Let g be a semisimple Lie algebra and θ be an involution of
g. A quantum symmetric pair consists of the quantized enveloping algebra
U for g and a left coideal subalgebra B which is a quantum
analog of U(gθ). This talk includes an introduction to the
coideal subalgebras B. The quantum symmetric space corresponding to
U,B is the set of right B invariants inside the associated quantized
function algebra. Using the quantum Peter-Weyl decomposition and the
classification of finite-dimensional spherical modules associated to U,B
one obtains a nice decomposition of the symmetric space as a left U
module. This in turn implies that the set of B bi-invariants of the
quantum symmetric space is a direct sum of one-dimensional eigenspaces for the
action of the center of U. When the restricted root system for θ is
reduced, we show that the zonal spherical functions, i.e. representations of
each eigenspace, correspond to Macdonald polynomials under the expected
identification.
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