On the modular composition factors of the Steinberg representation

Let G be a finite group of Lie type and Stk be the Steinberg representation of G, defined over a field k. We are interested in the case where k has prime characteristic ℓ and Stk is reducible. Tinberg has shown that the socle of Stk is always simple. We give a new proof of this result in terms of the Hecke algebra of G with respect to a Borel subgroup and show how to identify the simple socle of Stk among the principal series representations of G. Furthermore, we determine the composition length of Stk when G=GLn(q) or G is a finite classical group and ℓ is a so-called linear prime.