The Iwasawa decomposition of a real symplectic matrix $S \in \text{Sp}(2n,\mathbb R)$ gives a convenient decomposition $S=KAN$ where $$K\in \text{Sp}(2n,\mathbb R) \cup \text{SO}(2n,\mathbb R)\\ A \in \mathcal{A}=\{D(\vec{\kappa}): \vec{\kappa}=(\kappa_{1},\kappa_{2}\dots,\kappa_{1}^{-1},\dots,\kappa^{-1}_{n})\}\\ N\in\mathcal{N}=\left\{\left( \begin{matrix} A&0 \\ C&A^{-T} \end{matrix} \right) \in \text{Sp}(2n,\mathbb R): A_{ii}=1,A_{i,j<i}=0, A^{T}C=C^{T}A \right\}$$.
Does the same decomposition hold for:
- (1) symplectic matrices defined over the integers, i.e., $\text{Sp}(2n,\mathbb Z)$; and
- (2) for symplectic matrices defined over the integers $\mod k$, i.e., $\text{Sp}(2n,\mathbb Z_k)$, where $\mathbb Z_k=\{0,\dots,k-1\}$?
If not, is there some similar decomposition with the same or similar properties?