Some new examples in 1-cohomology

This paper gives some new examples in the 1-cohomology theory of finite groups of Lie type, obtained from both computer calculations and the use of several theoretical results. In particular, the paper gives the first known examples of 1-cohomology groups of dimension greater than 2 for absolutely irreducible faithful modules of a finite group. The computer calculations were made originally while checking special cases of Lusztigʹs conjecture on characteristic p representations of algebraic groups, and we take this opportunity to announce in print some results in that direction. (They reinforce Lusztigʹs conjecture, even in a strong form suggested by Kato.)