### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 012, 33 pages      arXiv:1705.09469      https://doi.org/10.3842/SIGMA.2018.012

### $k$-Dirac Complexes

Tomáš Salač
Mathematical Institute, Charles University, Sokolovská 49/83, Prague, Czech Republic

Received June 01, 2017, in final form February 06, 2018; Published online February 16, 2018

Abstract
This is the first paper in a series of two papers. In this paper we construct complexes of invariant differential operators which live on homogeneous spaces of $|2|$-graded parabolic geometries of some particular type. We call them $k$-Dirac complexes. More explicitly, we will show that each $k$-Dirac complex arises as the direct image of a relative BGG sequence and so this fits into the scheme of the Penrose transform. We will also prove that each $k$-Dirac complex is formally exact, i.e., it induces a long exact sequence of infinite (weighted) jets at any fixed point. In the second part of the series we use this information to show that each $k$-Dirac complex is exact at the level of formal power series at any point and that it descends to a resolution of the $k$-Dirac operator studied in Clifford analysis.

Key words: Penrose transform; complexes of invariant differential operators; relative BGG complexes; formal exactness; weighted jets.

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References

1. Baston R.J., Quaternionic complexes, J. Geom. Phys. 8 (1992), 29-52.
2. Baston R.J., Eastwood M.G., The Penrose transform. Its interaction with representation theory, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1989.
3. Bureš J., Damiano A., Sabadini I., Explicit resolutions for the complex of several Fueter operators, J. Geom. Phys. 57 (2007), 765-775.
4. Bureš J., Souček V., Complexes of invariant operators in several quaternionic variables, Complex Var. Elliptic Equ. 51 (2006), 463-485.
5. Čap A., Salač T., Parabolic conformally symplectic structures I: definition and distinguished connections, Forum Math., to appear, arXiv:1605.01161.
6. Čap A., Salač T., Parabolic conformally symplectic structures II: parabolic contactification, Ann. Mat. Pura Appl., to appear, arXiv:1605.01897.
7. Čap A., Salač T., Parabolic conformally symplectic structures III: invariant differential operators and complexes, arXiv:1701.01306.
8. Čap A., Slovák J., Parabolic geometries. I. Background and general theory, Mathematical Surveys and Monographs, Vol. 154, Amer. Math. Soc., Providence, RI, 2009.
9. Čap A., Slovák J., Souček V., Bernstein-Gelfand-Gelfand sequences, Ann. of Math. 154 (2001), 97-113, math.DG/0001164.
10. Čap A., Souček V., Relative BGG sequences: I. Algebra, J. Algebra 463 (2016), 188-210, arXiv:1510.03331.
11. Čap A., Souček V., Relative BGG sequences: II. BGG machinery and invariant operators, Adv. Math. 320 (2017), 1009-1062, arXiv:1510.03986.
12. Colombo F., Sabadini I., Sommen F., Struppa D.C., Analysis of Dirac systems and computational algebra, Progress in Mathematical Physics, Vol. 39, Birkhäuser Boston, Inc., Boston, MA, 2004.
13. Colombo F., Souček V., Struppa D.C., Invariant resolutions for several Fueter operators, J. Geom. Phys. 56 (2006), 1175-1191.
14. Franek P., Generalized Dolbeault sequences in parabolic geometry, J. Lie Theory 18 (2008), 757-774, arXiv:0710.0093.
15. Goodman R., Wallach N.R., Symmetry, representations, and invariants, Graduate Texts in Mathematics, Vol. 255, Springer, Dordrecht, 2009.
16. Hörmander L., An introduction to complex analysis in several variables, D. Van Nostrand Co., Inc., Princeton, N.J. - Toronto, Ont. - London, 1966.
17. Krump L., A resolution for the Dirac operator in four variables in dimension 6, Adv. Appl. Clifford Algebr. 19 (2009), 365-374.
18. Morimoto T., Lie algebras, geometric structures and differential equations on filtered manifolds, in Lie Groups, Geometric Structures and Differential Equations - One Hundred Years After Sophus Lie (Kyoto/Nara, 1999), Adv. Stud. Pure Math., Vol. 37, Math. Soc. Japan, Tokyo, 2002, 205-252.
19. Nacinovich M., Complex analysis and complexes of differential operators, in Complex Analysis (Trieste, 1980), Lecture Notes in Math., Vol. 950, Springer, Berlin - New York, 1982, 105-195.
20. Sabadini I., Struppa D.C., Sommen F., Van Lancker P., Complexes of Dirac operators in Clifford algebras, Math. Z. 239 (2002), 293-320.
21. Salač T., $k$-Dirac operator and the Cartan-Kähler theorem, Arch. Math. (Brno) 49 (2013), 333-346, arXiv:1304.0956.
22. Salač T., $k$-Dirac operator and parabolic geometries, Complex Anal. Oper. Theory 8 (2014), 383-408, arXiv:1201.0355.
23. Salač T., $k$-Dirac operator and the Cartan-Kähler theorem for weighted differential operators, Differential Geom. Appl. 49 (2016), 351-371, arXiv:1601.08077.
24. Salač T., Resolution of the $k$-Dirac operator, Adv. Appl. Clifford Algebr. 28 (2018), 28:3, 19 pages, arXiv:1705.10168.
25. Spencer D.C., Overdetermined systems of linear partial differential equations, Bull. Amer. Math. Soc. 75 (1969), 179-239.
26. Ward R.S., Wells Jr. R.O., Twistor geometry and field theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1990.
27. Wells Jr. R.O., Differential analysis on complex manifolds, Graduate Texts in Mathematics, Vol. 65, 2nd ed., Springer-Verlag, New York - Berlin, 1980.