The decomposition of permutation module for infinite Chevalley groups

Abstract

Let G be a connected reductive group defined over F q , the finite field with q elements. Let B be an Borel subgroup defined over F q . In this paper, we completely determine the composition factors of the induced module M(tr) = kG ⊗ kB tr (tr is the trivial B-module) for any field k.

Abstract induced modules for reductive algebraic groups with Frobenius maps

“…The idea is similar to the proof of [7,Proposition 4.4]. However the discussion is more complicated.…”

“…Similar to [7, Lemma 4.5], we have the following key lemma whose proof is identical to that of [7,Lemma 4.5] as long as one replace C J there with D(θ) J . Lemma 3.5.…”

“…We need the following claim whose proof is identical to that of [7,Proposition 4.3],as long as one replace C J there with D(θ) J .…”

“…Xi constructed a submodule filtration of the abstract induced module from the trivial character of a Borel subgroup whose subquotients indexed by the subsets of the set of simple reflections, and they turned out to be pairwisely nonisomorphic. Moreover the authors proved these subquotients are irreducible (see [6] for the cross characteristic case and see [7] for the defining characteristic case). The first author also made an attempt to study the submodule structure of some induced modules in [4] and [5].…”

The Principal Representations of Reductive Algebraic Groups with Frobenius Maps

“…So it seems to be difficult to study the abstract representations of G. However in the paper [12], Nanhua Xi studied the abstract representations of G over k by taking the direct limit of the finite dimensional representations of G q a and has got many interesting results. 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).…”