Resumen de Abstract irreducible representations of reductive algebraic groups with Borel-stable line

Xiaoyu Chen

• Let p be a prime number and k=F¯p, the algebraic closure of the finite field Fp of p elements. Let G be a connected reductive group defined over Fp and B be a Borel subgroup of G (not necessarily defined over Fp). Let kH be the group algebra of a group H over k. We show that for each (one-dimensional) character θ of B (not necessarily rational), there is a unique (up to isomorphism) irreducible kG-module L(θ) containing θ as a kB-submodule, and moreover, L(θ) is isomorphic to a parabolic induction from a finite-dimensional irreducible kL-module for some Levi subgroup L of G. Thus, we have classified and constructed all (abstract) irreducible kG-modules with B-stable line (i.e. an one-dimensional kB-submodule). As a byproduct, we give a new proof of a result of Borel and Tits on the classification of finite-dimensional irreducible kG-modules.