I need some help with this problem.
Let $g$ be an element of the orthogonal group and $s$ a reflection. Then the dimension of the fixed point set of $g$ and $gs$ differ by $\pm 1$. Since that lead to confusion, I mean $ \operatorname{dim}(\operatorname{ker}(1-gs))= \operatorname{dim}(\operatorname{ker}(1-g)) \pm 1$
Actually I only need this result in the special case of a finite reflection group defined by a root system. Does anyone have an idea? And it does not simply follow from the deletion condition.