The Second Cohomology of Simple SL₃-Modules

Abstract

Let G be the simple, simply connected algebraic group SL ₃ defined over an algebraically closed field K of characteristic p > 0. In this article, we find H ²(G, V) for any irreducible G-module V. When p > 7, we also find H ²(G(q), V) for any irreducible G(q)-module V for the finite Chevalley groups G(q) = SL(3, q) where q is a power of p.

Key Words:

• Algebraic groups
• Cohomology
• Finite groups of Lie type
• Representation theory

2000 Mathematics Subject Classification:

• Primary 20G10, 20G30, 20J06
• Secondary 13D07

ACKNOWLEDGMENTS

Some of the work in this article was prepared towards the author's PhD qualification under the supervision of Prof. M. W. Liebeck, with financial support from the EPSRC. We would like to thank Prof. Liebeck for his help in producing this article. Also we thank the anonymous referee for many helpful suggestions and corrections, and the editor P. Tiep for his suggestion to extend the work to cover the finite groups of Lie type.

Notes

Recent work by the author together with B. Parshall and L. Scott answers Question 3.10 in the affirmative, with no condition on p but depending on the root system. That result is then used to give answers to Question 3.8 also.

Communicated by P. Tiep.