Skip to main content

Demazure character formula for semi-infinite flag varieties

Abstract

We prove that every Schubert variety of a semi-infinite flag variety is projectively normal. This gives us an interpretation of a Demazure module of a global Weyl module of a current Lie algebra as the (dual) space of global sections of a line bundle on a semi-infinite Schubert variety. Moreover, we give geometric realizations of Feigin–Makedonskyi’s generalized Weyl modules, and the \(t = \infty \) specialization of non-symmetric Macdonald polynomials.

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

References

  1. 1.

    Beck, J., Nakajima, H.: Crystal bases and two-sided cells of quantum affine algebras. Duke Math. J. 123(2), 335–402 (2004)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Bjorner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol. 231. Springer, New York, pp. xiv+363 (2005)

  3. 3.

    Braverman, A., Gaitsgory, D.: Geometric Eisenstein series. Invent. Math. 150(2), 287–384 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Braverman, A., Finkelberg, M.: Private note (2012)

  5. 5.

    Braverman, A., Finkelberg, M.: Semi-infinite Schubert varieties and quantum \(K\)-theory of flag manifolds. J. Am. Math. Soc. 27(4), 1147–1168 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Braverman, A., Finkelberg, M.: Weyl modules and \(q\)-Whittaker functions. Math. Ann. 359(1–2), 45–59 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Braverman, A., Finkelberg, M.: Twisted zastava and \(q\)-Whittaker functions. J. Lond. Math. Soc. 96(2), 309–325 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Chari, V., Ion, B.: BGG reciprocity for current algebras. Compos. Math. 151(7), 1265–1287 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Chari, V., Loktev, S.: Weyl, Demazure and fusion modules for the current algebra of \(\mathfrak{sl}_{r+1}\). Adv. Math. 207(2), 928–960 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Chari, V., Pressley, A.: Weyl modules for classical and quantum affine algebras. Rep. Theory 5, 191–223 (2001). (electronic)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Cherednik, I.: Nonsymmetric Macdonald polynomials. Int. Math. Res. Notices 10, 483–515 (1995)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Cherednik, I., Orr, D.: Nonsymmetric difference Whittaker functions. Math. Z. 279(3–4), 879–938 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Demazure, M.: Désingularisation des variétés de Schubert généralisées. Ann. Sci. École Norm. Sup. 4(7), 53–88 (1974)

    Article  MATH  Google Scholar 

  14. 14.

    Feigin, B., Finkelberg, M., Kuznetsov, A., Mirković, I.: Semi-infinite flags. II. Local and global intersection cohomology of quasimaps’ spaces. In Differential topology, infinite-dimensional Lie algebras, and applications, volume 194 of Amer. Math. Soc. Transl. Ser. 2, pp.113–148. American Mathematical Society, Providence, RI (1999)

  15. 15.

    Feigin, B., Frenkel, E.: Affine Kac-Moody algebras and semi-infinite flag manifold. Commun. Math. Phys. 128, 161–189 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Feigin, E., Makedonskyi, I.: Generalized Weyl modules, alcove paths and Macdonald polynomials. Select. Math. 23(4), 2863–2897 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Feigin, E., Makedonskyi, I., Orr, D.: Generalized Weyl modules and nonsymmetric \(q\)-Whittaker functions (2016). arXiv:1605.01560

  18. 18.

    Finkelberg, M., Mirković, I.: Semi-infinite flags. I. Case of global curve \(\mathbf{P}^1\). In Differential topology, infinite-dimensional Lie algebras, and applications, volume 194 of Amer. Math. Soc. Transl. Ser. 2, pages 81–112. American Mathematical Society, Providence, RI (1999)

  19. 19.

