Abstract induced modules for reductive algebraic groups with Frobenius maps

Abstract

Let G be a connected reductive group defined over the finite field Fq of q elements, and B be a Borel subgroup of G defined over Fq. We show that the abstract induced module M(θ) = kG ⊗ kB θ (here kH is the group algebra of H over the field k) has a composition series (of finite length) if the characteristic of k is not equal to that of Fq. In the case k = Fq and θ is a rational character, we give the necessary and sufficient condition for the existence of the composition series (of finite length) of M(θ). We determine all the composition factors whenever the composition series exist. This gives a large class of abstract infinite dimensional irreducible kG-modules.

Alvis-Curtis Duality for Representations of Reductive Groups with Frobenius Maps

“…K is the sum of all M J (θ) K with J K ⊂ J(θ). Analogous to the proof of Theorem 2.1(see [3]), E J (θ) K is also irreducible and pairwise non-isomorphic. Let O(L J ) be the subcategory of kG-modules containing objects whose subquotients are E J (θ) K .…”

“…for any θ ∈ T. Each θ ∈ T can be regarded as a character of B by the homomorphism B → T and we denote by k θ the corresponding B-module. The abstract induced module M(θ) = kG ⊗ kB k θ was studied in the paper [3] and all the composition factors of M(θ) are given in that paper. For convenience to the reader, we recall the main results here.…”

“…Assume that θ is a character of T, where T is a maximal torus contained in B also defined over F q . The abstract induced module M(θ) = kG ⊗ kB k θ was studied in the paper [3] for any field k with char k = char Fq or k = Fq and θ is rational. All the composition factors of M(θ) are given when char k = char Fq .…”

The Principal Representations of Reductive Algebraic Groups with Frobenius Maps

“…Later, motivated by Xi's idea, the structure of the permutation module k[G/B] (B is a Borel subgroup of G ) was studied in [6] for the cross characteristic case and [7] for the defining characteristic case. The paper [8] studied the general abstract induced module M(θ) = kG ⊗ kB k θ for any field k with char k = char Fq or k = Fq , where T is a maximal splitting torus contained in a F -stable Borel subgroup B and θ ∈ T (character group of T). The induced module M(θ) has a composition series (of finite length) if char k = char Fq .…”

“…The induced module M(θ) has a composition series (of finite length) if char k = char Fq . In the case k = Fq and θ is a rational character, M(θ) has such composition series if and only if θ is antidominant (see [8] for details). In both cases, the composition factors has the form E(θ) J with J ⊂ I(θ) (see Section 1 for explicit definition).…”