Affine Hecke Algebras and Orthogonal Polynomials

Over the last fifteen years or so, there has emerged a satisfactory and coherent

theoryoforthogonalpolynomialsinseveralvariables,attachedtorootsystems,

and depending on two or more parameters. At the present stage of its develop-

ment, it appears that an appropriate framework for its study is provided by the

notionofanaffinerootsystem:toeachirreducibleaffinerootsystem S thereare

associated several families of orthogonal polynomials (denoted by E λ , P λ , Q λ ,

P (ε)

λ

in this book). For example, when S is the non-reduced affine root system

of rank 1 denoted here by (C ∨

1 ,C 1 ), the polynomials P λ are the Askey-Wilson

polynomials [A2] which, as is well-known, include as special or limiting cases

all the classical families of orthogonal polynomials in one variable.

I have surveyed elsewhere [M8] the various antecedents of this theory: sym-

metric functions, especially Schur functions and their generalizations such as

zonal polynomials and Hall-Littlewood functions [M6]; zonal spherical func-

tions on p-adic Lie groups [M1]; the Jacobi polynomials of Heckman and

Opdam attached to root systems [H2]; and the constant term conjectures of

Dyson, Andrews et al. ([D1], [A1], [M4], [M10]). The lectures of Kirillov [K2]

also provide valuable background and form an excellent introduction to the

subject.

Thetitleofthismonographisthesameasthatofthelecture[M7].Thatreport,

for obvious reasons of time and space, gave only a cursory and incomplete

overview of the theory. The modest aim of the present volume is to fill in the

gaps in that report and to provide a unified foundation for the theory in its

present state.

The decision to treat all affine root systems, reduced or not, simultaneously

onthesamefootinghasresultedinanunavoidablycomplexsystemofnotation.

In order to formulate results uniformly it is necessary to associate to each affine

root system S another affine root system S ? (which may or may not coincide

with S), and to each labelling ( § 1.5) of S a dual labelling of S ? .