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 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
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
on the quantized function algebra associated to g (see Theorem
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
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
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
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