Iwasawa decomposition of the general symplectic groups over the ring of adeles

Question

This post is going to be quite a silly one, as its subject is something I believe to be the case, but which I am unable to prove to be the case (due to my lack of experience with algebraic and specifically reductive groups); nor have I been able to find a reference that asserts this to be the case.

I will begin by defining my terms:

• Let $G := \textrm{GSp}_{2n}(\mathbb{A})$ denote the rank $2n$ general symplectic group over the ring of adeles.
• Let $$ P := \left\{ \left(\begin{array}{cc} A & B\\ C & D \end{array}\right) \in G\quad \middle|\quad C=0 \right\} $$ denote the Siegel parabolic subgroup of $G$.
• The unipotent subgroup of $P$ can then be seen to be of the form $$ U = \left\{ \left(\begin{array}{cc} \textrm{Id}_n & X\\ 0 & \textrm{Id}_n \end{array}\right) \in G\quad \middle|\quad X \in \textrm{Sym}_n(\mathbb{A}) \right\}, $$ where $\textrm{Sym}_n(\mathbb{A})$ denotes the additive group of $n \times n$ matrices with entries in $\mathbb{R}$, and $\textrm{Id}_n$ denotes the $n \times n$ identity matrix.
• We define the standard Levi component of $P$ to be: $$ M := \left\{ \left(\begin{array}{cc} Y & 0\\ 0 & u (Y^t)^{-1} \end{array}\right) \in G\quad \middle|\quad Y \in \textrm{GL}_n(\mathbb{A}),\ u \in \mathbb{A}^{\times} \right\}, $$
• We define the standard maximal compact subgroup of $\textrm{Sp}_{2n}(\mathbb{R})$ as: $$ K_{\infty} := \left\{ g \in \textrm{Sp}_{2n}(\mathbb{R})\ \middle|\ g \cdot i \textrm{Id}_n = i \textrm{Id}_n \right\}, $$ where the group action of $\textrm{Sp}_{2n}(\mathbb{R})$ is the usual one.
• Lastly, for any natural number $q$, we define the following family of open subgroups of the $\textrm{GSp}_{2n}(\mathbb{Z}_p)$: $$ K_p(q) := \left\{ g \in \textrm{GSp}_{2n}(\mathbb{Z}_p)\ \middle|\ g \equiv \left(\begin{array}{cc} \textrm{Id}_n & 0\\ 0 & a \textrm{Id}_n \end{array}\right)\quad (\textrm{mod}\ q),\quad\textrm{for some}\ a \in \mathbb{Z}_p^{\times} \right\}, $$ and hence: $$ K_0(q) := \prod_p K_p(q), $$ with the product taken over the set of all rational primes $p$.

My question is whether or not we have the following "Iwasawa decomposition" of $G$:

$$ G = U M K_{\infty} K_0(q), $$ for any natural number $q$.

My reason for believing the above to be true is that it is a natural generalisation of a statement about $\textrm{GSp}_4(\mathbb{A})$ that certainly is true.

I would much appreciate it if someone could refer me to a place where the above is demonstrated to be true, or if you could let me know if and why it isn't true.