positions of the [[‘$1$’]{}s ]{}, and $\eta_{n-d+1} < \cdots < \eta_{n}$ denote the positions of the [[‘$2$’]{}s ]{}.

Our first objective is to describe the space $\hat{Y}_{\sigma}$. Let $V$ denote the coordinate subspace $V = \langle x_{n-r+1}, \ldots, x_n \rangle$, and $S$ denote the coordinate subspace $S = \langle x_{n-d+1}, \ldots, x_n \rangle$. (Eventually $d$ and $S$ will play the same role as they did in section \[sec:grassmannian\].) We take our base flag $V^0$ to be the coordinate two-step flag $$V^0 = {\{0\}}\subset S \subset V \subset {\mathbb{C}}^n.$$

Now ${\mathfrak{g}}/{\mathfrak{p}}$ has a basis descending from the standard basis $$\big\{ E_{jk}
\ \big|\ \mbox{$j \leq n-r$, $k>n-d$, and ($k>n-d$ or $j \leq n-r$)}
\big\}$$ where $E_{jk}$ is the image under ${\mathfrak{g}}\to {\mathfrak{g}}/{\mathfrak{p}}$ of the $n \times n$ matrix whose only non-zero entry is a [[‘$1$’]{} ]{}in the $(j,k)$-position. These basis vectors naturally partition ${\mathfrak{g}}/{\mathfrak{p}}$ into three blocks: the upper left block is spanned by those $E_{jk}$ such that $1 \leq j \leq n-r$, $n-r < k \leq n-d$; the lower right block is spanned by those $E_{jk}$ such that $n-r < j \leq n-d$, $n-d < k \leq n$, and the upper right block is spanned by those $E_{jk}$ such that 