# A question on standard parabolic subgroup

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 $$M_P=(U_Q \cap M_P)(M_Q )(K \cap M_P)$$.

• What do you mean by $G=PK$? Your $P$ is an algebraic subgroup of $G$, hence it is an algebraic group over the number field $F$, while $K$ is a maximal compact subgroup of some topological group - what group? In order to define $PK$, you groups $P$ and $K$ must be subgroups of the same group - what group? Please edit your question. – Mikhail Borovoi Aug 27 '19 at 19:45
• OP probably means that $K$ is a maximal compact subgroup of $G$''. Then $G=PK$ is an Iwasawa decomposition of $G$. If OP, by $M$, meant Levis in the corresponding groups, then the last assertion is true. That assertion may probably be derived from the Iwasawa decomposition of $M_P$. – Subhajit Jana Aug 28 '19 at 8:01
• @SubhajitJana The issue is you do not have an Iwasawa decomposition over arbitrary fields. E.g., one often puts the discrete topology on $G(\mathbb Q)$, in which case compact subgroups are just the finite subgroups. – Kimball Aug 28 '19 at 10:57
• In what group is $K$ contained? The number field $F$ has infinitely many completions $F_v$. Do you mean that $K$ is a maximal compact subgroup in $G(F_v)$ for some $v$? Is your $F_v$ isomorphic to $\mathbb R$, or to $\mathbb C$, or is it a nonarchmedean local field? Please edit your question again. – Mikhail Borovoi Aug 28 '19 at 17:14
• Is your field $F$ indeed a number field, or you mean a local field? – Mikhail Borovoi Aug 28 '19 at 17:17

Firstly on the level of algebraic groups, note that since $$Q\subset P$$ are standard, we can fix a Levi decomposition $$T_0U_0=B_0$$ of a minimal parabolic of $$G$$ and require that our Levi decompositions $$Q= M_QU_Q$$ and $$P=M_PU_P$$ satisfy $$T_0\subset M_Q\subset M_P$$. Then $$M\cap Q= M_Q\cdot (U_Q\cap M)$$ is a parabolic subgroup of $$M_P$$.
On the level of adelic points, we have the maximal compact subgroup $$K\subset G(\mathbb{A})$$ satisfying $$G(\mathbb{A})=P(\mathbb{A})K$$ for any standard parabolic subgroup $$P$$. We are free to choose $$K$$ so that $$K_M=M(\mathbb{A})\cap K$$ is a good maximal compact subgroup of $$M(\mathbb{A})$$ (this is, for example, stated in Moeglin-Waldsurger I.1.4). We then have an Iwasawa decomposition $$M(\mathbb{A})=(M\cap Q)(\mathbb{A})K_M.$$ This is as close to what you ask as I can imagine since $$(M\cap Q)(\mathbb{A}) = M_Q(\mathbb{A})\cdot (U_Q\cap M)(\mathbb{A})$$.