Equivalence of Demazure and BottSamelson Resolutions via Factorization
Abstract
Let $G$, $B$, and $H$ denote a complex semisimple 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 BottSamelson 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 $\tilde{\mathbf k}(X_w)$.
 Publication:

arXiv eprints
 Pub Date:
 February 2015
 arXiv:
 arXiv:1503.00077
 Bibcode:
 2015arXiv150300077C
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Representation Theory;
 22E30;
 14M15
 EPrint:
 13 pages