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On the endomorphism algebra of generalised Gelfand-Graev representations

Author: Matthew C. Clarke
Journal: Trans. Amer. Math. Soc. 364 (2012), 5509-5524
MSC (2010): Primary 20G40
Published electronically: April 25, 2012
MathSciNet review: 2931337
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Abstract: Let $ G$ be a connected reductive algebraic group defined over the finite field $ \mathbb{F}_q$, where $ q$ is a power of a good prime for $ G$, and let $ F$ denote the corresponding Frobenius endomorphism, so that $ G^F$ is a finite reductive group. Let $ u \in G^F$ be a unipotent element and let $ \Gamma _u$ be the associated generalised Gelfand-Graev representation of $ G^F$. Under the assumption that $ G$ has a connected centre, we show that the dimension of the endomorphism algebra of $ \Gamma _u$ is a polynomial in $ q$, with degree given by $ \dim C_G(u)$. When the centre of $ G$ is disconnected, it is impossible, in general, to parametrise the (isomorphism classes of) generalised Gelfand-Graev representations independently of $ q$, unless one adopts a convention of considering separately various congruence classes of $ q$. Subject to such a convention we extend our result.

References [Enhancements On Off] (What's this?)

  • [Ale79] A. V. Alekseevskii.
    Component groups of centralizers of unipotent elements in semisimple algebraic groups.
    Trudy Tbiliss. Math. Inst. Razmadze Akad. Nauk. Gruzin SSR, 62:5-27, 1979. MR 557505 (81k:20063)
  • [BMM93] M. Broué, G. Malle, and J. Michel.
    Generic blocks of finite reductive groups.
    Astérisque, 212:7-92, 1993. MR 1235832 (95d:20072)
  • [Bru10] O. Brunat.
    Counting $ p'$-characters in finite reductive groups.
    J. Lond. Math. Soc., 81:544-562, 2010. MR 2650783
  • [BS84] W. M. Beynon and N. Spaltenstein.
    Green functions of finite Chevalley groups of type $ {E}_n$ ($ n=6,7,8$).
    J. Algebra, 88:584-614, 1984. MR 747534 (85k:20136)
  • [Car93] R. Carter.
    Finite Groups of Lie Type.
    Wiley Classics Library. Wiley, 1993. MR 1266626 (94k:20020)
  • [DLM03] F. Digne, G. Lehrer, and J. Michel.
    The space of unipotently supported class functions on a finite reductive group.
    J. Algebra, 260:111-137, 2003. MR 1973579 (2004g:20020)
  • [DM91] F. Digne and J. Michel.
    Representations of Finite Groups of Lie Type, volume 21 of London Math. Soc. Student Texts.
    Cambridge University Press, 1991. MR 1118841 (92g:20063)
  • [Dud10] O. Dudas.
    Géométrie des variétés de Deligne-Lusztig, décompositions, comologie modulo $ \ell $ et représentations.
    Ph.D. thesis, Bresançon,, 2010.
  • [Gec99] M. Geck.
    Character sheaves and generalized Gelfand-Graev characters.
    Proc. London Math. Soc., 78:139-166, 1999. MR 1658164 (2000a:20024)
  • [Gec03] M. Geck.
    An Introduction to Algebraic Geometry and Algebraic Groups, volume 10 of Oxford Graduate Texts in Mathematics.
    Oxford University Press, 2003. MR 2032320 (2004m:20001)
  • [GM99] M. Geck and G. Malle.
    On the existence of a unipotent support for the irreducible characters of a finite group of Lie type.
    Trans. Amer. Math. Soc., 352:429-456, 1999. MR 1475683 (2000c:20064)
  • [GR09a] S. M. Goodwin and G. R $ \ddot {\mathrm {o}}$hrle.
    On conjugacy of unipotent elements in finite groups of Lie type.
    J. Group Theory, 12:235-245, 2009. MR 2502217 (2010a:20101)
