

A typical element of the Siegel parabolic is given by $g$. I could get write the $d_{ij}$ in terms of $A$, but I think it would make it messier to look at.

The first coset rep is just the identity, so that just leaves $g$. The second set are of the following form with $z \in \mathbb{F}_{p}$.

Multiplied by a typical parabolic element:

The third set is given by reps of the following form with $y,z \in \mathbb{F}_p$.

Multiplied by a typical parabolic element:

Finally, the last type is given by the following form with $x,y,z \in \mathbb{F}_p$.

Multiplied by a typical parabolic element:

