# The weird projection from SO(2n)/B to maximal isotropic grassmannian

Take the generalized flag variety $SO(2n,\mathbb{C})/B$, considered as the moduli of isotropic flags (according to the form $\langle e_i, e_{2n+1-j}\rangle=\delta_{ij}$) $$F_1\subset F_2\subset\cdots F_n\subset\mathbb{C}^{2n}$$ with $\dim(F_i)=i$ and $\dim(F_n\cap\mathrm{Span}(e_1,\ldots,e_n))$ even.

Let $P$ be the maximal parabolic subgroup of $SO(2n)$ omitting one of the two nodes at the forked end. Then $G/P$ is the moduli of isotropic subspaces of dimension $n$ in $\mathbb{C}^{2n}$.

For one of those end nodes, the description of the projection $G/B\rightarrow G/P$ is obvious; one just sends the point corresponding to the full isotropic flag to the point corresponding to the maximal subspace $F_n$.

In explicit linear algebra terms, what is the projection $G/B\rightarrow G/P$ for the maximal parabolic corresponding to the other end node on the forked end? In other words, what is the other way to get a maximal isotropic subspace out of a full isotropic flag?

• A full isotropic flag is not $F_1\subset F_2 \subset \dots \subset F_{n-1}\subset F_n \subset \mathbb{C}^{2n}$. A full isotropic flag is $F_1\subset F_2 \subset \dots \subset F_{n-1}$ together with a pair $F_{n-1}\subset F'_n$ and $F_{n-1}\subset F''_n$ such that $F'_n$ and $F''_n$ are distinct isotropic subspaces. – Jason Starr Aug 3 '15 at 1:19
• "You won't believe this 1 weird projection from..." – David Roberts Aug 3 '15 at 2:06
• @JasonStarr Thanks for the hint (and, really, one should dispense with $F_{n-1}$ altogether and just think of it as the moduli with two Lagrangian subspaces which define $F_{n-1}$ as their intersection). Followup question which I'll try to answer here in a few days if no one knows: Can this viewpoint allow us to dispense with the annoying subtle parity conditions in the linear algebra definition of type D Schubert varieties? – Alexander Woo Aug 3 '15 at 4:53
• Followup to the above: No it can't. This is stated in the appendix to Buch, Kresch, and Tamvakis, A giambelli formula for even orthogonal Grassmannians. An explicit example in D4 is the pair 34127856 and 46718235 (usual embedding into S8), which are incomparable in Bruhat order but comparable using dimensions of subspaces. – Alexander Woo Jul 17 '16 at 4:02

The other projection goes to the other Lagrangian subspace containing $F_{n-1}$. If you look at $F_{n-1}^\perp/F_{n-1}$, this is a 2-dimensional space with symmetric non-degenerate bilinear form. Thus, it has exactly 2 isotropic lines (there are coordinates where the matrix of the form is $\begin{bmatrix}0 & 1\\ 1&0\end{bmatrix}$, in which case the coordinate lines are the only isotropic ones), whose preimages are the spaces $F_n$ and $F_n'$ canonically.