arXiv 1406.57442014-06math.AG

Demazure根和球形变种：水平SL的例子（2）-动作

Demazure roots and spherical varieties: the example of horizontal SL(2)-actions

Kevin Langlois, Alexander Perepechko

备注：: Added Section 4; modified main result, Theorem 5.18 now; other changes

摘要：

设$G$是连通约化群，$X$是仿射$G$ -球面簇.我们证明了$G$规范化的$X$上的$\mathbb{G}_{a}$ -作用的分类可以归结为半直积$\mathbb{G}_{a}\rtimes G$作用下的拟仿射齐性空间的描述，具有以下性质.诱导的$G$ -作用是球面的，开轨道的补轨道要么是空的，要么是余维为1的$G$ -轨道。这些齐性空间由$G$的特征格$\mathbb{X}(G)$的子集${\rm Rt}(X)$参数化，我们称之为$X$的Demazure根集。当$G$是${\rm SL}_{2}$和代数环面的半直积时，我们给出了集合${\rm Rt}(X)$的完整描述;我们特别证明了${\rm Rt}(X)$可以明确地作为$\mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{X}(G)$中多面体的有限并集和$\mathbb{X}(G)$的子格的交集而得到.我们猜想，${\rm Rt}(X)$可以描述在一个类似的组合方式为任意的仿射球面簇$X$。

英文摘要：

Let $G$ be a connected reductive group, and let $X$ be an affine $G$-spherical variety. We show that the classification of $\mathbb{G}_{a}$-actions on $X$ normalized by $G$ can be reduced to the description of quasi-affine homogeneous spaces under the action of a semi-direct product $\mathbb{G}_{a}\rtimes G$ with the following property. The induced $G$-action is spherical and the complement of the open orbit is either empty or a $G$-orbit of codimension one. These homogeneous spaces are parametrized by a subset ${\rm Rt}(X)$ of the character lattice $\mathbb{X}(G)$ of $G$, which we call the set of Demazure roots of $X$. We give a complete description of the set ${\rm Rt}(X)$ when $G$ is a semi-direct product of ${\rm SL}_{2}$ and an algebraic torus; we show particularly that ${\rm Rt}(X)$ can be obtained explicitly as the intersection of a finite union of polyhedra in $\mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{X}(G)$ and a sublattice of $\mathbb{X}(G)$. We conjecture that ${\rm Rt}(X)$ can be described in a similar combinatorial way for an arbitrary affine spherical variety $X$.

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摘要：

英文摘要：

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