Is there a standard algorithm for finding the double coset representatives of $H_1$ \ $G/H_2$, where the groups are finite of Lie type?
Specifically, I need to compute the representatives when $G=Sp_4(\mathbb{F}_q)$ (I'm using $J$ the anti diagonal with top two entries $1$, and the other two $-1$), $H_1$ is the parabolic with $4=2+2$, and $H_2=SL_2(\mathbb{F}_q)\ltimes H$, where $H$ is the group of matrices of the form: $\begin{bmatrix} 1&x&y&z \\\\ 0&1&0&y \\\\ 0&0&1&-x \\\\ 0&0&0&1 \end{bmatrix}$ which is isomorphic to the Heisenberg group, and $SL_2$ is embedded in $Sp_4$ as: $\begin{bmatrix} 1&&& \\\\ &a&b& \\\\ &c&d& \\\\ &&&1 \end{bmatrix}$