# Question on Iwasawa decomposition of unitary groups over adele ring

Let $$E/F$$ be a quadratic extension of number fields and $$V,\langle,\rangle$$ is a hermition vector space over $$E$$. Let $$\mathbb{A}$$ be the adele ring of $$F$$.

Assume that there is a hermitian line $$e \in V$$ such that $$\langle e,e\rangle=1$$ and let $$V'$$ be the orthogonal complement of $$\langle e\rangle$$ in $$V$$.

Let $$P_0'$$ be a minimal parabolic subgroup of $$U(V')$$ and $$K= \prod_v K_v$$ a ‘good’ maximal compact subgroup of $$U(V')$$ such that $$U(V')=P_0'K$$. (Here, the definition of ‘good’ is from Section I.1.4 of the book "Spectral decomposition and Eisenstein series" by Waldspurger and Mœglin. That is, for any standard parabolic subgroup $$P'$$ of $$U(V')$$, $$P'(\mathbb{A}) \cap K=(M'(\mathbb{A}) \cap K)(U'(\mathbb{A}) \cap K))$$ and $$M'(\mathbb{A}) \cap K$$ is still a maximal compact subgroup of $$M'(\mathbb{A})$$.)

Then I have two questions.

1. What does $$K(\mathbb{A})$$ look like? I guessed $$K_v=U(V')(\mathfrak{o}_v)$$ for non-archimean $$v$$ and $$K_v=U(p,q)$$ for some $$(p,q)$$ such that $$p+q=n$$, when $$v$$ is archimedean. But if so, it is independent of the choice of $$P_0'$$. Is this right?

2. If we choose a good maximal compact subgroup $$K'(\mathbb{A})$$ of $$U(V')$$, then can we choose a good maximal compact subgroup $$K$$ of $$U(V)$$ such that $$K' \subset K$$? (Here, $$P_0$$ is a minimal parabolic subgroup of $$U(V)$$ contained in the parabolic subgroup of $$U(V)$$ which stabilizes the flag of $$V'$$ which defines $$P_0'$$.)

• TeX note: please use \langle\rangle for inner products, not <>. Compare $\langle e, e\rangle = 1$ \langle e, e\rangle = 1 to $<e, e> = 1$ <e, e> = 1. I have edited accordingly. Apr 21, 2021 at 22:11