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Dirac operator in contact symplectic parabolic geometry. (English) Zbl 1024.53021
Slovák, Jan (ed.) et al., The proceedings of the 21th winter school “Geometry and physics”, Srní, Czech Republic, January 13-20, 2001. Palermo: Circolo Matemàtico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 69, 97-107 (2002).
The symplectic Dirac operator was introduced by B. Kostant [Symplectic spinors, Symp. Math. 14, 139-152 (1974; Zbl 0321.58015)] in geometric quantization. K. Habermann [Ann. Global Anal. Geom. 13, 155-168 (1995; Zbl 0842.58042)] gave an equivalent definition of the Dirac operator, as the composition of the spinor derivative and Clifford multiplication. A version of this definition is described in the author’s paper [NATO Sci. Ser. II, Math. Phys. Chem. 25, 103-111 (2001; Zbl 1039.58017)] in the orthogonal case.
The present paper extends this approach and defines an analog for the Dirac operator in contact symplectic parabolic geometry. Moreover, by computing the Casimir elements used in J. Slovák and V. Souček [Sémin. Congr. 4, 251-276 (2000; Zbl 0998.53021)], one verifies the invariance of the defined Dirac operator in the context of the contact symplectic parabolic geometry.
For the entire collection see [Zbl 0994.00029].

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53D05 Symplectic manifolds (general theory)