论文标题
kac-moody群体反演子组的强烈完整性
Strong integrality of inversion subgroups of Kac-Moody groups
论文作者
论文摘要
令$ a $为对应的kac-moody代数$ \ frak {g} $ of $ {\ mathbb q} $的对应的概括cartan矩阵。令$ v = v^λ$为可集成的最高权重$ \ frak {g} $ - 模块，让$ v _ {\ mathbb z} = v^λ_ {\ mathbb z} $ be a $ {\ mathbb z} z $ form $ v $。令$ g $为关联的最小表示理论kac-moody group，让$ g（{\ mathbb z}）$为其积分子组。令$γ（{\ Mathbb z}）$为$ g $的Chevalley子组，即，在$ v $中稳定晶格$ v _ {\ mathbb z} $的子组。对于$ g $的亚组$ m $，我们说$ m $是不可或缺的，如果$ m \ cap g（{\ Mathbb z}）= m \capγ（{\ Mathbb z}）$，并且如果存在V^λ_{\ Mathbb z $ g的$ v \ in cd in cd in cd in cd in cd in v^y cd in cd in cd in cd in v \ in V _ {\ Mathbb {z}} $表示$ g \ in G（{\ Mathbb {z}}}）$。我们证明了$ g $的反转子群的强烈整合$ u _ {（w）} $，对于$ w \ in w $中的$ w \，$ u _ {（w）} $是由$ w^{ - 1} $将正真实根组生成的基团。我们用它来证明由通勤实际根组产生的单位子组$ u $ u $ $ g $的子组的强大完整性。当$ a $具有等级2时，这给出了子群体$ u_1 $和$ u_2 $的强烈完整性，其中$ u = u = u_ {1} {\ ligal {*}} \ u_ {2} $，每个$ u_ {i {i} $由``一半''积极的真实动力产生。
Let $A$ be a symmetrizable generalized Cartan matrix with corresponding Kac--Moody algebra $\frak{g}$ over ${\mathbb Q}$. Let $V=V^λ$ be an integrable highest weight $\frak{g}$-module and let $V_{\mathbb Z}=V^λ_{\mathbb Z}$ be a ${\mathbb Z}Z$-form of $V$. Let $G$ be an associated minimal representation-theoretic Kac--Moody group and let $G({\mathbb Z})$ be its integral subgroup. Let $Γ({\mathbb Z})$ be the Chevalley subgroup of $G$, that is, the subgroup that stabilizes the lattice $V_{\mathbb Z}$ in $V$. For a subgroup $M$ of $G$, we say that $M$ is integral if $M\cap G({\mathbb Z})=M\cap Γ({\mathbb Z})$ and that $M$ is strongly integral if there exists $v\in V^λ_{\mathbb Z}$ such that, for all $g\in M$, $g\cdot v\in V_{\mathbb{Z}}$ implies $g\in G({\mathbb{Z}})$. We prove strong integrality of inversion subgroups $U_{(w)}$ of $G$ where, for $w\in W$, $U_{(w)}$ is the the group generated by positive real root groups that are flipped to negative roots by $w^{-1}$. We use this to prove strong integrality of subgroups of the unipotent subgroup $U$ of $G$ generated by commuting real root groups. When $A$ has rank 2, this gives strong integrality of subgroups $U_1$ and $U_2$ where $U=U_{1}{\Large{*}}\ U_{2}$ and each $U_{i}$ is generated by `half' the positive real roots.
