On the variety of Borels in relative position $\vec{w}$

Abstract

Let $G$ be a connected semi-simple group defined over and algebraically closed field, $T$ a fixed Cartan, $B$ a fixed Borel containing $T$, $S$ a set of simple reflections associated to the simple positive roots corresponding to $(T,B)$, and let $\mathcal{B}\cong G/B$ denote the Borel variety. For any $s_i\in S$, $1\le i\le n$, let $\bar{O}(s_1,...,s_n)=\{(B_0,...,B_n)\in {\mathcal{B}}^{n+1}|(B_{i-1},B_i)\in \overline{O(s_i)},1\le i\le n\}$, where $O(s)$ denotes the subvariety of pairs of Borels in ${\mathcal{B}}^2$ in relative position $s$. We show that such varieties are smooth and indicate why this result is, in one sense, best possible. Our main results assume that $k$ has characteristic 0.

16 pages