On the extended weight monoid of a spherical homogeneous space and its applications to spherical functions

Given a connected simply connected semisimple group G and a connected spherical sub group K subset of G we determine the generators of the ex-tended weight monoid ofG/K, based on the homogeneous spherical datum of G/K .Let H subset of G be a reductive subgroup and let P subset of H be a parabolic sub group for which G/Pis spherical. A triple (G, H, P) with this property is called multiplicity free system and we determine the generators of the extended weight monoid of G/P explicitly in the cases where (G, H) is strictly indecomposable. The extended weight monoid of G/P describes the induction from H to G of an irreducible H-representation pi:H -> GL(V) whose lowest weight isa character ofP. The space of regular End(V)-valued functions on G that satisfy F (h(1)gh(2))=pi(h(1))F(g)pi(h(2)) for all h(1), h (2)is an element of Hand all g is an element of G is a module over the algebra of H-biinvariant regular functions on G. We show that under a mild assumption this module is freely and finitely generated. As a consequence the spherical functions of such a type pi can be described as a family of vector-valued orthogonal polynomials with properties similar to Jacobi polynomials