Let $W$ be a $2n$-dimensional complex vector space endowed with a non-degenerate, symmetric, bilinear form $Q$. We choose Euclidean coordinates on $W$ such that $Q$ is represented by symmetric matrix \begin{align*} \left[\begin{array}{ccc} 0 & I_n \\ I_n & 0 \end{array}\right], \end{align*} where $I_n$ the identity matrix of order $n$. Define $\operatorname{Gr}_{Q}(n,2n)$ to be the Grassmannian of isotropic $n$-planes in $W$( for any $n$-plane $V$, we have $Q|_V=0$). I'm having some trouble understanding how the isotropic Grassmannian $\operatorname{Gr}_{Q}(n,2n)$ has two irreducible connected components. Any information would be helpful!
2026-03-26 09:39:53.1774517993
Connected components of the isotropic Grassmannian
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