Let $G:=GL(n)$ be defined over $F:=\mathbb{Q}_p$ (for convenience, of course in general over a $p$-adic field $F$). Then it's well-known that we have the Iwasawa and Cartan decompositions, just as Lie groups:
Theorem (1)(Iwasawa decomposition) Let $B$ be the standard Borel subgroup (upper triangular) and $K:=GL(n,\mathbb{Z}_p)$. Then $G(F)=B(F)K=KB(F)$.
(2)(Cartan decomposition) Let $T$ be the subgroup of diagonal elements of $G$. Then $G(F)=KT(F)K$.
(3)(direct corollary of the Cartan decomposition)$K$ is a maximal compact open subgroup.
I know the proofs of these facts for $G=GL(n)$ because they are very elementary (linear algebra, row and column reduction of matrices). However, during a course lectured by my advisor to undergraduates (for which I'm a TA, ah ah ah), my advisor left as an exercise to the students to prove the above theorems for $G=Sp(2n)$. Here we let $T$ to be the diagonal subgroup of $G=Sp(2n)$ and $N$ the subgroup generated by $$ \begin{pmatrix} I_n & 0 \\ S & I_n \end{pmatrix}\left({ }^t S=S\right), \quad\begin{pmatrix} u & 0 \\ 0 & { }^t u^{-1} \end{pmatrix}, $$ where $u\in GL(n)$ is lower triangular. We define the Borel $B$ to be $B=TN$. In this context, we have to prove the above theorem for $G=Sp(2n)$.
I know that these two decompositions hold for $p$-adic classical groups: namely symplectic and special orthogonal groups. I first saw this fact on Bruhat's report in Algebraic Groups and Discontinuous Subgroups, Proceedings of Symposia in Pure Mathematics 009, AMS, page 66. However, I didn't try to find any reference for this, and for the case $SO(n)$ it requires some Bruhat-Tits theory since in general $SO(n)$ may neither be defined over integers nor split(the proof is more difficult, I think).
Now I have to find an adhoc proof for the case $G=Sp(2n)$ mentioned as above (maybe the onlything I can use is still linear algebra, some special elements in $G=Sp(2n)$?). So far I tried some elements but I can't write down a precise proof. Can anybody help me? And I also want to ask for references on the case $G=SO(n)$. These decompositions are too useful while studying representation theory of $p$-adic groups. Have they been proved for general reductive groups?