Coset Reps

Here I'll give the coset representatives that are given in Lemma 5.1.1 of Roberts-Schmidt.  Note that they use the symplectic form $\begin{pmatrix} 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ -1 & 0 & 0 & 0 \end{pmatrix}$ as well as the transpose on the first factor. Therefore, we have to change bases to get the correct elements for our group.  Recall this amounts to conjugating by $P = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{pmatrix}$.

They give elements $s_1$ and $s_2$ that arise from the Bruhat decomposition.  We change those to $S_1$ and $S_2$ for us. In fact, $S_1 = s_1$, but $S_2$ is not the same as $s_2$.

They give representatives for $K_0(p^{n})\backslash Sp_4(F_p)$, so their first set of representatives reduces to just the identity, which is the same in either basis.  Define the matrices of the following form where $z \in F_p$:

The second set of coset representatives are given by these matrices multiplied by $s_2$ on the right:

We now change this into our basis:

Now consider matrices of the following form for $y,z$ in $F_p$:

The representatives here are given by these matrices multiplied on the right by $s_2 s_1$.  We change bases:

Finally, consider matrices of the following form for $x,y,z \in F_p$:

The representatives are given by these matrices multiplied on the right by $s_2 s_1 s_2$. We change bases: