Vladimír Souček

Bernstein-Gelfand-Gelfand sequences

The Bernstein-Gelfand-Gelfand sequences extend the complexes of homogeneous vector bundles to curved Cartan geometries.

Fundaments of Hermitean Clifford analysis part II: splitting of h -monogenic equations

Hermitean Clifford analysis focuses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space, where h-monogenicity is expressed by means of two complex and mutually adjoint Dirac operators, which are invariant under the action of a Clifford realization of the unitary group. In part 1 of the article the fundamental elements of the Hermitean setting have been introduced in a natural way, i.e., by introducing a complex structure on the underlying vector space, eventually extended to the whole complex Clifford algebra . The two Hermitean Dirac operators are then shown to originate as generalized gradients when projecting the gradient on invariant subspaces. In this part of the article, the aim is to further unravel the conceptual meaning of h-monogenicity, by studying possible splittings of the corresponding first-order system into independent parts without changing the properties of the solutions. In this way further connections with holomorphic functions of several complex variables are established. As an illustration, we give a full characterization of h-monogenic functions for the case n = 2. During the final redaction of this article, we received the sad news that our friend, colleague and co-author Jarolím Bureš died on 1 October 2006.

The Howe dual pair in Hermitean Clifford analysis

Clifford analysis offers a higher dimensional function theory studying the null solutions of the rotation invariant, vector valued, first order Dirac operator ∂. In the more recent branch Hermitean Clifford analysis, this rotational invariance has been broken by introducing a complex structure J on Euclidean space and a corresponding second Dirac operator ∂J , leading to the system of equations ∂f = 0 = ∂Jf, expressing so-called Hermitean monogenicity. The invariance of this system is reduced to the unitary group. In this paper we show that this choice of equations is fully justified. Indeed, constructing the Howe dual for the action of the unitary group on the space of all spinor valued polynomials, the generators of the resulting Lie superalgebra reveal the natural set of equations to be considered in thiscontext, which exactly coincide with the chosen ones.

Symmetric Analogues of Rarita-Schwinger Equations

In this paper a generalization of the classicalRarita–Schwinger equations for spin 3/2 fields to the case of spin fieldswith values in irreducible representation spaces with weight k+1/2 isgiven. It corresponds to the study of serie of first orderconformal invariant operators, which are constructed from twisted Diracoperators. The representation character of polynomial solutions of the equations onflat space and their relations are described in details.

Dunkl operators and a family of realizations of (1\vert2)

In this paper, a family of radial deformations of the realization of the Lie superalgebra osp(1|2) in the theory of Dunkl operators is obtained. This leads to a Dirac operator depending on 3 parameters. Several function theoretical aspects of this operator are studied, such as the associated measure, the related Laguerre polynomials and the related Fourier transform. For special values of the parameters, it is possible to construct the kernel of the Fourier transform explicitly, as well as the related intertwining operator.

Ricci-corrected derivatives and invariant differential operators☆

Abstract We introduce the notion of Ricci-corrected differentiation in parabolic geometry, which is a modification of covariant differentiation with better transformation properties. This enables us to simplify the explicit formulae for standard invariant operators given in [A. Cap, J. Slovak, V. Soucek, Invariant operators on manifolds with almost hermitian symmetric structures, III. Standard operators, Differential Geom. Appl. 12 (2000) 51–84], and at the same time extend these formulae from the context of AHS structures (which include conformal and projective structures) to the more general class of all parabolic structures (including CR structures).

Gelfand–Tsetlin bases for spherical monogenics in dimension 3

The main aim of this paper is to recall the notion of the Gelfand-Tsetlin bases (GT bases for short) and to use it for an explicit construction of orthogonal bases for the spaces of spherical monogenics (i.e., homogeneous solutions of the Dirac or the generalized Cauchy-Riemann equation, respectively) in dimension 3. In the paper, using the GT construction, we obtain explicit orthogonal bases for spherical monogenics in dimension 3 having the Appell property and we compare them with those constructed by the first and the second author recently (by a direct analytic approach).

Generalized hypercomplex analysis and its integral formulas

The homological version of the Cauchy integral formula is formulated in the paper for solutions of corresponding equations in complexified hypercomplex analysis. Many different cases are treated in unified manner, including some higher order operators. The notion of index of n-cycles is defined in this complexified situation and its properties are studied.

Subcomplexes in curved BGG-sequences

BGG-sequences offer a uniform construction for invariant differential operators for a large class of geometric structures called parabolic geometries. For locally flat geometries, the resulting sequences are complexes, but in general the compositions of the operators in such a sequence are nonzero. In this paper, we show that under appropriate torsion freeness and/or semi-flatness assumptions certain parts of all BGG sequences are complexes. Several examples of structures, including quaternionic structures, hypersurface type CR structures and quaternionic contact structures are discussed in detail. In the case of quaternionic structures we show that several families of complexes obtained in this way are elliptic.

Gel’fand‐Tsetlin Procedure for the Construction of Orthogonal Bases in Hermitean Clifford Analysis

In this note, we describe the Gel’fand‐Tsetlin procedure for the construction of an orthogonal basis in spaces of Hermitean monogenic polynomials of a fixed bidegree. The algorithm is based on the Cauchy‐Kovalevskaya extension theorem and the Fischer decomposition in Hermitean Clifford analysis.