Advertisement

Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 133–149 | Cite as

Counting absolutely cuspidals for quivers

  • T. Bozec
  • O. SchiffmannEmail author
Article

Abstract

For an arbitrary quiver \(Q=(I,\Omega )\) and dimension vector \(\mathbf {d} \in \mathbb {N}^I\) we define the dimension of absolutely cuspidal functions on the moduli stacks of representations of dimension \(\mathbf {d}\) of a quiver Q over a finite field \(\mathbb {F}_q\), and prove that it is a polynomial in q, which we conjecture to be positive and integral. We obtain a closed formula for these dimensions of spaces of cuspidals for totally negative quivers.

Notes

Acknowledgements

We are grateful to B. Davison and A. Okounkov for some stimulating discussion and correspondences, and to B. Deng and J. Xiao for explanations concerning their work [6]. The work of the first author started during his postdoctoral appointment at MIT, before being supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). The second author was partially supported by ANR (Grant 13-BS01- 0001-01).

References

  1. 1.
    Borcherds, R.: Generalized Kac–Moody algebras. J. Algebra 115, 501–512 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bozec, T.: Quivers with loops and generalized crystals. Compos. Math. 152(10), 1999–2040 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bozec, T., Schiffmann, O., Vasserot, E.: On the number of points of nilpotent quiver varieties over finite fields, (2017). arXiv:1701.01797
  4. 4.
    Deligne, P.: Comptage de faisceaux l-adiques, in De la géométrie aux formes automorphes (I) (en l’honneur du soixantième anniversaire de Gérard Laumon, Astérisque 369, 285–312 (2015)Google Scholar
  5. 5.
    Davison, B., Meinhardt, S.: Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras, (2016). preprint arXiv:1601.02479
  6. 6.
    Deng, B., Xiao, J.: A new approach to Kac’s theorem on representations of valued quivers. Math. Z. 245, 183–199 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hausel, T.: Kac’s conjecture from Nakajima quiver varieties. Invent. Math. 181, 21–37 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hausel, T., Letellier, E., Rodriguez-Villegas, F.: Positivity for Kac polynomials and DT-invariants of quivers. Ann. Math. (2) 177(3), 1147–1168 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hua, J.: Counting representations of quivers over finite fields. J. Algebra 226(2), 1011–1033 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hua, J., Xiao, J.: On Ringel-Hall algebras of tame hereditary algebras. Algebr. Represent. Theory 5(5), 527–550 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kac, V.: Infinite root systems, representations of graphs and invariant theory. Invent. Math. 56(1), 57–92 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kang, S.-J.: Quantum deformations of generalized Kac-Moody algebras and their modules. J. Algebra 175(3), 1041–1066 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Kontsevich, M.: Notes on motives in finite characteristic, (2007). arXiv:math/0702206
  14. 14.
    Lafforgue, L.: Chtoucas de Drinfeld et correspondance de Langlands. Invent. Math. 147(1), 1–242 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Maulik, D., Okounkov, A.: Quantum groups and quantum cohomology, (2012). arXiv:1211.1287
  16. 16.
    Mozgovoy, S.: Motivic Donaldson-Thomas invariants and McKay correspondence, (2011). arXiv:1107.6044
  17. 17.
    Okounkov, A.: On some interesting Lie algebras, Conference in honor of Victor Kac, IMPA (2013). https://www.youtube.com/watch?v=H8rCJ7ls1K4. Accessed Nov 2015
  18. 18.
    Schiffmann, O., Vasserot, E.: Cohomological Hall algebras of quivers: generators, (2017). arXiv:1705.07488
  19. 19.
    Sevenhant, B., Van den Bergh, M.: A relation between a conjecture of Kac and the structure of the Hall algebra. J. Pure Appl. Algebra 160, 319–332 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut Camille JordanUniversité Lyon 1, Bât. BraconnierVilleurbanne cedexFrance
  2. 2.Département de MathématiquesUniversité de Paris-Sud Paris-SaclayOrsay CedexFrance

Personalised recommendations