Introduction

Harmonic analysis on symmetric spaces studies invariant differential operators

and their joint eigenspaces in connection with Lie groups. The discovery of quantum

groups in the 1980's inspired the growing subject of harmonic analysis on quan-

tum symmetric spaces. In particular, zonal spherical functions on most compact

quantum symmetric spaces have been identified with Macdonald or Macdonald-

Koornwinder polynomials ([N], [NS], [S], [L5], [DN], [NDS], and [DS]). In this pa-

per, the focus is on quantum invariant differential operators. We prove a quantum

analog of Harish-Chandra's fundamental result: the Harish-Chandra map induces

an isomorphism between the ring of invariant differential operators on a symmetric

space and invariants of an appropriate polynomial ring under the restricted Weyl

group. We further establish a quantum version of a related theorem due to Helga-

son: the image of the center under this Harish-Chandra map is the entire invariant

ring if and only if the underlying irreducible symmetric pair is not one of four ex-

ceptional types, EIII, EIV, EVII, or EIX. Moreover, we exhibit a nice basis for the

quantum invariant differential operators which corresponds to a special commuting

family of difference operators associated to Macdonald polynomials.

Let 9 be a semisimple complex Lie algebra. A classical (infinitesimal) symmetric

pair is a pair of Lie algebras g,ge where g6 is the Lie subalgebra fixed by an

involution 9. One can associate an infinitesimal symmetric pair to each symmetric

space. In this algebraic framework, invariant differential operators on the symmetric

space correspond to ad g° invariant elements of U(g). Thus a quantum invariant

differential operator should be an element of the quantum analog of the fixed ring

of U(g) under the action of

ge.

More precisely, let U denote the simply connected

quantized enveloping algebra of g over the algebraic closure C of C(q). Quantum

symmetric pairs are defined using left coideal subalgebras B of U which can be

viewed as quantum analogs of the enveloping algebra of ge ([L3, Section 7]). Our

investigation of quantum invariant differential operators is an analysis of the ring

UB of invariants in U with respect to the right adjoint action of B. In analogy to

the classical case, quantum zonal spherical functions are eigenvectors with respect

to the action of

UB

on the quantized function algebra associated to g (see Theorem

1.4).

Let a denote the eigenspace for 9 with eigenvalue —1 inside a maximally split

Cartan subalgebra \) of g. In the classical case, the Harish-Chandra map associated

to

g,ge

is the projection from U(g) onto the enveloping algebra of a, defined using

the Iwasawa decomposition of g. This picture can be lifted to the quantum setting

as follows. The Cartan subalgebra

UQ

of U is the group algebra of a multiplica-

tive group f isomorphic to the weight lattice of g. The restricted root system X

associated to the pair g,ge spans the vector space a*. A quantum analog for a is

a multiplicative subgroup A of f isomorphic to the weight lattice of 2X1. Using a

l