论文标题
带有Borel稳定线的还原基团的不可还原模块
Irreducible Modules of Reductive Groups with Borel-stable Line
论文作者
论文摘要
令$ p $为素数，$ \ bbbk = \ bar {\ mathbb {f}} _ p $，有限字段$ \ mathbb {f} _p $ p $ elements的代数关闭。令$ {\ bf g} $为$ \ mathbb {f} _p $和$ {\ bf b} $定义的连接还原组为$ {\ bf g} $的孔子子组（不一定在$ \ thebb {f} _p $上定义为$ {\ bf g} $）。我们表明，对于$ {\ bf b} $的每个（一维）$θ$（不一定是理性的），都有一个唯一的（不可记录的）$ \ bbbk {\ bf g} $ - 模块 - 模块 - module $ \ mathbb {l}（θ） b} $ - 子模块，此外，$ \ mathbb {l}（θ）$是同构对抛物线诱导的同构，来自有限维的$ \ bbbk {\ bf l} $ - 用于某些LEVI子组$ $ {\ bf l} $ {\ bf l} $ of $ {\ bf l} $的模块。因此，我们已经对所有（摘要）$ \ bbbk {\ bf g} $ - 带有$ {\ bf b} $稳定线的模块（即一维$ \ bbbk {\ bf bf b b} $ - suppoule）。作为副产品，我们给出了一个新的证据，证明了borel的结果和山雀，以分类有限维$ \ bbbk {\ bf g} $ - 模块。
Let $p$ be a prime number and $\Bbbk=\bar{\mathbb{F}}_p$, the algebraic closure of the finite field $\mathbb{F}_p$ of $p$ elements. Let ${\bf G}$ be a connected reductive group defined over $\mathbb{F}_p$ and ${\bf B}$ be a Borel subgroup of ${\bf G}$ (not necessarily defined over $\mathbb{F}_p$). We show that for each (one-dimensional) character $θ$ of ${\bf B}$ (not necessarily rational), there is a unique (up to isomorphism) irreducible $\Bbbk{\bf G}$-module $\mathbb{L}(θ)$ containing $θ$ as a $\Bbbk{\bf B}$-submodule, and moreover, $\mathbb{L}(θ)$ is isomorphic to a parabolic induction from a finite-dimensional irreducible $\Bbbk{\bf L}$-module for some Levi subgroup ${\bf L}$ of ${\bf G}$. Thus, we have classified and constructed all (abstract) irreducible $\Bbbk{\bf G}$-modules with ${\bf B}$-stable line (i.e. an one-dimensional $\Bbbk{\bf B}$-submodule). As a byproduct, we give a new proof of a result of Borel and Tits on the classification of finite-dimensional irreducible $\Bbbk{\bf G}$-modules.
