SOME RECURRENCE FORMULAS FOR SPHERICAL POLYNOMIALS ON TUBE DOMAINS

For a tube domain G/K we study the tenser products of two spherical re presentations of the maximal compact group K and the product of the co rresponding spherical polynomials. When one of these is a fundamental representation, we prove that the spherical representations appear wit h multiplicity at most one and we then find all the coefficients in th e recurrence formula for the product of the spherical polynomials. Thi s generalizes the previous result of L. Vretare and proves for certain cases a conjecture of R. Stanley on Jack symmetric polynomials.