Ellipticity of the symplectic twistor complex

Svatopluk Krýsl

Archivum Mathematicum (2011)

  • Volume: 047, Issue: 4, page 309-327
  • ISSN: 0044-8753

Abstract

top
For a Fedosov manifold (symplectic manifold equipped with a symplectic torsion-free affine connection) admitting a metaplectic structure, we shall investigate two sequences of first order differential operators acting on sections of certain infinite rank vector bundles defined over this manifold. The differential operators are symplectic analogues of the twistor operators known from Riemannian or Lorentzian spin geometry. It is known that the mentioned sequences form complexes if the symplectic connection is of Ricci type. In this paper, we prove that certain parts of these complexes are elliptic.

How to cite

top

Krýsl, Svatopluk. "Ellipticity of the symplectic twistor complex." Archivum Mathematicum 047.4 (2011): 309-327. <http://eudml.org/doc/246918>.

@article{Krýsl2011,
abstract = {For a Fedosov manifold (symplectic manifold equipped with a symplectic torsion-free affine connection) admitting a metaplectic structure, we shall investigate two sequences of first order differential operators acting on sections of certain infinite rank vector bundles defined over this manifold. The differential operators are symplectic analogues of the twistor operators known from Riemannian or Lorentzian spin geometry. It is known that the mentioned sequences form complexes if the symplectic connection is of Ricci type. In this paper, we prove that certain parts of these complexes are elliptic.},
author = {Krýsl, Svatopluk},
journal = {Archivum Mathematicum},
keywords = {Fedosov manifolds; Segal-Shale-Weil representation; Kostant’s spinors; elliptic complexes; Fedosov manifold; Segal-Shale-Weil representation; Kostant's spinor; elliptic complexes},
language = {eng},
number = {4},
pages = {309-327},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Ellipticity of the symplectic twistor complex},
url = {http://eudml.org/doc/246918},
volume = {047},
year = {2011},
}

TY - JOUR
AU - Krýsl, Svatopluk
TI - Ellipticity of the symplectic twistor complex
JO - Archivum Mathematicum
PY - 2011
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 047
IS - 4
SP - 309
EP - 327
AB - For a Fedosov manifold (symplectic manifold equipped with a symplectic torsion-free affine connection) admitting a metaplectic structure, we shall investigate two sequences of first order differential operators acting on sections of certain infinite rank vector bundles defined over this manifold. The differential operators are symplectic analogues of the twistor operators known from Riemannian or Lorentzian spin geometry. It is known that the mentioned sequences form complexes if the symplectic connection is of Ricci type. In this paper, we prove that certain parts of these complexes are elliptic.
LA - eng
KW - Fedosov manifolds; Segal-Shale-Weil representation; Kostant’s spinors; elliptic complexes; Fedosov manifold; Segal-Shale-Weil representation; Kostant's spinor; elliptic complexes
UR - http://eudml.org/doc/246918
ER -

References

top
  1. Borel, A., Wallach, N., Continuous cohomology, discrete subgroups, and representations of reductive groups. Second edition, cond edition, Math. Surveys Monogr. 67 (2000), xviii+260 pp. (2000) MR1721403
  2. Branson, T., 10.1006/jfan.1997.3162, J. Funct. Anal. 151 (2) (1997), 334–383. (1997) Zbl0904.58054MR1491546DOI10.1006/jfan.1997.3162
  3. Cahen, M., Schwachhöfer, L., Special symplectic connections, J. Differential Geom. 83 (2) (2009), 229–271. (2009) Zbl1190.53019MR2577468
  4. Casselman, W., 10.4153/CJM-1989-019-5, Canad. J. Math. 41 (3) (1989), 385–438. (1989) Zbl0702.22016MR1013462DOI10.4153/CJM-1989-019-5
  5. Fedosov, B., A simple geometrical construction of deformation quantization, J. Differential Geom. 40 (2) (1994), 213–238. (1994) Zbl0812.53034MR1293654
  6. Gelfand, I., Retakh, V., Shubin, M., 10.1006/aima.1998.1727, Adv. Math. 136 (1) (1998), 104–140. (1998) Zbl0945.53047MR1623673DOI10.1006/aima.1998.1727
  7. Green, M. B., Hull, C. M., 10.1016/0370-2693(89)91009-5, Phys. Lett. B 225 (1989), 57–65. (1989) MR1006387DOI10.1016/0370-2693(89)91009-5
  8. Habermann, K., Habermann, L., 10.1007/978-3-540-33421-7_4, Lecture Notes in Math., vol. 1887, Springer-Verlag, Berlin, 2006. (2006) Zbl1102.53032MR2252919DOI10.1007/978-3-540-33421-7_4
  9. Hotta, R., Elliptic complexes on certain homogeneous spaces, Osaka J. Math. 7 (1970), 117–160. (1970) Zbl0197.47703MR0265519
  10. Howe, R., 10.1090/S0002-9947-1989-0986027-X, Trans. Amer. Math. Soc. 313 (2) (1989), 539–570. (1989) Zbl0674.15021MR0986027DOI10.1090/S0002-9947-1989-0986027-X
  11. Kostant, B., Symplectic Spinors, Symposia Mathematica, vol. XIV, Cambridge Univ. Press, 1974, pp. 139–152. (1974) Zbl0321.58015MR0400304
  12. Krýsl, S., Howe duality for metaplectic group acting on symplectic spinor valued forms, accepted in J. Lie Theory. 
  13. Krýsl, S., Symplectic spinor forms and the invariant operators acting between them, Arch. Math. (Brno) 42 (Supplement) (2006), 279–290. (2006) MR2322414
  14. Krýsl, S., 10.1007/s00605-009-0158-3, Monatsh. Math. 161 (4) (2010), 381–398. (2010) MR2734967DOI10.1007/s00605-009-0158-3
  15. Schmid, W., Homogeneous complex manifolds and representations of semisimple Lie group, Representation theory and harmonic analysis on semisimple Lie groups. (Sally, P., Vogan, D., eds.), vol. 31, American Mathematical Society, Providence, Rhode-Island, Mathematical Surveys and Monographs, 1989. (1989) MR1011899
  16. Shale, D., 10.1090/S0002-9947-1962-0137504-6, Trans. Amer. Math. Soc. 103 (1962), 149–167. (1962) Zbl0171.46901MR0137504DOI10.1090/S0002-9947-1962-0137504-6
  17. Stein, E., Weiss, G., 10.2307/2373431, Amer. J. Math. 90 (1968), 163–196. (1968) Zbl0157.18303MR0223492DOI10.2307/2373431
  18. Tondeur, P., 10.1007/BF02566901, Comment. Math. Helv. 36 (1961), 234–244. (1961) MR0138068DOI10.1007/BF02566901
  19. Vaisman, I., Symplectic Curvature Tensors, Monatshefte für Math., vol. 100, Springer-Verlag, Wien, 1985, pp. 299–327. (1985) MR0814206
  20. Weil, A., 10.1007/BF02391012, Acta Math. 111 (1964), 143–211. (1964) MR0165033DOI10.1007/BF02391012
  21. Wells, R., 10.1007/978-0-387-73892-5, Grad. Texts in Math., vol. 65, Springer, New York, 2008. (2008) Zbl1131.32001MR2359489DOI10.1007/978-0-387-73892-5

NotesEmbed ?

top

You must be logged in to post comments.