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Bulletin of the London Mathematical Society Advance Access originally published online on December 21, 2007
Bulletin of the London Mathematical Society 2007 39(6):1019-1028; doi:10.1112/blms/bdm080
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© 2007 London Mathematical Society

Projective modules for Frobenius kernels and finite Chevalley groups

Zongzhu Lin

Department of Mathematics
Kansas State University
Manhattan, KS 66506
USA
zlin@math.ksu.edu

Daniel K. Nakano

Department of Mathematics
University of Georgia
Athens, GA 30602
USA

Received 18 February 2006. Revision received 11 May 2007.

Let G be a connected reductive algebraic group with a Frobenius morphism F: G -> G defined over a finite field Fpr. The main result of the paper is to prove that any rational G-module M which is projective when restricted to the Frobenius kernel Gr = Ker(F) is also projective over the split and twisted finite Chevalley groups. In 1987, Parshall conjectured this statement for r = 1 in the split case. The authors verified this in 1999 with the possible exclusion of primes 2 and 3 in non-simply laced cases. The converse of the main result is also discussed for split groups in this paper.

2000 Mathematics Subject Classification 17B55, 20G (primary), 17B50 (secondary).

The research of the first author was supported in part by NSF grant DMS-0200673, and that of the second author in part by NSF grant DMS-0400548.


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