TITLE: Orbits of SL_n acting on matrices of determinant p
QUESTION [9 upvotes]: Fix a positive integer $n$ and let $S$ be the set of $n$ by $n$ matrices
with entries in $\mathbf{Z}_p$ (the $p$-adic integers) whose determinant is $p$.
The group $G:=\mathrm{SL}_n(\mathbf{Z}_p)$ acts freely on $S$ via left multiplication.

    Is it possible to write down an explicit list of representatives for the orbits of
    $G$ on $S$?  If so, what is this list?  If not, is there an algorithm for computing
    a complete list of orbit representatives?

For example, for $n=2$, I think that this is basically the "Hecke Operator at $p$" computation, and the $p+1$ orbits of $G$ on $S$ have representatives 
$$
\left(\begin{matrix} p & b \\\ 0 & 1 \end{matrix}\right),\ b=0,1,\ldots, p-1\ \text{and}\  
\left(\begin{matrix} 0 & -1 \\\ p & 0 \end{matrix}\right)
$$ 
This is probably a question whose answer is well-known to many people, so I apologize
in advance for my ignorance!

REPLY [7 votes]: To complement Qiaochu Yuan's answer, check out Lemma 9.3.2 in Goldfeld's book Automorphic Forms and L-functions for the group GL(n,R). On the two sides of (9.3.3) you can see the decompositions of the set of integral $n\times n$ matrices of determinant $N$ into double cosets and right cosets of $\mathrm{SL}_n(\mathbb{Z})$. The right hand side corresponds to your question, and yes, it gives rise to the $N$-th Hecke operator, see (9.3.5). Of course this is over $\mathbb{Z}$, but localizing at $p$ gives what you need: in that case $N$ is a $p$-power without loss of generality, since other prime powers are $p$-adic units.