Affine spherical homogeneous spaces with good quotient by a maximal unipotent subgroup

TL;DR

This paper investigates the conditions under which the factorization morphism of an affine spherical homogeneous space by a maximal unipotent subgroup is equidimensional, linking it to properties of the space's weight semigroup.

Contribution

It establishes a precise criterion connecting the equidimensionality of the morphism to the simplicity of the weight semigroup in affine spherical homogeneous spaces.

Findings

The morphism is equidimensional if and only if the weight semigroup satisfies a specific simple condition.

Provides a characterization of affine spherical homogeneous spaces with good quotients by maximal unipotent subgroups.

Links geometric properties of the quotient to algebraic properties of the weight semigroup.

Abstract

For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies some simple condition.