Skip to main content

A Weyl Module Stratification of Integrable Representations

Abstract

We construct a filtration on an integrable highest weight module of an affine Lie algebra whose adjoint graded quotient is a direct sum of global Weyl modules. We show that the graded multiplicity of each global Weyl module there is given by the corresponding level-restricted Kostka polynomial. This leads to an interpretation of level-restricted Kostka polynomials as the graded dimension of the space of conformal coinvariants. In addition, as an application of the level one case of the main result, we realize global Weyl modules of current algebras of type \({\mathsf{ADE}}\) in terms of Schubert subvarieties of thick affine Grassmanian, as predicted by Boris Feigin.

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

References

  1. 1

    Borel, A., Wallach, N.: Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups, Volume 67 of Mathematical Surveys and Monographs. American Mathematical Society, 2 edn, Providence, RI (2000)

  2. 2

    Chari V., Fourier G., Khandai T.: A categorical approach to weyl modules. Transform. Groups 15, 517–549 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3

    Chari V., Greenstein J.: Current algebras, highest weight categories and quivers. Adv. Math. 216(2), 811–840 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4

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

    MathSciNet  MATH  Article  Google Scholar 

  5. 5

    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  MATH  Article  Google Scholar 

  6. 6

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

  7. 7

    Cherednik I., Feigin B.: Rogers–Ramanujan type identities and Nil-DAHA. Adv. Math. 248, 1050–1088 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8

    Feigin B., Feigin E.: Homological realization of restricted Kostka polynomials. Int. Math. Res. Not. 33, 1997–2029 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9

    Feigin B., Frenkel E., Reshetikhin N.: Gaudin model, Bethe ansatz and critical level. Commun. Math. Phys. 166(1), 27–62 (1994)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  10. 10

    Fiebig P.: Centers and translation functors for the category \({{\mathscr{O}}}\) over Kac–Moody algebras. Math. Z. 243(4), 689–717 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11

    Fishel S., Grojnowski I., Teleman C.: The strong Macdonald conjecture and Hodge theory on the loop Grassmannian. Ann. Math. (2) 168(1), 175–220 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  12. 12

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

    MathSciNet  MATH  Article  Google Scholar 

  13. 13

    Frenkel, I.B., Kac V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62(1), 23–66 (1980/81)

  14. 14

    Hatayama G., Kirillov Anatol N., Kuniba A., Okado M., Takagi T., Yamada Y.: Character formulae of \({\widehat{\rm sl}_n}\)-modules and inhomogeneous paths. Nucl. Phys. B 536(3), 575–616 (1999)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  15. 15

    Heckenberger I., Kolb S.: On the Bernstein–Gelfand–Gelfand resolution for Kac–Moody algebras and quantized enveloping algebras. Transform. Groups 12(4), 647–655 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  16. 16

    Hong, J., Kang, S.-J.: Introduction to Quantum Groups and Crystal Bases, Volume 42 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2002)

  17. 17

    Kac V.G., Kazhdan D.A.: Structure of representations with highest weight of infinite-dimensional Lie algebras. Adv. Math. 34(1), 97–108 (1979)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18

    Kac Victor G.: Infinite-Dimensional Lie Algebras. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  19. 19

    Kang, S.-J., Kashiwara, M., Misra, K.C., Miwa, T., Nakashima, T., Nakayashiki A.: Affine crystals and vertex models. In: Infinite analysis, Part A, B (Kyoto, 1991), Volume 16 of Adv. Ser. Math. Phys., pp. 449–484. World Sci. Publ., River Edge, NJ (1992)

  20. 20

    Kashiwara M.: On crystal bases of the Q-analogue of universal enveloping algebras. Duke Math J. 63(2), 465–516 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21

    Kashiwara M., Miwa T., Stern E.: Decomposition of q-deformed Fock spaces. Sel. Math. (N.S.) 1(4), 787–805 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  22. 22

    Kashiwara, M.: The flag manifold of Kac–Moody Lie algebra. In: Proceedings of the JAMI Inaugural Conference, supplement to Amer. J. Math. the Johns Hopkins University Press (1989)

  23. 23

    Kashiwara, M.: Kazhdan–Lusztig conjecture for a symmetrizable Kac–Moody Lie algebra. In: The Grothendieck Festschrift, Vol. II, Volume 87 of Progr. Math., pp. 407–433. Birkhäuser Boston, Boston, MA, (1990)

  24. 24

    Kashiwara M.: The crystal base and Littelmann’s refined Demazure character formula. Duke Math. J. 71(3), 839–858 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  25. 25

