Reduced Standard Modules and Cohomology

First cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts, including Wiles’s famous paper (1995). Internal to group theory, 1-cohomology plays a role in the general theory of maximal subgroups of finite groups, as developed by Aschbacher and Scott (1985).One can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible. In this case, R. Guralnick (1986) conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups. This paper provides the first general positive results on this conjecture, proving that the generic 1-cohomology H1 gen(G,L) := lim q→∞ H1(G(q), L) (see Cline, Parshall, Scott, and van der Kallen) (1977) of a finite group G(q) of Lie type, with absolutely irreducible coefficients L (in the defining characteristic of G), is bounded by a constant depending only on the root system. In all cases, we are able to improve this result to a bound on H1(G(q), L) itself, still depending only on the root system. The generic H1 result, and related results for Ext, emerge here as a consequence of a general study, of interest in its own right, of the homological properties of certain rational modules ∆(λ),∇red(λ), indexed by dominant weights λ, for a reductive group G. The modules ∆red(λ) and ∇red(λ) arise naturally from irreducible representations of the quantum enveloping algebra Uζ (of the same type as G) at a pth root of unity, where p > 0 is the characteristic of the defining field for G. Finally, we apply our Ext1-bounds, results of Bendel, Nakano, and Pillen (2006), as well as results of Sin (1993), (1992), (1994) on the Ree and Suzuki groups to obtain the (nongeneric) bounds on H1(G(q), L).