Curved Casimir Operators and the BGG Machinery

TL;DR

This paper extends the Casimir operator to invariant differential operators on parabolic geometries, explores their properties, and connects them to BGG sequences, providing new tools for geometric analysis.

Contribution

It introduces a natural extension of the Casimir operator to parabolic geometries and develops the associated invariant operators and splitting operators.

Findings

Construction of invariant differential operators from the Casimir operator

Explicit computation of operator actions on natural bundles

Connection of curved Casimir operators to BGG sequences

Abstract

We prove that the Casimir operator acting on sections of a homogeneous vector bundle over a generalized flag manifold naturally extends to an invariant differential operator on arbitrary parabolic geometries. We study some properties of the resulting invariant operators and compute their action on various special types of natural bundles. As a first application, we give a very general construction of splitting operators for parabolic geometries. Then we discuss the curved Casimir operators on differential forms with values in a tractor bundle, which nicely relates to the machinery of BGG sequences. This also gives a nice interpretation of the resolution of a finite dimensional representation by (spaces of smooth vectors in) principal series representations provided by a BGG sequence.