Abstract-Induced Modules for Reductive Algebraic Groups With Frobenius Maps

Abstract

Let ${\textbf{G}}$ be a connected reductive algebraic group defined over a finite field $\mathbb{F}_q$ of $q$ elements and $\textbf{B}$ be a Borel subgroup of ${\textbf{G}}$ defined over $\mathbb{F}_q$. Let $\mathbb{k}$ be a field and we assume that $\mathbb{k}=\bar{\mathbb{F}}_q $ when $\textrm{char}\ \mathbb{k}=\textrm{char} \ \mathbb{F}_q$. We show that the abstract-induced module $\mathbb{M}(\theta )=\mathbb{k}{\textbf{G}}\otimes _{\mathbb{k}\textbf{B}}\theta $ (here $\mathbb{k}\textbf{H}$ is the group algebra of $\textbf{H}$ over the field $\mathbb{k}$ and $\theta $ is a character of $\textbf{B}$ over $\mathbb{k}$) has a composition series (of finite length) if $\textrm{char}\ \mathbb{k}\ne \textrm{char} \ \mathbb{F}_q$. In the case $\mathbb{k}=\bar{\mathbb{F}}_q$ and $\theta $ is a rational character, we give a necessary and sufficient condition for the existence of a composition series (of finite length) of $\mathbb{M}(\theta )$. We determine all the composition factors whenever a composition series exists. Thus we obtain a large class of abstract infinite-dimensional irreducible $\mathbb{k}{\textbf{G}}$-modules.

The decomposition of certain abstract-induced modules over reductive algebraic groups

“…The proof can be found in [5,Proposition 2.5]. For the convenience of later discussion, we give some details about the expression of the element ṡi u i ẇη(θ) J , where u i ∈ U αi \{id} (the neutral element of U) and w satisfies that ℓ(ww J ) = ℓ(w) + ℓ(w J ).…”

“…(a) The above theorem is not true for general θ when F = k. However when F = k = Fq , Theorem 6.1 is also valid when θ is antidominant (see [5,Theorem 4.1]). Moreover, paper [5] has showed that M(θ) has such a composition series if and only if θ is antidominant (see [5,Theorem 5.1,5.2]). (b) When θ is trivial, we will prove that all the kG-modules E(tr) J (J ⊂ I) are irreducible in Section 7 (see Theorem 7.1).…”

“…Around 2013, Nanhua Xi provided a new and fundamental method to study the abstract representations of G over k by taking the direct limit of the finite-dimensional representations of G q a and got many interesting results (see [12]). In particular, he showed that the infinite Steinberg module is irreducible when char k = 0 or char F. Later, Ruotao Yang removed this restriction on char k and proved the irreducibility of Steinberg module for any field k. Now let B be a Borel subgroup of G. Inspired by by Xi's results, we study the abstract-induced module M(θ) = kG⊗ kB θ in [5], where θ is a character of B over k. We show that M(θ) has a composition series (of finite length) if char k = char F q . In the case k = Fq and θ is a rational character, we give a necessary and sufficient condition for the existence of a composition series (of finite length) of M(θ).…”

Certain complex representations of $SL_2(\bar{\mathbb{F}}_q)$

“…Later, motivated by Xi's idea, the structure of the permutation module k[G/B] (B is a fixed Borel subgroup of G ) was studied in [4] for the cross characteristic case and in [5] for the defining characteristic case. We studied the general abstract induced module M(θ) = kG ⊗ kB k θ in [6] for any field k with char k = char Fq or k = Fq , where T is a maximal torus contained in a Borel subgroup B and θ is a character of T which can also be regarded as a character of B through the homomorphism B → T. 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 [6] for details).…”

“…We studied the general abstract induced module M(θ) = kG ⊗ kB k θ in [6] for any field k with char k = char Fq or k = Fq , where T is a maximal torus contained in a Borel subgroup B and θ is a character of T which can also be regarded as a character of B through the homomorphism B → T. 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 [6] for details). In both cases, the composition factors of M(θ) are E(θ) J with J ⊂ I(θ) (see Section 2 for the explicit setting).…”