Demazure分解与Bott-Samelson分解的等价性
Equivalence of Demazure and Bott-Samelson Resolutions via Factorization
摘要：
令$G$、$B$和$H$分别表示复半单代数群、$G$的Borel子群和$B$中的极大复环面。选择$G$的紧致实形式$K$，使得$T=K\cap H$是$T$中的极大环面。$G$的标志空间有两种模型：复商$X=G/B$和实商$K/T$。这些模型是光滑等价的，通过映射$\tilde{\mathbf k}\colon G/B\to K/T$诱导的因式分解$G$相对于Iwasawa分解$G=KAN$，其中$N$是$B$和$H=TA$的零根。同样，舒伯特子簇$\overline{X_w}\subset X$的分解也有两种模型：$\overline{X_w}$的Demazure分解是通过复代数商构造的，而$\mathbf k(\overline{X_w})$的Bott-Samelson分解是通过紧群的实商构造的。本文利用因子分解的方法证明了这两种分解的等价性和相容性。作为应用，我们可以计算$X_w$上的标准复代数坐标到$\tilde{\mathbf k}(X_w)$上的Lu实代数坐标之间的变量变换映射。
英文摘要：
Let $G$, $B$, and $H$ denote a complex semi-simple algebraic group, a Borel subgroup of $G$, and a maximal complex torus in $B$, respectively. Choose a compact real form $K$ of $G$ such that $T=K\cap H$ is a maximal torus in $T$. Then there are two models for the flag space of $G$: the complex quotient $X=G/B$ and the real quotient $K/T$. These models are smoothly equivalent via the map $\tilde{\mathbf k}\colon G/B\to K/T$ induced by factorization in $G$ relative to the Iwasawa decomposition $G=KAN$, where $N$ is the nilradical of $B$ and $H=TA$. Likewise, there are two models for resolutions of the Schubert subvarieties $\overline{X_w}\subset X$: the Demazure resolution of $\overline{X_w}$ which is constructed via a complex algebraic quotient and the Bott-Samelson resolution of $\mathbf k(\overline{X_w})$ which is constructed as a real quotient of compact groups. This paper makes explicit the equivalence and compatibility of these two resolutions using factorization. As an application, we can compute the change of variables map relating the standard complex algebraic coordinates on $X_w$ to Lu's real algebraic coordinates on $\tilde{\mathbf k}(X_w)$.
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