Invariant Polynomials for Multiplicity Free Actions

This work concerns linear multiplicity free actions of the complex groups GC = GL(n,C), GL(n,C) × GL(n,C) and GL(2n,C) on the vector spaces V = Sym(n,C), Mn(C) and Skew(2n,C). We relate the canonical invariants in C[V ⊕ V ∗] to spherical functions for Riemannian symmetric pairs (G,K) where G = GL(n,R), GL(n,C) or GL(n,H) respectively. These in turn can be expressed using three families of classical zonal polynomials. We use this fact to derive a combinatorial algorithm for the generalized binomial coefficients in each case. Many of these results were obtained previously by Knop and Sahi using different methods.