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Curved Casimir Operators and the BGG Machinery
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- Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
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- Symmetry, Integrability and Geometry: Methods and Applications, 2007, том 3
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Curved Casimir Operators and the BGG Machinery
Інший ID:
2000 Mathematics Subject Classification: 22E46; 53A40; 53C15; 58J70
Посилання:
Curved Casimir Operators and the BGG Machinery / A. Cap, V. Soucek // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 18 назв. — англ.
Підтримка:
This paper is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson. The idea to study the Casimir operator on tractor bundle valued forms grew out of questions by M. Cowling and by P. Julg, who conjectured Corollary 1 for the case of the trivial representation. We are very grateful to them for drawing our attention to this problem. Our thanks also go to the anonymous referees for helpful suggestions and corrections. Most of the work was done during meetings of authors at the Erwin Schr¨odinger Institute for Mathematical Physics in Vienna. First author supported by project P19500–N13 of the Fonds zur F¨orderung der wissenschaftlichen Forschung (FWF). The second author thanks the grant GACR Nr. 201/05/2117 and the institutional grant MSM 0021620839 for their support.
Дата:
2007
Переглядів:
665
Завантажень:
382
Анотація:
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.
