Extensions for finite Chevalley groups I

AbstractLet G be a connected semisimple algebraic group defined and split over the field Fp with p elements, and k be the algebraic closure of Fp. Assume further that G is almost simple and simply connected and let G(Fq) be the finite Chevalley group consisting of Fq-rational points of G where q=pr for a non-negative integer r. In this paper, formulas are found relating extensions between simple kG(Fq)-modules and extensions over G (considered as an algebraic group over k). One of these formulas, which only holds for primes p⩾3(h−1) (where h is the Coxeter number of G), is then used to show the vanishing of self-extensions between simple kG(Fq)-modules except for certain simple modules when r=1 and the underlying root system is of type A1 or Cn