Let $G$ be a connected reductive group over a number field $F$ and $P_0$ its minimal parabolic subgroup. Then we call a parabolic subgroup $P$ of $G$ is standard if $P_0 \subset P$.

Let $K$ be a fixed good maximal compact subgroup of $G(\mathbb{A}_F)$ such that $G=PK$ for all standard parabolic subgroup $P=UM$. (here $U,M$ are the unipotent radical and Levi of $P$.

If $Q=U_Q M_Q$ is another standard parabolic group of $G$ such that $Q \subset P$, I am wondering whether $M_P=(U_Q \cap M_P)(M_Q )(K \cap M_P)$.

Any comments are highly appreciated!

number field, or you mean a local field? $\endgroup$ – Mikhail Borovoi Aug 28 '19 at 17:17