Mod p Reducibility of Unramified Representations of Finite Groups of Lie Type

Dedicated to the memory of Professor A. I. Kostrikin The main problem under discussion is to determine, for quasi‐simple groups of Lie type G , irreducible representations ϕ of G that remain irreducible under reduction modulo the natural prime p . The method is new. It works only for p >3 and for representations ϕ that can be realized over an unramified extension of Q p , the field of p ‐adic numbers. Under these assumptions, the main result says that the trivial and the Steinberg representations of G are the only representations in question provided G is not of type A1. This is not true for G =SL(2, p ). The paper contains a result of independent interest on infinitesimally irrreducible representations ρ of G over an algebraically closed field of characteristic p . Assuming that G is not of rank 1 and G ≠ G 2 (5), it is proved that either the Jordan normal form of a root element contains a block of size d with 1< d < p ‐1 or the highest weight of ρ is equal to p ‐1 times the sum of the fundamental weights. 2000 Mathematical Subject Classification : 20C33, 20G15.