Extended weight semigroups of affine spherical homogeneous spaces of non-simple semisimple algebraic groups

TL;DR

This paper computes the extended weight semigroups for all strictly irreducible affine spherical homogeneous spaces of non-simple semisimple algebraic groups, providing a comprehensive classification and analysis of their representation spectra.

Contribution

It provides the first complete computation of extended weight semigroups for these spaces, including highest weight functions for indecomposable elements.

Findings

Computed extended weight semigroups for all cases

Identified highest weight functions for indecomposable elements

Completed classification for all strictly irreducible affine spherical spaces

Abstract

The extended weight semigroup of a homogeneous space G/H of a connected semisimple algebraic group G characterizes the spectra of the representations of G on the spaces of regular sections of homogeneous linear bundles over G/H, including the space of regular functions on G/H. We compute the extended weight semigroups for all strictly irreducible affine spherical homogeneous spaces G/H, where G is a simply connected non-simple semisimple complex algebraic group and H a connected closed subgroup of it. In all the cases we also find the highest weight functions corresponding to the indecomposable elements of this semigroup. Among other things, our results complete the computation of the weight semigroups for all strictly irreducible simply connected affine spherical homogeneous spaces of semisimple complex algebraic groups.