This question relates to the question and the question.
Let $B^-$ be the Borel subgroup of $Sp(4)$ consisting of all lower triangular matrices. Let $X = \left(\begin{array}{cccc} x_{1,1} & 0 & 0 & 0\\ x_{2,1} & x_{2,2} & 0 & 0\\ x_{3,1} & x_{3,2} & x_{3,3} & 0\\ x_{4,1} & x_{4,2} & x_{4,3} & x_{4,4} \end{array}\right) \in B^-$. Since $X^T J X = J$, where \begin{align} J = \left(\begin{array}{cccc} 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0\\ 0 & -1 & 0 & 0\\ -1 & 0 & 0 & 0 \end{array}\right), \end{align} the matrices in $B^-$ are of the form: \begin{align*} X=\left(\begin{array}{cccc} x_{1,1} & 0 & 0 & 0\\ x_{21} & x_{2,2} & 0 & 0\\ x_{3,1} & x_{3,2} & \frac{1}{x_{22}} & 0 \\ x_{41} & x_{4,2} & x_{43} & \frac{1}{x_{11}} \end{array}\right), \end{align*} where $x_{1,1}\, x_{4,2} + x_{2,1}\, x_{3,2} - x_{2,2}\, x_{3,1}=0$ and $x_{1,1}\, x_{4,3} + x_{2,1}\, x_{3,3}=0$.
The corner minors in the case of $SL_4$ are $$ x_{41}, x_{31}x_{42}-x_{32}x_{41}, x_{21}x_{32}x_{43}+x_{22} x_{33}x_{41} -x_{22} x_{31} x_{43} - x_{21} x_{33} c_{42}. $$
These minors are determined by the entries of the matrix $t$ which satisfies $X = u \overline{w}_0 t u'$, where $u, u' \in U$, $U$ is the unipotent subgroup of $SL_4$, $\overline{w}_0 \in SL_4$ is the standard representative of the longest word in the Weyl group of $SL_4$.
In the case of $Sp(4)$, what are the corner minors of \begin{align*} X=\left(\begin{array}{cccc} x_{1,1} & 0 & 0 & 0\\ x_{21} & x_{2,2} & 0 & 0\\ x_{3,1} & x_{3,2} & \frac{1}{x_{22}} & 0 \\ x_{41} & x_{4,2} & x_{43} & \frac{1}{x_{11}} \end{array}\right)? \end{align*}
Thank you very much.
