Parabolic Verma Modules and Invariant Differential Operators

TL;DR

This paper advances the classification of invariant differential operators by analyzing parabolic Verma modules and their relation to parabolic subalgebras in non-compact semisimple Lie groups, with applications to conformal algebras.

Contribution

It provides a systematic study of parabolic Verma modules and their connection to invariant differential operators, focusing on real and complex semisimple Lie algebras.

Findings

Relation between parabolic subalgebras of real and complex Lie algebras established.

Explicit analysis of conformal algebra in 4D Minkowski space.

Study of minimal parabolics in classical Lie algebras.

Abstract

In the present paper we continue the project of systematic classification and construction of invariant differential operators for non-compact semisimple Lie groups. This time we make the stress on one of the main building blocks, namely the Verma modules and the corresponding parabolic subalgebras. In particular, we start the study of the relation between the parabolic subalgebras of real semisimple Lie algebras and of their complexification. Two cases are given in more detail: the conformal algebra of 4D Minkowski space-time and the minimal parabolics of classical real semisimple Lie algebras.