Abstract-Induced Modules for Reductive Algebraic Groups With Frobenius Maps

Let G be a connected reductive algebraic group defined over a finite field F-q of q elements and B be a Borel subgroup of G defined over F-q. Let k be a field and we assume that k = (F) over bar (q) when char k = char F-q. We show that the abstract-induced module M(theta) = kG circle times(kB) theta (here kH is the group algebra of H over the field k and theta is a character of B over k) has a composition series (of finite length) if char k not equal char F-q. In the case k = (F) over bar (q) and theta is a rational character, we give a necessary and sufficient condition for the existence of a composition series (of finite length) of M(theta). We determine all the composition factors whenever a composition series exists. Thus we obtain a large class of abstract infinite-dimensional irreducible kG-modules.