Skip to main content
SpringerLink
Log in

A geometric Schur functor

  • Published:
Selecta Mathematica Aims and scope Submit manuscript

Abstract

We give geometric descriptions of the category \(C_k(n,d)\) of rational polynomial representations of \(GL_n\) over a field \(k\) of degree \(d\) for \(d\le n\), the Schur functor and Schur–Weyl duality. The descriptions and proofs use a modular version of Springer theory and relationships between the equivariant geometry of the affine Grassmannian and the nilpotent cone for the general linear groups. Motivated by this description, we propose generalizations for an arbitrary connected complex reductive group of the category \(C_k(n,d)\) and the Schur functor.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. While Mirković works with \({\mathcal {D}}\)-modules, the same argument works in the constructible context with arbitrary coefficients.

References

  1. Achar, P., Henderson, A.: Geometric Satake, Springer correspondence, and small representations. Preprint. arXiv:1108.4999 (2012)

  2. Achar, P., Henderson, A., Juteau, D., Riche, S.: Weyl group actions on the Springer sheaf. Proceedings of London Mathematical Society first published online December 11, 2013. doi:10.1112/plms/pdt055

  3. Achar, P., Henderson, A., Juteau, D., Riche, S.: Modular generalized Springer correspondence I: the general linear group. Preprint. arXiv:1307.2702 (2013)

  4. Achar, P., Henderson, A., Riche, S.: Geometric Satake, Springer correspondence, and small representations II. Preprint. arXiv:1205.5089 (2012)

  5. Achar, P., Mautner, C.: Sheaves on nilpotent cones, Fourier transform, and a geometric Ringel duality. Preprint. arXiv:1207.7044 (2012)

  6. Beĭlinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers. Analyse et topologie sur les espaces singuliers. I (Luminy, 1981), volume 100 of Astérisque, pp. 5–171. Soc. Math. France, Paris (1982)

  7. Beĭlinson, A., Drinfel’d, V.: Quantization of Hitchin integrable system and Hecke eigensheaves. Preprint. http://www.math.uchicago.edu/mitya/langlands/hitchin/BD-hitchin.pdf

  8. Brylinski, J.-L.: Transformations canoniques, dualité projective, théorie de Lefschetz, transformations de Fourier et sommes trigonométriques. Astérisque, 140–141, 3–134. Géométrie et analyse microlocales (1986)

  9. Chriss, N., Ginzburg, V.: Representation theory and complex geometry. Birkhäuser Boston Inc., Boston (1997)

    MATH  Google Scholar 

  10. Carter, R.W., Lusztig, G.: On the modular representations of the general linear and symmetric groups. Math. Z. 136, 193–242 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  11. Doty, S.R., Erdmann, K., Nakano, D.K.: Extensions of modules over Schur algebras, symmetric groups and Hecke algebras. Algebr. Represent. Theory 7(1), 67–100 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Donkin, S.: On Schur algebras and related algebras, II. J. Algebr. 111(2), 354–364 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. Donkin, S.: On tilting modules for algebraic groups. Math. Z. 212(1), 39–60 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ginsburg, V.: Intégrales sur les orbites nilpotentes et représentations des groupes de Weyl. C. R. Acad. Sci. Paris Sér. I Math. 296(5), 249–252 (1983)

  15. Ginzburg, V.: Perverse sheaves on a loop group and Langlands duality. Preprint. arXiv:alg-geom/9511007 (1995)

  16. Green, J.A.: Polynomial representations of \({\rm GL}_{n}\). Lecture Notes in Mathematics, vol. 830. Springer, Berlin (1980)

    MATH  Google Scholar 

  17. Hotta, R., Kashiwara, M.: The invariant holonomic system on a semisimple Lie algebra. Invent. Math. 75(2), 327–358 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hotta, R.: On Springer’s representations. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28(3), 863–876 (1982) 1981

    Google Scholar 

  19. Jantzen, J.C.: Representations of algebraic groups, volume 107 of Mathematical Surveys and Monographs, 2nd edn. American Mathematical Society, Providence (2003)

  20. Juteau, D.: Modular Springer correspondence and decomposition matrices. PhD thesis, Université Paris 7 Denis Diderot. arXiv:0901.3671 (2007)

  21. Kleshchev, A.S.: Branching rules for modular representations of symmetric groups I. J. Algebr. 178(2), 493–511 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kashiwara, M., Schapira, P.: Sheaves on manifolds, volume 292 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1994)

  23. Laszlo, Y., Olsson, M.: Perverse \(t\)-structure on Artin stacks. Math. Z. 261(4), 737–748 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lusztig, G.: Green polynomials and singularities of unipotent classes. Adv. Math. 42, 169–178 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lusztig, G.: Singularities, character formulas, and a \(q\)-analog of weight multiplicities. Analyse et topologie sur les espaces singuliers. II, III (Luminy, 1981), volume 101–102 of Astérisque, pp. 208–229. Soc. Math. France, Paris (1983)

  26. Martin, S.: Schur Algebras and Representation Theory, volume 112 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  27. Mautner, C.: Sheaf Theoretic Methods in Modular Representation Theory. PhD thesis, University of Texas at Austin (2010)

  28. Mirković, I.: Character sheaves on reductive Lie algebras. Mosc. Math. J. 4(4), 897–910, 981 (2004)

    Google Scholar 

  29. Mirković, I., Vilonen, K.: Geometric Langlands duality and representations of algebraic groups over commutative rings. Ann. Math. (2) 166(1), 95–143 (2007)

    Article  MATH  Google Scholar 

  30. Mirković, I., Vybornov, M.: Quiver varieties and Beilinson-Drinfeld Grassmannians of type A. Preprint. arXiv:0712.4160 (2007)

  31. Ngô, B.C.: Faisceaux pervers, homomorphisme de changement de base et lemme fondamental de Jacquet et Ye. Ann. Sci. École Norm. Sup. (4) 32(5), 619–679 (1999)

    MATH  Google Scholar 

Download references

Acknowledgments

This paper has been a long time coming and so the author had a number of years to benefit from useful conversations and deep insights from many people. He would like to thank in particular: David Ben-Zvi for continued support and advice, Daniel Juteau whose thesis was a source of inspiration for much of this paper, Pramod Achar for encouragement, Geordie Williamson for comments on a draft of the paper and an anonymous referee for a careful reading and many helpful comments. Thanks as well to Dennis Gaitsgory, Joel Kamnitzer, David Helm, David Nadler, Catharina Stroppel, Zhiwei Yun, and Xinwen Zhu.

Author information

Authors and Affiliations

  1. Max Planck Institut für Mathematik, Vivatsgasse 7, 53111 , Bonn, Germany

    Carl Mautner

Authors
  1. Carl Mautner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carl Mautner.

Additional information

The author was supported by an NSF postdoctoral research fellowship.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mautner, C. A geometric Schur functor. Sel. Math. New Ser. 20, 961–977 (2014). https://doi.org/10.1007/s00029-014-0147-9

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00029-014-0147-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation