标题： 通过分解的Demazure和Bott-Samelson解的等价性 显示英文标题

标题： Equivalence of Demazure and Bott-Samelson Resolutions via Factorization

摘要： 设$G$、$B$和$H$分别表示一个复半单代数群、$G$的一个Borel子群以及$B$中的一个极大复环面。 选择一个紧致实形式$K$的$G$，使得$T=K\cap H$是$T$中的一个极大环面。 然后，$G$的旗空间有两种模型：复数商$X=G/B$和实数商$K/T$。 这些模型通过由 $\tilde{\mathbf k}\colon G/B\to K/T$ 引入的映射在 $G$ 中的分解下光滑等价，相对于Iwasawa分解 $G=KAN$，其中 $N$ 是 $B$ 和 $H=TA$ 的幂零radical。 同样，存在两种关于舒伯特子簇$\overline{X_w}\subset X$的消解模型：通过复代数商构造的$\overline{X_w}$的戴门措尔消解，以及作为紧致群实商构造的$\mathbf k(\overline{X_w})$的博特-萨姆尔森消解。本文通过分解明确阐述了这两种消解之间的等价性和相容性。作为应用，我们可以计算将$X_w$上的标准复代数坐标与$\tilde{\mathbf k}(X_w)$上卢的实代数坐标相关联的变量变换映射。 显示英文摘要

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 $\tilde{\mathbf k}(X_w)$.