Reduced standard modules and cohomology

Abstract:

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 $H^1_{\operatorname {gen}}(G,L) \coloneq \underset {q\to \infty }{\lim } H^1(G(q),L)$ (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 $H^1(G(q),L)$ itself, still depending only on the root system. The generic $H^1$ result, and related results for $\operatorname {Ext}^1$, emerge here as a consequence of a general study, of interest in its own right, of the homological properties of certain rational modules $\Delta ^{\mathrm {red}}(\lambda ), \nabla _{\mathrm {red}}(\lambda )$, indexed by dominant weights $\lambda$, for a reductive group $G$. The modules $\Delta ^{\mathrm {red}}(\lambda )$ and $\nabla _{\mathrm {red}}(\lambda )$ arise naturally from irreducible representations of the quantum enveloping algebra $U_\zeta$ (of the same type as $G$) at a $p$th root of unity, where $p>0$ is the characteristic of the defining field for $G$. Finally, we apply our $\operatorname {Ext}^1$-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 (non-generic) bounds on $H^1(G(q),L)$.