Invariant differential operators on a class of multiplicity free spaces

TL;DR

This paper characterizes the algebra of invariant differential operators on certain multiplicity free spaces, showing it is a quotient of a Smith algebra with explicitly described relations.

Contribution

It provides explicit generators and relations for the algebra of invariant differential operators on multiplicity free spaces with a one-dimensional quotient.

Findings

The algebra $D(V)^{G'}$ is a quotient of a Smith algebra.

Explicit generators and relations for $D(V)^{G'}$ are given.

The structure of the algebra is fully described in terms of a two-sided ideal.

Abstract

If $(G,V)$ is a multiplity free space with a one dimensional quotient we give generators and relations for the non-commutative algebra $D(V)^{G'}$ of invariant differential operators under the semi-simple part $G'$ of the reductive group $G$. More precisely we show that $D(V)^{G'}$ is the quotient of a Smith algebra by a completely described two-sided ideal.