Unramified representations of reductive groups over finite rings

Stasinski, Alexander

doi:10.1090/S1088-4165-09-00350-1

Unramified representations of reductive groups over finite rings
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by Alexander Stasinski

Represent. Theory 13 (2009), 636-656

DOI: https://doi.org/10.1090/S1088-4165-09-00350-1

Published electronically: November 9, 2009

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Abstract:

Lusztig has given a construction of certain representations of reductive groups over finite local principal ideal rings of characteristic $p$, extending the construction of Deligne and Lusztig of representations of reductive groups over finite fields. We generalize Lusztig’s results to reductive groups over arbitrary finite local rings. This generalization uses the Greenberg functor and the theory of group schemes over Artinian local rings.

References

Roger W. Carter, Finite groups of Lie type, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR 794307
P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. MR 393266, DOI 10.2307/1971021
Michel Demazure, Schémas en groupes réductifs, Bull. Soc. Math. France 93 (1965), 369–413 (French). MR 197467, DOI 10.24033/bsmf.1629
François Digne and Jean Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. MR 1118841, DOI 10.1017/CBO9781139172417
Paul Gérardin, Construction de séries discrètes $p$-adiques, Lecture Notes in Mathematics, Vol. 462, Springer-Verlag, Berlin-New York, 1975. Sur les séries discrètes non ramifiées des groupes réductifs déployés $p$-adiques. MR 0396859, DOI 10.1007/BFb0082161
Marvin J. Greenberg, Schemata over local rings, Ann. of Math. (2) 73 (1961), 624–648. MR 126449, DOI 10.2307/1970321
Marvin J. Greenberg, Schemata over local rings. II, Ann. of Math. (2) 78 (1963), 256–266. MR 156855, DOI 10.2307/1970342
Gregory Hill, Regular elements and regular characters of $\textrm {GL}_n({\scr O})$, J. Algebra 174 (1995), no. 2, 610–635. MR 1334228, DOI 10.1006/jabr.1995.1143
Gregory Hill, Semisimple and cuspidal characters of $\textrm {GL}_n({\scr O})$, Comm. Algebra 23 (1995), no. 1, 7–25. MR 1311772, DOI 10.1080/00927879508825204
Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
George Lusztig, Representations of finite Chevalley groups, CBMS Regional Conference Series in Mathematics, vol. 39, American Mathematical Society, Providence, R.I., 1978. Expository lectures from the CBMS Regional Conference held at Madison, Wis., August 8–12, 1977. MR 518617, DOI 10.1090/cbms/039
G. Lusztig, Representations of reductive groups over finite rings, Represent. Theory 8 (2004), 1–14. MR 2048585, DOI 10.1090/S1088-4165-04-00232-8
Schémas en groupes. I: Propriétés générales des schémas en groupes, Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3); Dirigé par M. Demazure et A. Grothendieck. MR 0274458