Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 3 (2007), 111, 17 pages      arXiv:0708.3180      https://doi.org/10.3842/SIGMA.2007.111
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

Curved Casimir Operators and the BGG Machinery

Andreas Cap a, b and Vladimír Soucek c
a) Fakultät für Mathematik, Universität Wien, Nordbergstr. 15, A-1090 Wien, Austria
b) International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, A-1090 Wien, Austria
c) Mathematical Institute, Charles University, Sokolovská 83, Praha, Czech Republic

Received August 24, 2007, in final form November 16, 2007; Published online November 22, 2007

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.

Key words: induced representation; parabolic geometry; invariant differential operator; Casimir operator; tractor bundle; BGG sequence.

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