# Lagrangian subvariety in coadjoint orbit

In Chriss and Ginzburg's Representation Theory and Complex Geometry, Theorem 3.3.6 says that

"Let $\mathbb{O}$ be a coadjoint orbit in $\mathfrak{g}^*$. Let $x\in \mathbb{O}$ be such that $x|_{\mathfrak{n}}=0$. Then $\mathbb{O}\cap (x+\mathfrak{b}^\perp)$ is a Lagrangian subvariety in $\mathbb{O}$ with repsect to the natural symplectic structure on coadjoint orbits."

They claim that the case when $x\notin \mathfrak{b}^\perp$ is "less difficult and less interesting" and then proceed to proving the nilpotent case.

It is not hard to show that $\mathbb{O}\cap (x+\mathfrak{b}^\perp)$ is coisotropic in the general case by following the proof in the book. However, I was not able to come up with a proof for a statement in the form of Lemma 3.3.8 for the general case, namely to prove that $\dim(\mathbb{O}\cap (x+\mathfrak{b}^\perp))\leq \frac{1}{2}\dim \mathbb{O}$. Can someone please give me a hint? Thanks a lot.

• Maybe try to prove that the symplectic form vanishes after pullback to the subvariety? – Ben Webster Mar 7 '17 at 7:11
• @BenWebster: Good point. It is not necessary to compute the dimension. – Daps Mar 7 '17 at 15:46