Orthogonal Polynomials Associated with Root Systems

Abstract

The orthogonal polynomials that are the subject of these lectures are Laurent polynomials in several variables. They depend rationally on two parameters q and t, and there is a family of them attached to each root system R. For particular values of the parameters q and t, these polynomials reduce to objects familiar in representation theory:

• (i) when q = t,they are independent of q and are the Weyl characters for the root system R.

• (ii) when q = 0 they are (up to a scalar factor) the polynomials that give the values of zonal spherical functions on a semisimple p-adic Lie group G relative to a maximal compact subgroup K, such that the restricted root system of (G,K) is the dual root system R.

• (iii) when q and t both tend to 1, in such a way that (1 – t)/(1 – q) tends to a definite limit k , then (for certaion values of k) our polynomials guive the values of zonal spherical functions on a real (compact or noncompact) symmetric space G/K arising from finite-dimensional spherical representations of G, that is to say representations having a non zero K-fixed vector. Here the root system R is the restricted root system of G/K, and the parameter k is half the root multiplicity (assumed to be the same for all restricted roots).