k-Dirac operator and parabolic geometries

The principal group of a Klein geometry has canonical left action on the homogeneous space of the geometry and this action induces action on the spaces of sections of vector bundles over the homogeneous space. This paper is about construction of differential operators invariant with respect to the induced action of the principal group of a particular type of parabolic geometry. These operators form sequences which are related to the minimal resolutions of the k-Dirac operators studied in Clifford analysis

Penrose transform and monogenic sections

summary:The Penrose transform gives an isomorphism between the kernel of the $2$-Dirac operator over an affine subset and the third sheaf cohomology group on the twistor space. In the paper we give an integral formula which realizes the isomorphism and decompose the kernel as a module of the Levi factor of the parabolic subgroup. This gives a new insight into the structure of the kernel of the operator

kk-Dirac operator and the Cartan-Kähler theorem

summary:We apply the Cartan-Kähler theorem for the k-Dirac operator studied in Clifford analysis and to the parabolic version of this operator. We show that for $k=2$ the tableaux of the first prolongations of these two operators are involutive. This gives us a new characterization of the set of initial conditions for the 2-Dirac operator

Generalized Cartan geometries and invariant differential operators

We are getting familiar with difficulties with invariance of differential operators in case of parabolic geometries and fully characterize first order invariant operators. We define, so called curved Casimir operator. It is generalization of Casimir operator from representation theory. We give a new prove of characterization of first order invariant operators. We investigate more thoroughly behavior of curved Casimir operator on section of tractor bandle in conformal case and give list of various apllication

Zobecněné Dolbeaultovy komplexy v Cliffordově analýze

In the thesis we study particular sequences of invariant differ- ential operators of first and second order which live on homogeneous spaces of a particular type of parabolic geometries. We show that they form a reso- lution of the kernel of the first operator and that they descend to resolutions of overdetermined, constant coefficient, first order systems of PDE's called the k-Dirac operators. This gives uniform description of resolutions of the k-Dirac operator studied in Clifford analysis. We give formula for second order operators which appear in the resolutions. 1Hlavním tématem této disertační práce jsou konkrétní posloup- nosti invariantních diferenciálních operátorů prvního a druhého řádu, ktere žijí na plochém modelu jednoho typu parabolické geometrie. V práci je ukazáno, že tyto posloupnosti tvoří resolventu jádra prvního operátoru a že přirozeným způsobem indukují resolventy přeurčených systemů parcialních diferenciálních rovnic prvního řadu s konstantními koefiecienty. Tyto systémy jsou v literatuře známy jako k-Diracovy operátory. Takto dostáváme ucelený popis resolvent těchto systémů. V práci je navíc uveden vzorec pro operátory druhého řádu z těchto resolvent. 1Matematický ústav UKMathematical Institute of Charles UniversityFaculty of Mathematics and PhysicsMatematicko-fyzikální fakult