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

Abstract:

Given a connected simply connected semisimple group $G$ and a connected spherical subgroup $K\subseteq G$ we determine the generators of the extended weight monoid of $G/K$, based on the homogeneous spherical datum of $G/K$.

Let $H\subseteq G$ be a reductive subgroup and let $P\subseteq H$ be a parabolic subgroup for which $G/P$ is 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 $\pi :H\to \operatorname {GL}(V)$ whose lowest weight is a character of $P$. The space of regular $\operatorname {End}(V)$-valued functions on $G$ that satisfy $F(h_{1}gh_{2})=\pi (h_{1})F(g)\pi (h_{2})$ for all $h_{1},h_{2}\in H$ and all $g\in G$ 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.