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Projective structure, $$\widetilde{\operatorname{SL}}(3,\mathbb{R})$$ and the symplectic Dirac operator. (English) Zbl 1389.53082
The authors describe a realization of symplectic spinor fields and the symplectic Dirac operator $$D_s$$ in the framework of (the double covering of) the geometry of projective structure in real dimension two. Some recent results on symmetries of the symplectic Dirac operator are also given in [H. De Bie, M. Holíková and P. Somberg, Adv. Appl. Clifford Algebr. 27, No. 2, 1103–1132 (2017; Zbl 1367.53042)]. They introduce the homogeneous projective structure in real dimension two and describe its basic geometrical and representation theoretical properties. They work on the simple metaplectic components of the Segal-Shale-Weil representation (twisted by a character of the central generator of the Levi factor) as an inducing representation for generalized Verma modules associated to the Lie algebra $$\mathfrak{sl} (3,\mathbb R)$$ and its maximal parabolic subalgebra. In this way they produce the symplectic Dirac operator as an $$\widetilde{\mathrm{SL}} (3,\mathbb R)$$-equivariant differential operator on the double covering of the real projective space $$\mathbb{RP}^2$$.

##### MSC:
 53C30 Differential geometry of homogeneous manifolds 53D05 Symplectic manifolds (general theory) 81R25 Spinor and twistor methods applied to problems in quantum theory
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