论文标题
Nilfibre的规范组件，用于抛物线属于$ a $的抛物线伴随动作
The Canonical Component of the nilfibre for Parabolic adjoint action in type $A$
论文作者
论文摘要
这项工作是[Fittouhi和Joseph的延续，抛物线的伴奏动作，Weierstrass部分和Nilfibre type $ a $]的组件。让$ p $是一个不可约的简单代数$ g $，$ p'$的抛物线亚组，其派生的群体和$ \ m rathfrak m $是其Lie代数的nilradical。理查森的定理意味着subalgebra $ \ mathbb c [\ mathfrak m]^{p'} $，由$ p $ semi-invariants in $ \ mathbb c [\ mathfrak m] $跨越了。如果限制映射诱导$ \ Mathfrak m $ $ p'$在$ \ mathfrak m $上的动作的线性subvarietio $ e+v $，则称为Weierstrass部分，如果限制映射诱导了$ \ Mathbb c [\ Mathfrak M]^{P'} $ \ Mathbb c'} $ \ Mathbb C [e e e e e e+v]的同构。因此，只有在后者是多项式的情况下，才可以存在Weierstrass部分，但是即使存在它的存在也远未确保。 $ \ mathfrak m $中的Weierstrass节的存在是由一般组合建筑建立的。值得注意的是$ e \ in \ mathscr n $，是具有线性独立根的根向量的总和。 Weierstraass节$ E+V $在不同选择的抛物性选择方面看起来截然不同，但是在所有情况下都具有统一的结构，并且存在。它称为“规范Weierstrass部分”。 它在[Fittouhi和Joseph，Loc。 cit。然后，这些根矢量的线性跨度$ e_ {vs} $位于$ \ mathscr n^e $中，其关闭仅为$ \ Mathscr n^e $。然而，这个结果表明，$ \ mathscr n^e $不必承认密集的$ p $轨道。但是，仅在获得示例的特殊情况下进行了验证，该定理表明$ \ Mathscr n^e $可能未能承认一个密集的$ p $轨道。在这里给出了一般证明。最后，定义了从组成到不同非阴性整数集的地图。它的图像显示为确定规范的WeierStrass部分。
This work is a continuation of [Fittouhi and Joseph, Parabolic adjoint action, Weierstrass Sections and components of the nilfibre in type $A$]. Let $P$ be a parabolic subgroup of an irreducible simple algebraic group $G$, $P'$ its derived group and $\mathfrak m$ be the nilradical to its Lie algebra. A theorem of Richardson implies that the subalgebra $\mathbb C[\mathfrak m]^{P'}$, spanned by the $P$ semi-invariants in $\mathbb C[\mathfrak m]$, is polynomial. A linear subvariety $e+V$ of $\mathfrak m$ is is called a Weierstrass section for the action of $P'$ on $\mathfrak m$, if the restriction map induces an isomorphism of $\mathbb C[\mathfrak m]^{P'}$ onto $\mathbb C[e+V]$. Thus a Weierstrass section can exist only if the latter is polynomial, but even when this holds its existence is far from assured. The existence of a Weierstrass section $e+V$ in $\mathfrak m$ was established by a general combinatorial construction. Notably $e \in \mathscr N$ and is a sum of root vectors with linearly independent roots. The Weierstraass section $e+V$ looks very different for different choices of parabolics but nevertheless has a uniform construction and exists in all cases. It is called the "canonical Weierstrass section". It was announced in [Fittouhi and Joseph, loc. cit.] that one may augment $e$ to an element $e_{VS}$ by adjoining root vectors. Then the linear span $E_{VS}$ of these root vectors lies in $\mathscr N^e$ and its closure is just $\mathscr N^e$. Yet this result shows that $\mathscr N^e$ need not admit a dense $P$ orbit. However this theorem was only verified in the special case needed to obtain the example showing that $\mathscr N^e$ may fail to admit a dense $P$ orbit. Here a general proof is given. Finally a map from compositions to the set of distinct non-negative integers is defined. Its image is shown to determine the canonical Weierstrass section.
