Skip to main content

Restriction formula for stable basis of the Springer resolution


We give restriction formula for stable basis of the Springer resolution and generalize it to cotangent bundles of partial flag varieties. By a limiting process, we get the restriction formula of Schubert varieties.

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


  1. Bernšteĭn, I.N., Gel’fand, I.M., Gel’fand, S.I.: Schubert cells, and the cohomology of the spaces \(G/P\). Uspehi Mat. Nauk 28(3(171)), 3–26 (1973)

    MathSciNet  MATH  Google Scholar 

  2. Billey, S.C.: Kostant polynomials and the cohomology ring for \(G/B\). Duke Math. J. 96(1), 205–224 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  3. Chriss, N., Ginzburg, V.: Representation Theory and Complex Geometry. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston (2010)

    Book  MATH  Google Scholar 

  4. Fulton, W., Andersen, D.: Equivariant cohomology in Algebraic geometry. Columbia University, Spring (2007)

  5. Ginzburg, V.: Characteristic varieties and vanishing cycles. Invent. Math. 84(2), 327–402 (1986)

    MathSciNet  Article  Google Scholar 

  6. Lenart, C., Zainoulline, K.: Elliptic Schubert calculus via formal root polynomials. arXiv preprint arXiv:1408.5952

  7. Liu, C.C.M.: Localization in Gromov–Witten theory and orbifold Gromov–Witten theory. Hand book of moduli, vol. ii. Adv. Lect. Math.(ALM) 25, (2013)

  8. Maulik, D., Okounkov, A.: Quantum groups and quantum cohomology. arXiv preprint arXiv:1211.1287 (2012)

  9. Nakajima, H.: Lectures on Hilbert schemes of points on surfaces. University Lecture Series, vol. 18. American Mathematical Society, Providence, RI (1999)

  10. Nakajima, H.: More lectures on Hilbert schemes of points on surfaces. arXiv preprint arXiv:1401.6782, (2014)

  11. Negut, A.: The \(m/n\) Pieri rule. arXiv preprint arXiv:1407.5303 (2014)

  12. Rimányi, R., Tarasov, V., Varchenko, A.: Partial flag varieties, stable envelopes and weight functions. arXiv preprint arXiv:1212.6240 (2012)

  13. Shenfeld, D.: Abelianization of Stable Envelopes in Symplectic Resolutions. Ph.D. thesis (2013)

  14. Smirnov, A.: Polynomials associated with fixed points on the instanton moduli space. arXiv preprint arXiv:1404.5304 (2014)

  15. Springer, T.A.: Linear Algebraic Groups. Modern Birkhäuser Classics, 2nd edn. Birkhäuser Boston Inc, Boston (2009)

    Google Scholar 

  16. Su, C.: Equivariant quantum cohomology of cotangent bundle of \(G/P\). Adv. Math. 289, 362–383 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  17. Tymoczko, J.S.: Divided difference operators for partial flag varieties. arXiv preprint arXiv:0912.2545 (2009)

Download references


I wish to express my deepest thanks to my advisor Prof. Andrei Okounkov for teaching me stable basis and his patience and invaluable guidance. The author also thanks Chiu-Chu Liu, Michael McBreen, Davesh Maulik, Andrei Negut, Andrey Smirnov, Zijun Zhou, Zhengyu Zong for many stimulating conversations and emails. A lot of thanks also go to my friend Pak-Hin Lee for editing a previous version of the paper. The author would also like to thank the referee for valuable comments.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Changjian Su.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Su, C. Restriction formula for stable basis of the Springer resolution. Sel. Math. New Ser. 23, 497–518 (2017).

Download citation

  • Published:

  • Issue Date:

  • DOI:

Mathematics Subject Classification

  • Primary 05Exx
  • Secondary 22Exx