    Fourier, G., Littelmann, P.: Weyl modules, Demazure modules, KR-modules, crystals, fusion products and limit constructions. Adv. Math. 211(2), 566–593 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Givental, A., Lee, Y.-P.: Quantum \(K\)-theory on flag manifolds, finite-difference Toda lattices and quantum groups. Invent. Math. 151(1), 193–219 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Ion, B.: Nonsymmetric Macdonald polynomials and Demazure characters. Duke Math. J. 116(2), 299–318 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Ishii, M., Naito, S., Sagaki, D.: Semi-infinite Lakshmibai–Seshadri path model for level-zero extremal weight modules over quantum affine algebras. Adv. Math. 290, 967–1009 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Joseph, A.: On the Demazure character formula. Ann. Sci. École Norm. Sup. (4) 18(3), 389–419 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Kashiwara, M.: Crystal bases of modified quantized enveloping algebra. Duke Math. J. 73(2), 383–413 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Kashiwara, M.: On level-zero representations of quantized affine algebras. Duke Math. J. 112(1), 117–175 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Kashiwara, M.: Level zero fundamental representations over quantized affine algebras and Demazure modules. Publ. Res. Inst. Math. Sci. 41(1), 223–250 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Kumar, S.: Kac-Moody groups, their flag varieties and representation theory. Progress in Mathematics, vol. 204. Birkhäuser Boston Inc, Boston, MA, pp. xvi+606 (2002)

  28. 28.

    Lenart, C., Naito, S., Sagaki, D., Schilling, A., Shimozono, M.: A uniform model for Kirillov–Reshetikhin crystals I. Int. Math. Res. Notices 1848–1901, 2015 (2015)

    MATH  Google Scholar 

  29. 29.

    Lenart, C., Naito, S., Sagaki, D., Schilling, A., Shimozono, M.: A uniform model for Kirillov–Reshetikhin crystals II. Alcove model, path model, and P=X. Int. Math. Res. Notices 14, 4259–4319 (2017)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Lenart, C., Naito, S., Sagaki, D., Schilling, A., Shimozono, M.: A uniform model for Kirillov–Reshetikhin crystals III: nonsymmetric Macdonald polynomials at \(t=0\) and Demazure characters. Transf. Group 22, 1041–1079 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    Naito, S., Nomoto, F., Sagaki, D.: An explicit formula for the specialization of nonsymmetric macdonald polynomials at \(t = \infty \). To appear. Trans. Amer. Math. Soc. 370(4), 2739–2783 (2015)

    Article  MATH  Google Scholar 

  32. 32.

    Naito, S., Sagaki, D.: Crystal structure on the set of Lakshmibai–Seshadri paths of an arbitrary level-zero shape. Proc. Lond. Math. Soc. 96(3), 582 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  33. 33.

    Naito, S., Sagaki, D.: Demazure submodules of level-zero extremal weight modules and specializations of Macdonald polynomials. Math. Z. 283, 937–978 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  34. 34.

    Naoi, K.: Weyl modules, Demazure modules and finite crystals for non-simply laced type. Adv. Math. 229(2), 875–934 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Sanderson, Y.B.: On the connection between Macdonald polynomials and Demazure characters. J. Algebra Combin. 11(3), 269–275 (2000)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The author would like to thank Michael Finkelberg for attracting his attention to [16] and sent me his unpublished note [4]. He also would like to thank Satoshi Naito for various comments and suggestions on the topic presented in this paper, Shrawan Kumar for discussion on semi-infinite flag varieties, and Evgeny Feigin and Daniel Orr for preventing him from some incorrect references. The original version of this paper was written during the author’s stay at MIT in the academic year 2015/2016. The author would like to thank George Lusztig and MIT for their hospitality. Finally, the author would like to express his thanks to the referee who have kindly made many remarks on the previous version of this paper.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Syu Kato.

Additional information

Research supported in part by JSPS Grant-in-Aid for Scientific Research (B) 26287004 and Kyoto University Jung-Mung program.

Communicated by Jean-Yves Welschinger.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kato, S. Demazure character formula for semi-infinite flag varieties. Math. Ann. 371, 1769–1801 (2018). https://doi.org/10.1007/s00208-018-1652-5

Download citation