No abstract

In this paper we investigate a generalization of the classical Rarita-Schwinger equations for spin 3/2 fields to the case of functions taking values in irreducible representation spaces with weight k+1/2. These fields may be realised as functions taking values in spaces of spherical monogenics earlier considered in F. Sommen and N. Van Acker (1993, in ''Clifford Algebras and Their Applications in Mathematical Physics,'' Kluwer Academic, Dordrecht/Norwell, MA). In this paper we develop the main function theoretic results.

Hermitean Clifford analysis focusses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space. Monogenicity is expressed here by means of two complex mutually adjoint Dirac operators, which are invariant under the action of a Clifford realisation of the unitary group. In this contribution we present a deeper insight in the transition from the orthogonal setting to the Hermitean one. Starting from the orthogonal Clifford setting, by simply introducing a so-called complex structure J ∈ SO(2n; R), the fundamental elements of the Hermitean setting arise in a quite natural way. Indeed, the corresponding projection operators 1/2 (1 ± iJ) project the initial basis (eα, α = 1, . . . , 2n) onto the Witt basis and moreover give rise to a direct sum decomposition of C C 2n into two components, where the SO(2n; R)-elements leaving those two subspaces invariant, commute with the complex structure J. They generate a subgroup which is doubly covered by a subgroup of Spin(2n; R), denoted SpinJ (2n; R), being isomorphic with the unitary group U(n; C). Finally the two Hermitean Dirac operators are shown to originate as generalized gradients when projecting the gradient on the invariant subspaces mentioned, which actually implies their invariance under the action of SpinJ (2n; R). The eventual goal is to extend the complex structure J to the whole Clifford algebra C C 2n , in order to conceptually unravel the true meaning of Hermitean monogenicity and its connections to orthogonal monogenicity.

We initiate a new study of differential operators with symmetries and combine this with the study of branching laws for Verma modules of reductive Lie algebras. By the criterion for discretely decomposable and multiplicity-free restrictions of generalized Verma modules [T. Kobayashi, Transf. Groups (2012)], we are brought to natural settings of parabolic geometries for which there exist unique equivariant differential operators to submanifolds. Then we apply a new method (F-method) relying on the Fourier transform to find singular vectors in generalized Verma modules, which significantly simplifies and generalizes many preceding works. In certain cases, it also determines the Jordan-Hölder series of the restriction for singular parameters. The F-method yields an explicit formula of such unique operators, for example, giving an intrinsic and new proof of Juhl's conformally invariant differential operators [Juhl, Progr. Math. 2009] and its generalizations to spinor bundles. This article is the first in the series, and the next ones include their extension to curved cases together with more applications of the F-method to various settings in parabolic geometries.