First cohomology for finite groups of Lie type: simple modules with small dominant weights
arxiv(2010)
摘要
Let k be an algebraically closed field of characteristic p > 0, and let
G be a simple, simply connected algebraic group defined over 𝔽_p.
Given r ≥ 1, set q=p^r, and let G(𝔽_q) be the corresponding
finite Chevalley group. In this paper we investigate the structure of the first
cohomology group H^1(G(𝔽_q),L(λ)) where L(λ) is the
simple G-module of highest weight λ. Under certain very mild
conditions on p and q, we are able to completely describe the first
cohomology group when λ is less than or equal to a fundamental dominant
weight. In particular, in the cases we consider, we show that the first
cohomology group has dimension at most one. Our calculations significantly
extend, and provide new proofs for, earlier results of Cline, Parshall, Scott,
and Jones, who considered the special case when λ is a minimal nonzero
dominant weight.
更多查看译文
AI 理解论文
溯源树
样例
生成溯源树,研究论文发展脉络
Chat Paper
正在生成论文摘要