Multiplicity free spaces with a one dimensional quotient

TL;DR

This paper classifies indecomposable multiplicity free spaces with a one dimensional quotient, exploring their algebraic structure and geometric properties, including regularity and parabolic type, extending previous work by Thierry Levasseur.

Contribution

It provides a classification of indecomposable such spaces and analyzes their algebraic and geometric characteristics, including their relation to Smith algebras and prehomogeneous vector spaces.

Findings

Classification of indecomposable multiplicity free spaces with a one dimensional quotient.

Identification of the algebra of invariant differential operators as a Smith algebra.

Analysis of the regularity and parabolic nature of these spaces.

Abstract

The multiplicity free spaces with a one dimensional quotient were introduced by Thierry Levasseur in [11]. Recently, the author has shown that the algebra of differential operators on such spaces which are invariant under the semi-simple part of the group is a Smith algebra ([17]). We give here the classification of these spaces which are indecomposable, up to geometric equivalence. We also investigate whether or not these spaces are regular or of parabolic type as a prehomogeneous vector space.