A generalization of Pillen’s theorem for principal series modules

Yoshii, Yutaka

doi:10.1090/S0002-9939-2012-11395-4

A generalization of Pillen’s theorem for principal series modules
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Proc. Amer. Math. Soc. 140 (2012), 3761-3768 Request permission

Abstract:

Let $G$ be a connected, semisimple and simply connected algebraic group defined and split over the finite field of order $p$. Pillen proved in 1997 that the highest weight vectors of some Weyl $G$-modules generate the principal series modules as submodules for the corresponding finite Chevalley groups. This result is generalized in this paper.

References

Henning Haahr Andersen, A sum formula for tilting filtrations, J. Pure Appl. Algebra 152 (2000), no. 1-3, 17–40. Commutative algebra, homological algebra and representation theory (Catania/Genoa/Rome, 1998). MR 1783982, DOI 10.1016/S0022-4049(99)00141-3
R. W. Carter and G. Lusztig, Modular representations of finite groups of Lie type, Proc. London Math. Soc. (3) 32 (1976), no. 2, 347–384. MR 396731, DOI 10.1112/plms/s3-32.2.347
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
Jens C. Jantzen, Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne, J. Reine Angew. Math. 317 (1980), 157–199 (German). MR 581341, DOI 10.1515/crll.1980.317.157
Jens Carsten Jantzen, Zur Reduktion modulo $p$ der Charaktere von Deligne und Lusztig, J. Algebra 70 (1981), no. 2, 452–474 (German). MR 623819, DOI 10.1016/0021-8693(81)90229-5
Jens Carsten Jantzen, Filtrierungen der Darstellungen in der Hauptserie endlicher Chevalley-Gruppen, Proc. London Math. Soc. (3) 49 (1984), no. 3, 445–482 (German). MR 759299, DOI 10.1112/plms/s3-49.3.445
Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
Olivier Mathieu, Filtrations of $G$-modules, Ann. Sci. École Norm. Sup. (4) 23 (1990), no. 4, 625–644. MR 1072820
Cornelius Pillen, Reduction modulo $p$ of some Deligne-Lusztig characters, Arch. Math. (Basel) 61 (1993), no. 5, 421–433. MR 1241047, DOI 10.1007/BF01207541
Cornelius Pillen, Loewy series for principal series representations of finite Chevalley groups, J. Algebra 189 (1997), no. 1, 101–124. MR 1432367, DOI 10.1006/jabr.1996.6886
Forrest Richen, Modular representations of split $BN$ pairs, Trans. Amer. Math. Soc. 140 (1969), 435–460. MR 242972, DOI 10.1090/S0002-9947-1969-0242972-1
Hideki Sawada, A characterization of the modular representations of finite groups with split $(B, N)$-pairs, Math. Z. 155 (1977), no. 1, 29–41. MR 450384, DOI 10.1007/BF01322605