On the principal representations of reductive groups in defining characteristic

Abstract

Let p be a prime number. Let k=F¯p, the algebraic closure of field Fp of p elements, and G be a connected reductive group defined over Fp and B be a Borel subgroup of G. Let θ be a (one dimensional) character of B (may not be rational). We give the necessary and sufficient condition for M(θ)=kG⊗kBθ being finite length.