Curved Casimir Operators and the BGG Machinery

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.Comment: This is a contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson, published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

Eigenvalues of conformally invariant operators on spheres

Spectrum of a certain class of first order conformally invariant operators on the sphere is explicitly computed. The class contains the (elliptic verions of) Rarita-Schwinger operator and its higher spin analogues.Comment: 14 page

The Gelfand-Tsetlin bases for Hodge-de Rham systems in Euclidean spaces

The main aim of this paper is to construct explicitly orthogonal bases for the spaces of k-homogeneous polynomial solutions of the Hodge-de Rham system in the Euclidean space R^m which take values in the space of s-vectors. Actually, we describe even the so-called Gelfand-Tsetlin bases for such spaces in terms of Gegenbauer polynomials. As an application, we obtain an algorithm how to compute an orthogonal basis of the space of homogeneous solutions of a generalized Moisil-Theodoresco system in R^m.Comment: submitte

Joseph ideals and harmonic analysis for osp(m|2n)

The Joseph ideal in the universal enveloping algebra U(so(m)) is the annihilator ideal of the so(m)-representation on the harmonic functions on R^m-2. The Joseph ideal for sp(2n) is the annihilator ideal of the Segal–Shale–Weil (metaplectic) representation. Both ideals can be constructed in a unified way from a quadratic relation in the tensor algebra of so(m) or sp(2n). In this paper, we construct two analogous ideals for the orthosymplectic Lie super-algebra sop(m|2n) and prove that they have unique characterizations that naturally extend the classical case. Then we show that these two ideals are the annihilator ideals of, respectively, the representation on the spherical harmonics and a generalization of the metaplectic representation to spo(2n|m). This proves that these ideals are reasonable candidates to establish the theory of Joseph-like ideals for Lie super-algebras. We also discuss the relation between the Joseph ideal of osp(m|2n) and the algebra of symmetries of the super-conformal Laplace operator, regarded as an intertwining operator between principal series representations for sop(m|2n) . As a side result, we obtain the proof of a conjecture of Eastwood about the Cartan product of irreducible representations of semisimple Lie algebras made in [10]

Bernstein-Gelfand-Gelfand sequences

This paper is devoted to the study of geometric structures modeled on homogeneous spaces G/P, where G is a real or complex semisimple Lie group and $P\subset G$ is a parabolic subgroup. We use methods from differential geometry and very elementary finite-dimensional representation theory to construct sequences of invariant differential operators for such geometries, both in the smooth and the holomorphic category. For G simple, these sequences specialize on the homogeneous model G/P to the celebrated (generalized) Bernstein-Gelfand-Gelfand resolutions in the holomorphic category, while in the smooth category we get smooth analogs of these resolutions. In the case of geometries locally isomorphic to the homogeneous model, we still get resolutions, whose cohomology is explicitly related to a twisted de Rham cohomology. In the general (curved) case we get distinguished curved analogs of all the invariant differential operators occurring in Bernstein-Gelfand-Gelfand resolutions (and their smooth analogs). On the way to these results, a significant part of the general theory of geometrical structures of the type described above is presented here for the first time.Comment: 45 page