arXiv 1702.056862017-02math.RT

半单群的Frobenius映射的主列表示

On the principal series representations of semisimple groups with Frobenius maps

Xiaoyu Chen

备注：: A stronger result which cover this paper is originated by the same author, so I withdraw this paper

摘要：

设${\bf G}$是$\Bbbk=\bar{\mathbb{F}}_q$上的单连通半单代数群，是$\mathbb{F}_q$（含$q=p^e$元素的有限域）的代数闭包，$F$是标准Frobenius映射.设${\bf B}$是$F$稳定的Borel子群，${\bf T}$是包含在${\bf B}$中的$F$稳定的极大环面.为任何$r>0$设置$G_{q^r}={\bf G}^{F^r}$和$B_{q^r}={\bf B}^{F^r}$。本文研究了原诱导模$\op{Ind}_{\bf B}^{\bf G}λ=\Bbbk{\bf G}\otimes_{\Bbbk{\bf B}}λ$（这里$\Bbbk{\bf H}$是群${\bf H}$的群代数，$λ$是${\bf T}$作为${\bf B}$ -模的一个有理特征标）.证明了若$λ$正则且占优，则对任何$r>0$都存在满射${\bf G}$ -模同态$\op{Ind}_{B_{q^r}}^{\bf G}λ\rightarrow \op{St}\otimes L(λ)$，其中$\op{St}$是Nanhua Xi定义的无穷维Steinberg模.因此，我们表明，$\op{Ind}_{\bf B}^{\bf G}λ$是不可约的，如果$λ$当且仅当$λ$是正规和反显性。此外，对于${\bf G}=SL_2(\bar{\mathbb{F}}_q)$和$0<λ<p$，我们证明了$\op{Ind}_{\bf B}^{\bf G}λ$有无穷多个组成因子，每个有限维。因此，我们发现某些$λ$的$\op{Ind}_{\bf B}^{\bf G}λ$有一个无限的子模过滤一般${\bf G}$。

英文摘要：

Let ${\bf G}$ be a simply connected semisimple algebraic group over $\Bbbk=\bar{\mathbb{F}}_q$, the algebraically closure of $\mathbb{F}_q$ (the finite field with $q=p^e$ elements), and $F$ be the standard Frobenius map. Let ${\bf B}$ be an $F$-stable Borel subgroup and ${\bf T}$ an $F$-stable maximal torus contained in ${\bf B}$. Set $G_{q^r}={\bf G}^{F^r}$ and $B_{q^r}={\bf B}^{F^r}$ for any $r>0$. This paper studies the original induced module $\op{Ind}_{\bf B}^{\bf G}λ=\Bbbk{\bf G}\otimes_{\Bbbk{\bf B}}λ$ (here $\Bbbk{\bf H}$ is the group algebra of the group ${\bf H}$, and $λ$ is a rational character of ${\bf T}$ regarded as a ${\bf B}$-module). We show that if $λ$ is regular and dominant, then there is a surjective ${\bf G}$-module homomorphism $\op{Ind}_{B_{q^r}}^{\bf G}λ\rightarrow \op{St}\otimes L(λ)$ for any $r>0$, where $\op{St}$ is the infinite dimensional Steinberg module defined by Nanhua Xi. As a consequence, we show that $\op{Ind}_{\bf B}^{\bf G}λ$ is irreducible if $λ$ if and only if $λ$ is regular and antidominant. Moreover, for ${\bf G}=SL_2(\bar{\mathbb{F}}_q)$ and $0<λ<p$, we show that $\op{Ind}_{\bf B}^{\bf G}λ$ have infinite many composition factors with each finite dimensional. Consequently, we find certain $λ$ for which $\op{Ind}_{\bf B}^{\bf G}λ$ has an infinite submodule filtration for the general ${\bf G}$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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