# Fixed points of an involution

Let $V=\mathbb C^{2n}$ with the standard basis $\{e_1,e_2, \cdots , e_{2n}\}$ and let $\sigma$ be the involution $e_i \mapsto -e_{2n+1-i}$. This induces an involution of the Grassmannian $G(n,2n)$ of $n$ dimensional subspaces of $\mathbb C^{2n}$. Then what are the fixed points of this involution ? Does it have a nice structure ?