On the extended weight monoid and its applications to orthogonal polynomials

TL;DR

This paper characterizes the generators of the extended weight monoid for certain homogeneous spaces and demonstrates how these structures lead to families of matrix-valued orthogonal polynomials.

Contribution

It explicitly determines the generators of the extended weight monoid for G/K and G/P in the context of spherical homogeneous spaces, linking algebraic structures to orthogonal polynomials.

Findings

Generators of the extended weight monoid are explicitly determined.

Modules of regular End(V)-valued functions are shown to be finitely generated.

Spherical functions are described as matrix-valued orthogonal polynomials.

Abstract

Given a connected simply connected semisimple group G and a connected spherical subgroup K we determine the generators of the extended weight monoid of G/K, based on the homogeneous spherical datum of G/K. Let H be a reductive subgroup of G and let P be a parabolic subgroup of H 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 V whose lowest weight is a character of P. The space of regular End(V)-valued functions on G that satisfy F(hgk)=hF(g)k for all h,k 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 V can be described as a family of matrix-valued orthogonal polynomials.