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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.

IzumiEternal
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  • Try Humphreys' textbook "Linear Algebraic Groups". His monograph "Conjugacy Classes in Algebraic Groups" has many more than this. – P Vanchinathan Nov 19 '16 at 07:52
  • @PVanchinathan, thank you very much for the comment. I have a copy of "Linear Algebraic Groups" by Humphreys. But I can't find "Conjugacy Classes in Algebraic Groups". So I am wondering where in "Linear Algebraic Groups" I can find the related result or proof. – IzumiEternal Nov 21 '16 at 03:45
  • It should be in the later chapters on structure of reductive groups. – P Vanchinathan Nov 21 '16 at 03:47

1 Answers1

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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.

freeRmodule
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    Thank you very much. If I am not mistaken, Cartan subalgebra is some subset of the Lie algebra. Then how can an element $x \in B \subseteq G$, be contained in the Lie algebra? Are you talking about some element-level correspondence between this algebraic group and its Lie algbera? Thanks again. – IzumiEternal Nov 21 '16 at 03:37
  • Sorry! I like thinking in terms of Cartan subalgebras. Just replace that with 'maximal torus'. – freeRmodule Nov 21 '16 at 16:28
  • I just edited it. – freeRmodule Nov 21 '16 at 16:31