Orthogonal polynomials associated with root systems

TL;DR

This paper introduces a new family of W-invariant orthogonal polynomials associated with root systems, generalizing symmetric functions and connecting to zonal spherical functions on symmetric spaces.

Contribution

It constructs parameter-dependent orthogonal polynomials linked to root systems, extending Macdonald's symmetric functions and encompassing spherical functions on symmetric spaces.

Findings

Polynomials are W-invariant and orthogonal in multiple variables.

Special parameter values recover zonal spherical functions.

Recovers Macdonald's symmetric polynomials for type A_n.

Abstract

Let R and S be two irreducible root systems spanning the same vector space and having the same Weyl group W, such that S (but not necessarily R) is reduced. For each such pair (R,S) we construct a family of W-invariant orthogonal polynomials in several variables, whose coefficients are rational functions of parameters $q,t_1,t_2,...,t_r$, where r (=1,2 or 3) is the number of W-orbits in R. For particular values of these parameters, these polynomials give the values of zonal spherical functions on real and p-adic symmetric spaces. Also when R=S is of type $A_n$, they conincide with the symmetric polynomials described in I. G. Macdonald, Symmetric Functions and Hall Polynomials, 2nd edition, Oxford University Press (1995), Chapter VI.