Let $G$ be a connected reductive algebraic group. Let $T$ be a maximal torus of $G$.
For any $x \in G$, let $\mathscr B_x$ be the set of Borel subgroups of $G$ containing $x$. Then $\mathscr B_x$ is a subvariety of the flag variety of $G$.
I was told that for $x$ regular, $\dim \mathscr B_x =0$. But where can I find a proof for this?
In particular, when $x \in T$ is regular semisimple, then is it true that $$\mathscr B_x \cong W,$$ where $W$ is the Weyl group of $G$? Why?
Thanks to everyone.
I'll write out a quick proof because I don't think it's so complicated.
If $x$ is a regular element of a Borel $B$, then it necessarily is in a maximal torus of the Borel, and it also uniquely determines a maximal torus by regularity. So if a Borel contains $x$ is must be a standard Borel with respect to the maximal torus determined by $x$, and then each such Borel is equivalent to a choice of positive roots, and the Weyl group acts simply transitively on choices of positive root systems.