  • [GR09b] S. M. Goodwin and G. R $ \ddot {\mathrm {o}}$hrle.
    Rational points on generalized flag varieties and unipotent conjugacy in finite groups of Lie type.
    Trans. Amer. Math. Soc., 361:177-206, 2009. MR 2439403 (2009i:20093)
  • [Kaw85] N. Kawanaka.
    Generalized Gelfand-Graev representations and Ennola duality.
    In H. Morikawa, editor, Algebraic Groups and Related Topics, volume 6 of Advanced Studies in Pure Math., pages 179-206. Kinokuniya, Tokyo and North-Holland, Amsterdam, 1985. MR 803335 (87e:20075)
  • [Kaw86] N. Kawanaka.
    Generalized Gelfand-Graev representations for exceptional simple groups over a finite field I.
    Invent. Math, 84:575-616, 1986. MR 837529 (88a:20058)
  • [Lus86a] G. Lusztig.
    Character sheaves IV.
    Adv. Math., 59:1-63, 1986. MR 825086 (87m:20118b)
  • [Lus86b] G. Lusztig.
    Character sheaves V.
    Adv. Math., 61:103-155, 1986. MR 849848 (87m:20118c)
  • [Lus92] G. Lusztig.
    A unipotent support for irreducible representations.
    Adv. in Math., 94:139-179, 1992. MR 1174392 (94a:20073)
  • [Mac79] I. Macdonald.
    Symmetric Functions and Hall Polynomials.
    Clarendon Press, Oxford, 1979. MR 553598 (84g:05003)
  • [Miz80] K. Mizuno.
    On the conjugate classes of unipotent classes of the Chevalley groups $ {E}_7$ and $ {E}_8$.
    Tokyo J. Math, 3:391-461, 1980. MR 605099 (82m:20046)
  • [MS03] G. J. McNinch and E. Sommers.
    Component groups of unipotent centralizers in good characteristic.
    J. Algebra, 260:323-327, 2003. MR 1976698 (2004d:20054)
  • [Sho82] T. Shoji.
    On the Green polynomials of Chevalley groups of type $ {F}_4$.
    Comm. Algebra, 10:505-543, 1982. MR 647835 (83d:20030)
  • [Sho83] T. Shoji.
    On the Green polynomials of classical groups.
    Invent. Math., 74:239-267, 1983. MR 723216 (85f:20032)
  • [Sho87] T. Shoji.
    Green functions of reductive groups over a finite field.
    In P. Fong, editor, The Arcata Conference on Representations of Finite Groups, volume 47 of Proceedings of Symposia in Pure Math., pages 289-301. AMS, 1987. MR 933366 (88m:20014)
  • [Sho88] T. Shoji.
    Geometry of orbits and Springer correspondence.
    Astérique, 168:61-140, 1988. MR 1021493 (91b:20057)
  • [Spa82] N. Spaltenstein.
    Classes Unipotentes et Sous-Groupes de Borel, volume 946 of Lecture Notes in Math.
    Springer, 1982. MR 672610 (84a:14024)
  • [SS70] T. A. Springer and R. Steinberg.
    Conjugacy classes.
    In A. Borel et al., editor, Seminar on Algebraic Groups and Related Finite Groups, volume 131 of Lecture Notes in Math. Springer, 1970. MR 0268192 (42:3091)
  • [Ste67] R. Steinberg.
    Lectures on Chevalley Groups.
    Yale University, 1967.
  • [Yok67] T. Yokonuma.
    Sur la commutant d'une représentation d'une groupe Chevalley fini.
    C. R. Acad. Sci. Paris, 264:433-436, 1967. MR 0212102 (35:2977)
  • [Yok68] T. Yokonuma.
    Sur la commutant d'une représentation d'une groupe Chevalley fini.
    J. Fac. Sci. Univ. Tokyo, 15:115-129, 1968. MR 0248249 (40:1501)

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Additional Information

Matthew C. Clarke
Affiliation: Department of Mathematics, Trinity College, Cambridge, CB2 1TQ, United Kingdom

Received by editor(s): September 21, 2010
Received by editor(s) in revised form: January 11, 2011
Published electronically: April 25, 2012
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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