Mathematische Zeitschrift

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

Counting absolutely cuspidals for quivers

  • T. Bozec
  • O. SchiffmannEmail author


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.



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).


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Authors and Affiliations

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

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