Algebras of invariant differential operators on a class of multiplicity free spaces

TL;DR

This paper studies the algebra of invariant differential operators on certain multiplicity free spaces, showing they are quotients of Smith algebras, generalizing known cases like the Weil representation.

Contribution

It proves that the algebra of G'-invariant differential operators on these spaces is a quotient of a Smith algebra, extending previous results to broader classes of representations.

Findings

D(V)^{G'} is a quotient of a Smith algebra over its center.

Generalizes the algebraic structure known from the Weil representation.

Provides new structure results for regular prehomogeneous vector spaces of commutative parabolic type.

Abstract

Let G be a connected reductive algebraic group and let G'=[G,G] be its derived subgroup. Let (G,V) be a multiplicity free representation with a one dimensional quotient (see definition below). We prove that the algebra D(V)^{G'} of G'-invariant differential operators with polynomial coefficients on V, is a quotient of a so-called Smith algebra over its center. Over C this class of algebras was introduced by S.P. Smith as a class of algebras similar to the enveloping algebra U(sl(2)) of sl(2). Our result generalizes the case of the Weil representation, where the associative algebra generated by Q(x) and Q(?) (Q being a non degenerate quadratic form on V) is a quotient of U(sl(2)) Other structure results are obtained when (G,V) is a regular prehomogeneous vector space of commutative parabolic type.