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

    MathSciNet  MATH  Article  Google Scholar 

  26. 26

    Kashiwara M., Tanisaki T.: Kazhdan–Lusztig conjecture for affine Lie algebras with negative level. Duke Math. J. 77(1), 21–62 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27

    Kato S.: A homological study of Green polynomials. Ann Sci. Éc. Norm. Supér. (4) 48(5), 1035–1074 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  28. 28

    Kato S.: Demazure character formula for semi-infinite flag varieties. Math. Ann. 371(3), 1769–1801 (2018) arXiv:1605.0279

  29. 29

    Kato, S.: Frobenius splitting of thick flag manifolds of Kac–Moody algebras. Int. Math. Res. Not. IMRN, to appear (2018). https://doi.org/10.1093/imrn/rny174

  30. 30

    Kazhdan D., Lusztig G.: Tensor structures arising from affine Lie algebras. I. J Am. Math. Soc. 6, 905–947 (1993)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31

    Khoroshkin, A.: Highest weight categories and macdonald polynomials (2013). arXiv:1312.7053

  32. 32

    Kleshchev A.S.: Affine highest weight categories and affine quasihereditary algebras. Proc. Lond. Math. Soc. (3) 110(4), 841–882 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  33. 33

    Kumar S.: Demazure character formula in arbitrary Kac–Moody setting. Invent. Math. 89(2), 395–423 (1987)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  34. 34

    Kumar, S.: Kac–Moody Groups, Their Flag Varieties and Representation Theory, Volume 204 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, (2002)

  35. 35

    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. Transform. Groups 22(4), 1041–1079 (2015) arXiv:1511.00465

    MathSciNet  MATH  Article  Google Scholar 

  36. 36

    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. Not. IMRN (2016). arXiv:1402.2203

  37. 37

    Lusztig G.: Hecke algebras and Jantzen’s generic decomposition patterns. Adv. Math. 37(2), 121–164 (1980)

    MathSciNet  MATH  Article  Google Scholar 

  38. 38

    Naito S., Sagaki D.: Lakshmibai–Seshadri paths of level-zero shape and one-dimensional sums associated to level-zero fundamental representations. Compos. Math. 144(6), 1525–1556 (2008)

    MathSciNet  MATH  Article  Google Scholar 

  39. 39

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

    MathSciNet  MATH  Article  Google Scholar 

  40. 40

    Masato Okado. mimeo.

  41. 41

    Polo, P.: Projective versus injective modules over graded Lie algebras and a particular parabolic category \({{\mathscr{O}}}\) for affine Kac–Moody algebras. preprint.

  42. 42

    Schilling, A., Shimozono, M.: Bosonic formula for level-restricted paths. In: Combinatorial Methods in Representation Theory (Kyoto, 1998), Volume 28 of Adv. Stud. Pure Math., pp. 305–325. Kinokuniya, Tokyo (2000)

  43. 43

    Schilling A., Ole Warnaar S.: Inhomogeneous lattice paths, generalized Kostka polynomials and \({A_{n-1}}\) supernomials. Commun. Math. Phys. 202(2), 359–401 (1999)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  44. 44

    Soergel, W.: Kazhdan-Lusztig polynomials and a combinatoric[s] for tilting modules. Represent. Theory 1, 83–114 (electronic) (1997)

  45. 45

    Teleman C.: Lie algebra cohomology and the fusion rules. Commun. Math. Phys. 173(2), 265–311 (1995)

    ADS  MathSciNet  MATH  Article  Google Scholar 

  46. 46

    Zhu X.: Affine Demazure modules and T-fixed point subschemes in the affine Grassmannian. Adv. Math. 221(2), 570–600 (2009)

    MathSciNet  MATH  Article  Google Scholar 

Download references

Acknowledgements

S.K. thanks Michael Finkelberg for communicating Feigin’s insight and a kind invitation to Moscow in the fall 2016, and Masato Okado to show/explain his mimeo. S.L. thanks the Department of Mathematics of Kyoto University for their hospitality. We both thank Boris Feigin and Ivan Cherednik for fruitful and stimulating discussions. R.K. thanks Sergey Loktev for explaining his idea to use the free field realization in the Proof of Theorem 4.1.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Syu Kato.

Additional information

Publisher’s Note

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

With an appendix by Ryosuke Kodera.

Research supported in part by JSPS Grant-in-Aid for Scientific Research (B) JP26287004.

Research supported in part by Laboratory of Mirror Symmetry NRU HSE, RF Government Grant, ag. 14.641.31.0001.

Communicated by C. Schweigert

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kato, S., Loktev, S. A Weyl Module Stratification of Integrable Representations. Commun. Math. Phys. 368, 113–141 (2019). https://doi.org/10.1007/s00220-019-03327-5

Download citation