Let G be a connected reductive group defined over F q (the finite field with q elements, where q is a power of the prime number p), with the standard Frobenius map F . Let B be an F -stable Borel subgroup. Let k be a field (may not be algebraically closed) of characteristic 0 ≤ r = p. In this paper, we completely determine the composition factors of the permutation module module k[G/B] = kG ⊗ kB tr (here kH is the group algebra of the group H, and tr is the trivial Bmodule). In particular, we find a large family of infinite dimensional absolutely irreducible abstract representations of G over k.for some 1 ≤ t ′ ≤ t. We abbreviate w 1 = s i t ′ · · · s i 1 w ∈ Y J for convenience. Consider the following two cases.
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.
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.
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.
We introduce the principal representation category of reductive groups with Frobenius maps and show that this category is a highest weight category when the ground field is complex field C. We also study certain kind of bound quiver algebras whose representations are related to the principal representation category.
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