One particularly interesting consequence of the theory ofsymmetric spaces is that if one integrates a Schur function $s_lambda$(i.e., an irreducible character of the unitary group) over theorthogonal group, the integral vanishes unless all parts of thepartition $lambda$ are even (when the integral is 1). I'll discusssome ``quantum'' generalizations of this fact, in which the Schurfunctions are replaced by Macdonald polynomials; I'll also discussanalogous integrals related to other classical symmetric spaces.