Skip to main content

On the generalized \(\text {SO}(2n,{{\mathbb {C}}})\)-opers


Since their introduction by Beilinson–Drinfeld (Opers, 1993. arXiv math/0501398; Quantization of Hitchin’s integrable system and Hecke eigensheaves, 1991), opers have seen several generalizations. In Biswas et al. (SIGMA Symmetry Integr Geom Methods Appl 16:041, 2020), a higher rank analog was studied, named generalized B-opers, where the successive quotients of the oper filtration are allowed to have higher rank and the underlying holomorphic vector bundle is endowed with a bilinear form which is compatible with both the filtration and the oper connection. Since the definition did not encompass the even orthogonal groups, we dedicate this paper to study generalized B-opers whose structure group is \(\mathrm{SO}(2n,{\mathbb {C}})\) and show their close relationship with geometric structures on a Riemann surface.

This is a preview of subscription content, access via your institution.


  1. Atiyah, M.F.: Complex analytic connections in fibre bundles. Trans. Amer. Math. Soc. 85, 181–207 (1957)

    Article  MathSciNet  Google Scholar 

  2. Beilinson, A., Drinfeld, V. G.: Opers (1993). arXiv math/0501398

  3. Beilinson, A., Drinfeld, V.: Quantization of Hitchin’s integrable system and Hecke eigensheaves (1991)

  4. Biswas, I.: Invariants for a class of equivariant immersions of the universal cover of a compact Riemann surface into a projective space. J. Math. Pures Appl. 79, 1–20 (2000)

    Article  MathSciNet  Google Scholar 

  5. Biswas, I.: Coupled connections on a compact Riemann surface. J. Math. Pures Appl. 82, 1–42 (2003)

    Article  MathSciNet  Google Scholar 

  6. Biswas, I., Schaposnik, L.P., Yang, M.: Generalized \(B\)-opers. SIGMA Symmetry Integrability Geom. Methods Appl. 16, 041 (2020)

    MathSciNet  MATH  Google Scholar 

  7. Collier, B., Sanders, A.: (G, P)-opers and global Slodowy slices. Adv. Math. 377, 107490 (2021)

    Article  MathSciNet  Google Scholar 

  8. Drinfeld, V.G., Sokolov, V.V.: Lie algebras and equations of Korteweg-de Vries type, Current problems in mathematics, vol. 24, pp. 81–180, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (1984)

  9. Drinfeld, V.G., Sokolov, V.V.: Equations of Korteweg–de Vries type, and simple Lie algebras. Dokl. Akad. Nauk SSSR 258, 11–16 (1981)

    MathSciNet  Google Scholar 

  10. Frenkel, E.: Wakimoto modules, opers and the center at the critical level. Adv. Math. 195, 297–404 (2005)

    Article  MathSciNet  Google Scholar 

  11. Gunning, R.C.: Lectures on Riemann Surfaces, Mathematical Notes 2. Princeton University Press, Princeton (1966)

    MATH  Google Scholar 

  12. Yang M.: A comparison of generalized opers and (G,P)-opers, preprint (2021)

Download references


IB is supported by a J. C. Bose Fellowship. LPS is partially supported by the NSF CAREER Award DMS-1749013.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Indranil Biswas.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Biswas, I., Schaposnik, L.P. & Yang, M. On the generalized \(\text {SO}(2n,{{\mathbb {C}}})\)-opers. Ann Glob Anal Geom 60, 539–557 (2021).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification