\subsetneq \langle x_{n-r+1}, \ldots, x_{n-1} \rangle \subsetneq V$$ and $Q$ inherits the image of $${F^{\mathrm{std}}}_Q = V \subsetneq V + \langle x_{1} \rangle
\subsetneq V + \langle x_{1}, x_{2} \rangle \subsetneq \cdots
\subsetneq V + \langle x_{1}, \ldots, x_{n-r-1} \rangle \subsetneq {\mathbb{C}}^n$$ under the quotient map ${\mathbb{C}}^n \to Q$.

Under these identifications, $$\hat{X}_\lambda = 
\{\phi \in {\mathrm{Hom}}(V,Q)\ |\ \phi(({F^{\mathrm{std}}}_V)_l) \subset
({F^{\mathrm{std}}}_Q)_{\lambda_l}\ l=1, \ldots, r\}.$$

Let $\sigma$ denote the $01$-string corresponding to $\lambda$, and put $$\alpha_k = 
\begin{cases}
 \sum_{l=1}^k 1-\sigma_l, &\text{if $\sigma_k = 0$} \\
n-r+\sum_{l=1}^k \sigma_l, &\text{ if $\sigma_k = 1$.} 
\end{cases}$$ So $\alpha_k \leq n-r$ if $\sigma_k=0$, and $\alpha_k >n-r$ if $\sigma_k=1$. Then the flag $\hat{F}(\sigma)$ is $$\hat{F}(\sigma)
= {\{0\}}\subsetneq \langle x_{\alpha_1} \rangle
\subsetneq \langle x_{\alpha_1}, x_{\alpha_2}\rangle \subsetneq \cdots
\subsetneq \langle x_{\alpha_1}, \ldots, x_{\alpha_{r-1}} \rangle
\subsetneq {\mathbb{C}}^n.$$ To see this, note that $V \cap \hat{F}(\sigma)_l$ jumps in dimension exactly when $\sigma_l=1$, and $\alpha$ is the smallest permutation that accomplishes this.

By proposition \[prop:calcZ\], $\hat{X}_\sigma$ is is identified with ${\mathfrak{b}}(\hat{F}(\sigma))/{\mathfrak{p}}$. Let ${\mathfrak{n}}$ denote the orthogonal complement to ${\mathfrak{p}}$. Since $V$ and $\hat{F}(\sigma)$ are coordinate flags, we have ${\mathfrak{b}}(\hat{F}(\sigma))/{\mathfrak{p}}= {\mathfrak{b}}(\hat{F}(\sigma)) \cap {\mathfrak{n}}/{\mathfrak{p}}$. Thus it suffices to determine ${\mathfrak{b}}(\hat{F}(\sigma)) \cap {\mathfrak{n}}$. We see that an element $\phi \in {\mathfrak{n}}$ preserves $\hat{F}(\sigma)$ if 