论文标题
还原组和汉弗莱斯的谎言代数的模块化表示
Modular representations of Lie algebras of reductive groups and Humphreys' conjecture
论文作者
论文摘要
让$ g $连接为在代数封闭的特征$ p> 0 $的代数封闭字段定义的还连接的还原代数组，并假设$ p $是$ g $的根系是一个很好的素数，$ g $的根系是$ g $的元素，仅连接了$ g $的子组，lie代数代数$ \ m mathfrak $ \ m m mathfrak {g} = \ operatorneareareame aund aund a n lie lie and lie lie dund lie lie} ad $（g）$ - 不变的对称双线性表格。给定$ \ mathfrak {g} $上的线性函数$χ$，我们用$u_χ（\ mathfrak {g}）$表示$ \ mathfrak {g} $与$χ$相关的减少包络代数。由KAC-WEISFEILER的猜想（现为一个定理），任何不可约的$u_χ（\ Mathfrak {g}）$ - 模块的尺寸可除以$ p^{d（χ）} $，其中$ 2D（χ）$是含有$ g $ ubbit的$ g $ ubbit的$ g $ $ ubbit的尺寸。在本文中，我们对KAC，Humphreys和首先命名的作者在1990年代提出的自然问题给出了积极的答案，并表明任何代数$u_χ（\ Mathfrak {G}）$允许一个dimension $ p^{d（χ）} $的模块。
Let $G$ be connected reductive algebraic group defined over an algebraically closed field of characteristic $p > 0$ and suppose that $p$ is a good prime for the root system of $G$, the derived subgroup of $G$ is simply connected and the Lie algebra $\mathfrak{g} = \operatorname{Lie}(G)$ admits a non-degenerate Ad$(G)$-invariant symmetric bilinear form. Given a linear function $χ$ on $\mathfrak{g}$ we denote by $U_χ(\mathfrak{g})$ the reduced enveloping algebra of $\mathfrak{g}$ associated with $χ$. By the Kac-Weisfeiler conjecture (now a theorem), any irreducible $U_χ(\mathfrak{g})$-module has dimension divisible by $p^{d(χ)}$ where $2d(χ)$ is the dimension of the coadjoint $G$-orbit containing $χ$. In this paper we give a positive answer to the natural question raised in the 1990s by Kac, Humphreys and the first-named author and show that any algebra $U_χ(\mathfrak{g})$ admits a module of dimension $p^{d(χ)}$.
