Let $G$ be a semisimple Lie group and $B_-$ a Borel subgroup of $G$. Let $U$ be a unipotent subgroup of $G$.
In the paper, Proposition 1.1, it is said that $U^w=U \cap B_- w B_-$ is parametrized by $\mathbb R^m$. That is, very element in $U^w$ can be written as $x_{i_1}(t_1) \cdots x_{i_m}(t_m)$, where $w=s_{i_1} \cdots s_{i_m}$ is a reduced expression.
I don't know how to prove that $U^w=U \cap B_- w B_-$ is parametrized by $\mathbb R^m$. Are there some examples in the case $G=GL_n$ which explain this result? Thank you very much.