Fourier Transform

On the Affine Analogue of Jack's and Macdonald's Polynomials

Introduction. Jack's and Macdonald's polynomials are an important class of symmetric functions associated to root systems. In this paper we deene and study an analogue of Jack's and Macdonald's polynomials for aane root systems. Our approach is based on representation theory of aane Lie algebras and quantum aane algebras, and follows the ideas of our recent papers EK1,EK2,EK3]. We start with a ...

Constant Term Identities and Poincaré Polynomials

In 1982 Macdonald published his now famous constant term conjectures for classical root systems. This paper begins with the almost trivial observation that Macdonald’s constant term identities admit an extra set of free parameters, thereby linking them to Poincaré polynomials. We then exploit these extra degrees of freedom in the case of type A to give the first proof of Kadell’s orthogonality ...

Representation-theoretic Proof of the Inner Product and Symmetry Identities for Macdonald's Polynomials

This paper is a continuation of our papers EK1, EK2]. In EK2] we showed that for the root system A n1 one can obtain Macdonald's polynomials { a new interesting class of symmetric functions recently deened by I. Macdonald M1] { as weighted traces of intertwining operators between certain nite-dimensional representations of U q sl n. The main goal of the present paper is to use this construction...