Rationality of the Generalized Binomial Coefficients for a Multiplicity Free Action

Let V be a finite dimensional Hermitian vector space and K be a compact Lie subgroup of U(V ) for which the representation of K on C[V ] is multiplicity free. One obtains a canonical basis {pα} for the space C[VR] ofK-invariant polynomials on VR and also a basis {qα} via orthogonalization of the pα’s. The polynomial pα yields the homogeneous component of highest degree in qα. The coefficients that express the qα’s in terms of the pβ ’s are the generalized binomial coefficients of Yan. The main result in this paper shows that these numbers are rational.