论文标题
关于简单模块的简单模块的张量张量的完全降低性，简单的代数组
On complete reducibility of tensor products of simple modules over simple algebraic groups
论文作者
论文摘要
让$ g $成为一个简单连接的简单代数群，比代数封闭的特征$ p> 0 $的代数封闭的字段$ k $。理性$ g $ - 模型的类别不是半圣事。我们考虑了两个简单$ g $ -MODULES $ L（λ）$和$ L（μ）$的张量产品的问题。使用一些有关弱最大矢量的技术结果（即Frobenius内核$ G_1 $ g $的最大矢量）在张量产品中，我们可以减少最高权重$λ$和$μ$的情况。在这种情况下，我们还证明$ l（λ）\ otimes l（μ）$在$ g $ -MODULE中完全还原为$ l（λ）。
Let $G$ be a simply connected simple algebraic group over an algebraically closed field $k$ of characteristic $p>0$. The category of rational $G$-modules is not semisimple. We consider the question of when the tensor product of two simple $G$-modules $L(λ)$ and $L(μ)$ is completely reducible. Using some technical results about weakly maximal vectors (i.e. maximal vectors for the action of the Frobenius kernel $G_1$ of $G$) in tensor products, we obtain a reduction to the case where the highest weights $λ$ and $μ$ are $p$-restricted. In this case, we also prove that $L(λ)\otimes L(μ)$ is completely reducible as a $G$-module if and only if $L(λ)\otimes L(μ)$ is completely reducible as a $G_1$-module.
