On the principal representations of reductive groups in defining characteristic.

Let p be a prime number. Let k = F ¯ p , the algebraic closure of field F p of p elements, and G be a connected reductive group defined over F p 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) = k G ⊗ k B θ being finite length. [ABSTRACT FROM AUTHOR]