Invariant Differential Operators for Non-Compact Lie Groups: the Sp(n,R) Case

TL;DR

This paper systematically constructs invariant differential operators for the non-compact Lie algebra sp(6,R), revealing structures similar to conformal algebras in Minkowski space-time, with implications for lower dimensions.

Contribution

It provides a detailed construction of invariant differential operators for sp(6,R), extending the understanding of conformal Lie algebras and their representations.

Findings

Main multiplets and reduced multiplets identified for n=6

Data for invariant differential operators provided

Results applicable to lower n via reduction

Abstract

In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebras sp(n,R), in detail for n=6. Our choice of these algebras is motivated by the fact that they belong to a narrow class of algebras, which we call 'conformal Lie algebras', which have very similar properties to the conformal algebras of Minkowski space-time. We give the main multiplets and the main reduced multiplets of indecomposable elementary representations for n=6, including the necessary data for all relevant invariant differential operators. In fact, this gives by reduction also the cases for n<6, since the main multiplet for fixed n coincides with one reduced case for n+1.