Equivariant degenerations of spherical modules for groups of type A

TL;DR

This paper proves that the moduli scheme classifying certain affine G-varieties related to spherical modules is isomorphic to an affine space when the group G is of type A and the monoid is derived from a spherical G-module.

Contribution

It establishes that the moduli scheme M_S is an affine space specifically for spherical G-modules of type A, extending understanding of equivariant degenerations.

Findings

M_S is isomorphic to an affine space for type A groups.

The result applies to spherical G-modules with specific weight monoids.

Provides new insights into the structure of equivariant degenerations.

Abstract

Let G be a complex reductive algebraic group. Fix a Borel subgroup B of G, with unipotent radical U, and a maximal torus T in B with character group X(T). Let S be a submonoid of X(T) generated by finitely many dominant weights. V. Alexeev and M. Brion introduced a moduli scheme M_S which classifies pairs (X,f) where X is an affine G-variety and f is a T-equivariant isomorphism between the categorical quotient of X by U and the toric variety determined by S. In this paper, we prove that M_S is isomorphic to an affine space when S is the weight monoid of a spherical G-module with G of type A.