Skip to main content
SpringerLink
Log in

A reducible characteristic variety in type A

  • Chapter
  • First Online:
Book cover Representations of Reductive Groups

Part of the book series: Progress in Mathematics ((PM,volume 312))

Abstract

We show that simple highest weight modules for \( \mathfrak{s}\mathfrak{l}_{12}(\mathbb{C}) \) may have reducible characteristic variety. This answers a question of Borho–Brylinski and Joseph from 1984. The relevant singularity under Beilinson–Bernstein localization is the (in)famous Kashiwara–Saito singularity. We sketch the rather indirect route via the p-canonical basis, W-graphs and decomposition numbers for perverse sheaves that led us to examine this singularity.

Dedicated to David Vogan on the occasion of his60th birthday

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. P. Achar and S. Riche, Modular perverse sheaves on flag varieties I: tilting and parity sheaves (with an appendix by G. Williamson), preprint arXiv:1401.7245, to appear in Ann. Sci. Éc. Norm. Supér. 2014.

    Google Scholar 

  2. P. Achar and S. Riche, Modular perverse sheaves on flag varieties II: Koszul duality and formality, preprint arXiv:1401.7256, 2014.

    Google Scholar 

  3. S. Ariki, Robinson–Schensted correspondence and left cells, in Combinatorial methods in representation theory (Kyoto, 1998), 1–20, Adv. Stud. Pure Math., 28, Kinokuniya, Tokyo, 2000.

    Google Scholar 

  4. W. Borho and J.-L. Brylinski, Differential operators on homogeneous spaces. III: Characteristic varieties of Harish Chandra modules and of primitive ideals, Invent. Math., 80 (1985), 1–68.

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  6. W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput., 24 nos 3–4 (1997), 235–265. Computational algebra and number theory (London, 1993).

    Google Scholar 

  7. B. Elias and G. Williamson, Soergel calculus, preprint arXiv:1309.0865, 2013.

    Google Scholar 

  8. W. Fulton, Flags, Schubert polynomials, degeneracy loci, and determinantal formulas, Duke Math. J., 65 no. 3 (1992), 381–420.

    Article  MathSciNet  MATH  Google Scholar 

  9. O. Gabber. Equidimensionalité de la variété caractéristique, Exposé de O. Gabber redigé par T. Levasseur, 1982.

    Google Scholar 

  10. R. B. Howlett and V. M. Nguyen, W-graph ideals, J. Algebra 361 (2012), 188–212.

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Howlett and V. M. Nguyen, W-graph magma programs, software available at http://www.maths.usyd.edu.au/u/bobh/magma/, 2013.

  12. X. He and G. Williamson, Soergel calculus and Schubert calculus, preprint arXiv:1502.04914, 2014.

    Google Scholar 

  13. D. Juteau, C. Mautner, and G. Williamson, Parity sheaves, JAMS, 24 (2014), 2617–2638.

    MATH  Google Scholar 

  14. A. Joseph, On the variety of a highest weight module, J. Algebra 88 (1984), 238–278.

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Juteau, Decomposition numbers for perverse sheaves, Ann. Inst. Fourier (Grenoble) 59 no. 3 (2009), 1177–1229.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. Kazhdan and G. Lusztig, A topological approach to Springer’s representations, Adv. Math. 38 (1980), 222–228.

    Article  MathSciNet  MATH  Google Scholar 

  17. D. Kazhdan and G. Lusztig, Schubert varieties and Poincaré duality, in Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, pages 185–203. Amer. Math. Soc., Providence, R.I., 1980.

    Google Scholar 

  18. M. Kashiwara and Y. Saito, Geometric construction of crystal bases, Duke Math. J., 89 no. 1 (1997), 9–36.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Kashiwara and T. Tanisaki, The characteristic cycles of holonomic systems on a flag manifold related to the Weyl group algebra, Invent. Math. 77 (1984), 185–198.

    Article  MathSciNet  MATH  Google Scholar 

  20. A. Melnikov. Irreducibility of the associated varieties of simple highest weight modules in \( \mathfrak{s}\mathfrak{l}(n) \), C. R. Acad. Sci. Paris Sér. I Math. 316 no. 1 (1993), 53–57.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. W. Soergel, On the relation between intersection cohomology and representation theory in positive characteristic, J. Pure Appl. Algebra 152 (2000), 311–335.

    Article  MathSciNet  MATH  Google Scholar 

  23. T. A. Springer, Quelques applications de la cohomologie d’intersection, in Bourbaki Seminar, Vol. 1981/1982, Astérisque 92, Soc. Math. France, Paris, 1982, pp 249–273.

    Google Scholar 

  24. T. Tanisaki, Characteristic varieties of highest weight modules and primitive quotients, in Representations of Lie groups: analysis on homogeneous spaces and representations of Lie groups, Proc. Symp., Kyoto/Jap. and Hiroshima/Jap. 1986, Adv. Stud. Pure Math. 14 (1988), 1–30.

    Google Scholar 

  25. K. Vilonen and G. Williamson, Characteristic cycles and decomposition numbers, Math. Res. Lett. 20 no. 2 (2013), 359–366.

    Article  MathSciNet  MATH  Google Scholar 

  26. G. Williamson, Modular intersection cohomology complexes on flag varieties (with an appendix by Tom Braden), Math. Z. 272 (2012), 697–727.

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Woo and A. Yong, Governing singularities of Schubert varieties, J. Algebra 320 no. 2 (2008), 495–520.

    Google Scholar 

  28. A. Woo and A. Yong, A Gröbner basis for Kazhdan-Lusztig ideals, Amer. J. Math. 134 no. 4 (2012), 1089–1137.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This paper also owes a significant debt to Leticia Barchini who asked me repeatedly about Question 1.1, and answered questions during and following her visit to the MPI last year. Thanks also to Peter Trapa for some explanations and Anna Melnikov, Yoshihisa Saito, Toshiyuki Tanisaki and the referee for useful correspondence. The examples were found using Howlett and Nguyen’s software [HN13] for magma [BCP97] which produces the irreducible W-graphs for the symmetric group, implementing an algorithm described in [HN12, §6]. During a visit to MIT last year David Vogan asked me whether the results of [VW12] could produce new examples of reducible characteristic cycles, and asked about Question 1.1. It is a pleasure to dedicate this paper to David, thank him for his many wonderful contributions to Lie theory and to wish him a happy birthday!

Author information

Authors and Affiliations

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

    Geordie Williamson

Authors
  1. Geordie Williamson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Geordie Williamson .

Editor information

Editors and Affiliations

  1. Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario, Canada

    Monica Nevins

  2. Dept of Mathematics and Statistics, University of Utah, Salt Lake City, Utah, USA

    Peter E. Trapa

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Williamson, G. (2015). A reducible characteristic variety in type A . In: Nevins, M., Trapa, P. (eds) Representations of Reductive Groups. Progress in Mathematics, vol 312. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-23443-4_19

Download citation

Publish with us

Policies and ethics

Search

Navigation