Constant term identities and Poincare polynomials

TL;DR

This paper links Macdonald's constant term identities to Poincare polynomials, uses this connection to prove Kadell's orthogonality conjecture for type A, and introduces new parameters to explore these identities.

Contribution

It introduces extra parameters into Macdonald's identities and uses them to prove Kadell's orthogonality conjecture for type A, connecting constant term identities with Poincare polynomials.

Findings

Proof of Kadell's orthogonality conjecture for type A

Linking Macdonald's identities to Poincare polynomials

Introduction of free parameters in constant term identities

Abstract

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 Poincare polynomials. We then exploit these extra degrees of freedom in the case of type A to give the first proof of Kadell's orthogonality conjecture---a symmetric function generalisation of the q-Dyson conjecture or Zeilberger-Bressoud theorem. Key ingredients in our proof of Kadell's orthogonality conjecture are the polynomial lemma of Karasev and Petrov, the scalar product for Demazure characters and (0,1)-matrices.