Previous |  Up |  Next


projective structure; Segal-Shale-Weil representation; generalized Verma modules; symplectic Dirac operator; $(3,)$
Inspired by the results on symmetries of the symplectic Dirac operator, we realize symplectic spinor fields and the symplectic Dirac operator in the framework of (the double cover of) homogeneous projective structure in two real dimensions. The symmetry group of the homogeneous model of the double cover of projective geometry in two real dimensions is ${\widetilde{}}(3,)$.
[1] Collingwood, D.H., Shelton, B.: A duality theorem for extensions of induced highest weight modules. Pacific J. Math. 146 (2) (1990), 227–237. DOI 10.2140/pjm.1990.146.227 | MR 1078380 | Zbl 0733.17005
[2] De Bie, H., Holíková, M., Somberg, P.: Basic aspects of symplectic Clifford analysis for the symplectic Dirac operator. arXiv:1511.04189, 2015.
[3] De Bie, H., Somberg, P., Souček, V.: The metaplectic Howe duality and polynomial solutions for the symplectic Dirac operator. J. Geom. Phys. 75 (2014), 120–128. DOI 10.1016/j.geomphys.2013.09.005 | MR 3126939 | Zbl 1279.30051
[4] Habermann, K., Habermann, L.: Introduction to symplectic Dirac operators. Lecture Notes in Math., vol. 1887, Springer-Verlag, Berlin, 2006. DOI 10.1007/978-3-540-33421-7_4 | MR 2252919 | Zbl 1102.53032
[5] Kobayashi, T., Ørsted, B., Somberg, P., Souček, V.: Branching laws for Verma modules and applications in parabolic geometry. I. Adv. Math. 285 (2015), 1796–1852. DOI 10.1016/j.aim.2015.08.020 | MR 3406542 | Zbl 1327.53044
[6] Kostant, B.: Symplectic spinors. Symposia Mathematica, Progress in Mathematics, vol. XIV, Academic Press, London, 1974, pp. 139–152. MR 0400304 | Zbl 0321.58015
[7] Kostant, B.: Verma modules and the existence of quasi-invariant differential operators, Non-Commutative Harmonic Analysis. Lecture Notes in Math., vol. 466, Springer, Berlin, 1975, pp. 101–128. MR 0396853
[8] Křižka, L., Somberg, P.: Algebraic analysis of scalar generalized Verma modules of Heisenberg parabolic type I.: $A_n$-series. arXiv:1502.07095, to appear in Transformation Groups, 2015. MR 3449111
[9] Křižka, L., Somberg, P.: Algebraic analysis on scalar generalized Verma modules of Heisenberg parabolic type II.: $C_n, D_n$-series. (in preparation).
[10] Ørsted, B.: Generalized gradients and Poisson transforms. Global analysis and harmonic analysis, Sémin. Congr., vol. 4, Soc. Math. France, Paris, 2000, pp. 235–249. MR 1822363 | Zbl 0989.22018
[11] Somberg, P., Šilhan, J.: Higher symmetries of the symplectic Dirac operator. (in preparation).
[12] Torasso, P.: Quantification géométrique, opérateurs d’entrelacement et représentations unitaires de ${\widetilde{SL}}_3(\mathbb{R})$. Acta Math. 150 (1) (1983), 153–242. DOI 10.1007/BF02392971 | MR 0709141
[13] Wolf, J.A.: Unitary representations of maximal parabolic subgroups of the classical groups. Mem. Amer. Math. Soc., vol. 8, American Mathematical Society, Providence, 1976. MR 0444847 | Zbl 0344.22016
Partner of
EuDML logo