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

Abstract

Given a connected simply connected semisimple group G G and a connected spherical subgroup K ⊆ G K\subseteq G we determine the generators of the extended weight monoid of G / K G/K , based on the homogeneous spherical datum of G / K G/K . Let H ⊆ G H\subseteq G be a reductive subgroup and let P ⊆ H P\subseteq H be a parabolic subgroup for which G / P G/P is spherical. A triple ( G , H , P ) (G,H,P) with this property is called multiplicity free system and we determine the generators of the extended weight monoid of G / P G/P explicitly in the cases where ( G , H ) (G,H) is strictly indecomposable. The extended weight monoid of G / P G/P describes the induction from H H to G G of an irreducible H H -representation π : H → GL ⁡ ( V ) \pi :H\to \operatorname {GL}(V) whose lowest weight is a character of P P . The space of regular End ⁡ ( V ) \operatorname {End}(V) -valued functions on G G that satisfy F ( h 1 g h 2 ) = π ( h 1 ) F ( g ) π ( h 2 ) F(h_{1}gh_{2})=\pi (h_{1})F(g)\pi (h_{2}) for all h 1 , h 2 ∈ H h_{1},h_{2}\in H and all g ∈ G g\in G is a module over the algebra of H H -biinvariant regular functions on G 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 π \pi can be described as a family of vector-valued orthogonal polynomials with properties similar to Jacobi polynomials.