Low-Degree Cohomology of Finite Groups of Lie Type

Definitions and mathematical statement

Let $\Gamma$ be a finite group of Lie type, let $k$ be a coefficient field, and let $M$ be a simple $k\Gamma$-module. Its group cohomology is \[ H^i(\Gamma,M)=\operatorname{Ext}^i_{k\Gamma}(k,M), \] where $k$ denotes the trivial $k\Gamma$-module.

For a specified root datum with chosen simple coroots $\alpha_1^\vee,\ldots,\alpha_\ell^\vee$, the fundamental weights $\omega_j$ satisfy \[ \langle\omega_j,\alpha_i^\vee\rangle=\delta_{ij} \qquad(1\leq i,j\leq\ell). \] Here a fundamental dominant highest weight means one of these $\omega_j$, provided it belongs to the character lattice of the chosen algebraic group. A highest-weight description of a finite-group module requires an appropriate algebraic-group realization and a specified relationship between its highest-weight modules and the finite-group modules.

Requested computation. Determine \[ H^1(\Gamma,M)\qquad\text{and}\qquad H^2(\Gamma,M) \] for finite groups of Lie type $\Gamma$ and simple coefficient modules $M$ whose highest weight, in the intended highest-weight realization, is fundamental dominant, throughout the family intended by the source.

Unresolved scope. The supplied statement does not specify the coefficient field or its characteristic, the algebraic-group realization, the allowed forms of the finite groups, or the field-size and characteristic ranges. It also does not specify whether the coefficient modules are restrictions of simple rational algebraic-group modules or are obtained by another construction. Thus the displayed request preserves the source's target but does not define a fully quantified parameter family. Any interpretation using restrictions must specify when those restrictions are simple. These choices require clarification before a complete parameter-by-parameter formulation can be asserted.

Short English statement

Compute first and second group cohomology for finite groups of Lie type with simple coefficients labelled by fundamental dominant highest weights. The source leaves the coefficient field, highest-weight realization, group forms, and parameter ranges unspecified. The supplied registry labels the problem partially solved; that label and the accompanying dated literature assessment have not been independently verified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002016$, title Low-Degree Cohomology of Finite Groups of Lie Type, source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems. The proposed year is recorded as 2012; no individual proposer is supplied.
• Supplied report citation: Representations of Finite Groups (2012), Oberwolfach Reports, DOI 10.4171/owr/2012/16, pp. 963--1019. https://doi.org/10.4171/owr/2012/16. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions; it supplies no problem-specific page number.
• Supplied literature reference labelled First cohomology: https://arxiv.org/abs/1010.1203.
• Supplied literature reference labelled Second cohomology: https://arxiv.org/abs/1110.0228.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports computations of first and second cohomology for broad ranges of groups, primes, and fundamental or small dominant weights, while reporting no known uniform answer for all fundamental weights and parameters. This is supplied dated curation derived from the OWR material, not independently verified here; its time-sensitive status and cited claims require independent checking.
• The registry marks the statement as exact and reports a manual review finding it self-contained, mathematically coherent, and free of extraction artifacts. These are supplied verification claims; they do not resolve the scope questions recorded in this draft.