Abstract
Let G be a connected unipotent group over a finite field \({\mathbb F_q}\). In this article, we propose a definition of L-packets of complex irreducible representations of the finite group \({G(\mathbb F_q)}\) and give an explicit description of L-packets in terms of the so-called admissible pairs for G. We then apply our results to show that if the centralizer of every geometric point of G is connected, then the dimension of every complex irreducible representation of \({G(\mathbb F_q)}\) is a power of q, confirming a conjecture of Drinfeld. This paper is the first in a series of three papers exploring the relationship between representations of a group of the form \({G(\mathbb F_q)}\) (where G is a unipotent algebraic group over \({\mathbb F_q}\)), the geometry of G, and the theory of character sheaves.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Artin, M., Grothendieck, A., Verdier, J.-L.: SGA 4: Théorie des Topos et Cohomologie Étale des Schémas. In: Lecture Notes in Math. 269, 270 and 305, Springer (1972/1973)
Bakalov B., Kirillov A. Jr: Lectures on Tensor Categories and Modular Functors, University Lecture Series, vol. 21. American Mathematical Society, Providence (2001)
Begueri, L.: Dualité sur un corps local à corps résiduel algébriquement clos. Mém. Soc. Math. France (N.S.) no. 4 (1980/1981)
Beilinson, A.A., Bernstein, J., Deligne, P.: Faisceaux pervers. In: Analyse et topologie sur les espaces singuliers (I), Astérisque, vol. 100 (1982)
Bernstein J., Lunts V.: Equivariant Sheaves and Functors, Lecture Notes in Mathematics. vol. 1578. Springer, Berlin (1994)
Boyarchenko, M.: Base Change Maps for Unipotent Algebra Groups. Preprint, January 2006, arXiv:math.RT/0601133
Boyarchenko, M.: Representations of Unipotent Groups over Local Fields and Gutkin’s Conjecture. arXiv:1003.2742, submitted for publication
Boyarchenko, M.: Character Sheaves and Characters of Unipotent Groups over Finite Fields (2010, in preparation)
Boyarchenko, M., Drinfeld, V.: Character Sheaves on Unipotent Groups in Positive Characteristic: foundations. Preprint, October 2008, available online: arXiv:0810.0794, version 1 (version 2 is in preparation)
Boyarchenko, M., Drinfeld, V.: A motivated introduction to character sheaves and the orbit method for unipotent groups in positive characteristic., Preprint, September 2006, arXiv:math.RT/0609769, version 1
Breen, L.: Extensions du groupe additif sur le site parfait. In: Algebraic Surfaces (Orsay, 1976–1978), pp. 238–262, Lecture Notes in Mathematics, vol. 868. Springer, Berlin (1981)
Breen, L.: On the Classification of 2-Gerbes and 2-Stacks. Astérisque, vol. 225 (1994)
Deligne P.: La conjecture de Weil II. Publ. Math. IHES 52, 137–252 (1980)
Deligne, P., Boutot, J.-F., Illusie, L., Verdier, J.-L.: SGA \({4\frac{1}{2}}\): Cohomologie Étale. In: Lecture Notes in Mathematics, vol. 569. Springer, Heidelberg (1977)
Deshpande, T.: Heisenberg Idempotents on Unipotent Groups. To appear in Math. Res. Lett. arXiv:0907.3344, version 2
Drinfeld, V.: Lectures on the Geometric Aspects of Character Theory for Unipotent Groups (unpublished), University of Chicago (2005, July)
Ekedahl, T.: On the adic formalism. In: The Grothendieck Festschrift, vol. II, pp. 197–218, Progr. Math. 87, Birkhäuser Boston, Boston (1990)
Etingof P., Nikshych D., Ostrik V.: On fusion categories. Ann. Math. 162, 581–642 (2005)
Greenberg M.J.: Perfect closures of rings and schemes. Proc. Am. Math. Soc. 16, 313–317 (1965)
Grothendieck, A., Raynaud, M.: SGA 1: Revêtements étales et groupe fondamental. Documents Mathématiques (Paris) 3, Soc. Math. de France, Paris (2003) math.AG/0206203
Grothendieck, A., Bucur, I., Houzel, C., Illusie, L., Jouanolou, J.-P., Serre, J.-P.: SGA 5: cohomologie ℓ-adique et Fonctions L. In: Lecture Notes in Mathematics, vol. 589, Springer (1977)
Gutkin E.A.: Representations of algebraic unipotent groups over a self-dual field. Funkts. Analiz i Ego Prilozheniya 7, 80 (1973)
Halasi Z.: On the characters and commutators of finite algebra groups. J. Algebra 275, 481–487 (2004)
Higman G.: Enumerating p-groups. I. Inequalities. Proc. Lond. Math. Soc. 10(3), 24–30 (1960)
Isaacs I.M.: Characters of groups associated with finite algebras. J. Algebra 177, 708–730 (1995)
Kamgarpour M.: Stacky abelianization of algebraic groups. Transform. Groups 14(4), 825–846 (2009)
Kirillov A.A.: Unitary representations of nilpotent Lie groups. Uspehi Mat. Nauk 17(4(106)), 57–110 (1962)
Lang S.: Algebraic groups over finite fields. Am. J. Math. 78, 555–563 (1956)
Langlands, R.P.: Les débuts d’une formule des traces stable, Publ. Math. Univ. Paris VII 13, Paris (1983)
Laszlo Y., Olsson M.: The six operations for sheaves on Artin stacks II: Adic coefficients. Publ. Math. IHES 107, 169–210 (2008)
Lusztig, G.: Character sheaves and generalizations. In: Etingof, P., Retakh, V., Singer, I.M. (eds.) The Unity of Mathematics, Progress in Mathematics, vol. 244, pp. 443–455. Birkhäuser Boston, Boston (2006)
Milne, J.S.: Arithmetic Duality Theorems, 2nd edn. BookSurge, LLC, Charleston (2006), available at http://www.jmilne.org/math/
Mumford, D.: Bi-extensions of formal groups. In: Algebraic Geometry (Proceedings of Internat. Coll. Bombay, 1968), pp. 307–322. Oxford University Press (1969)
Serre, J.-P.: Groupes proalgébriques. Publ. Math. IHES 7 (1960)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Boyarchenko, M. Characters of unipotent groups over finite fields. Sel. Math. New Ser. 16, 857–933 (2010). https://doi.org/10.1007/s00029-010-0036-9
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-010-0036-9