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 as a projective variety ? 



Fixed points of an involution
In general, the fixed points of any map of the Grassmannian induced by a diagonalizable linear transformation is just a union of products of Grassmannians in the eigenspaces. Any subspace $V$ fixed by a diagonalizable transformation $A$ is the sum of the intersections of $V$ with the eigenspaces of $A$ (since the projection to each eigenspace is a polynomial in $A$). If we fix the dimension of each of these intersections, we get a map to the product of Grassmannians of the eigenspaces, which is obviously an isomorphism.


In this case, $\mathbb{R}^{2n}$ is the sum of the 1 and -1 eigenspaces, both having dimension $n$. Thus, the fixed points are a disjoint union of $Gr(k,n)\times Gr(n-k,n)$ for all $0\leq k\leq n$. 


