First cohomology groups of Chevalley groups in cross characteristic

• 成果类型: Article
• 署名作者: Guralnick, Robert M.; Pham Huu Tiep
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.174.1.16
• 发表日期: 2011
• 页码: 543-559
• 关键词: modules; REPRESENTATIONS; 1-cohomology; Examples; 2-blocks

摘要

Let G be a simple Chevalley group defined over F-q. We show that if r does not divide q and k is an algebraically closed field of characteristic r, then very few irreducible kG-modules have nonzero H-1(G, V). We also give an explicit upper bound for dim H-1(G, V) for V an irreducible kG-module that does not depend on q, but only on the rank of the group. Cline, Parshall and Scott showed that such a bound exists when r vertical bar q. We obtain extremely strong bounds in the case that a Borel subgroup has no fixed points on V.