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 ${\cal B}\cong G/B$ denote the Borel variety. For any $s_i\in S$, $1\leq i\leq n$, let $\bar{O}(s_1,..., s_n)= \{(B_0,..., B_{n})\in {\cal B}^{n+1} | (B_{i-1},B_{i})\in \bar{O(s_i)}, 1\leq i\leq n\}$, where $O(s)$ denotes the subvariety of pairs of Borels in ${\cal B}^2$ 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.