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For local fields $F$, we consider two case 1) $E$=quadratic extension of $F$ , 2) $E = F \times F$.

Let V be a 2-dim hermition space over E.

In 1) case, by Cartan decompostion $U(2)$ can be decomposed as $KMK$. (here, $K$ is a compact subgroup of $U(2)$ and M={$x \in E^\times | \left\vert x \right\vert \le 1 $}

In 2) case, $U(2)=GL_2 (F)$ the cartan decomposition of $U(2)$ is $KMK$ where $K$ is compact subgroup and $M$={ $\begin{pmatrix} x & 0 \\\ 0 & y \end{pmatrix} \in GL_{2}(F)$ | $|x|\le|y|$ }.

My question arises here.

If we consider $U(1)$, then, how can we think the Cartan decomposition of it in the above two cases? I don't know what the above $M$ should be in these case.

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Have you compared your question with the other question asked on this side? It is too elementary here. Please look at FAQ for some suggestion, where to put your question, eg. mathstackexchange. Also please use latex code there and here. Best, Marc –  plusepsilon.de Nov 23 '12 at 11:35
    
Sorry! After having seen your remark, I read the FAQ. I typed it again using latex code. But, I really want to know the answer. Since this is part I should know to write a paper, I asked it in mathstackexchanege but nobody answer to me. So, if you know some book or paper which deals the answer for this, would you recommend it for me? Then, I will be very appreciate to you. Best, –  Jude Nov 24 '12 at 15:02
    
Dear anonymous, If you have previously posted this question on Math.SE, it is good to post a link to here to that question (and there to this question). Also, I disagree with Marc Palm that this question is too elementary for this site, but it could be explained more clearly. (E.g. $K$ should probably be a maximal compact subgroup; also, do you really mean to work over a number field, rather than a local field. E.g. for a local field, $| x| \leq 1$ has a definite meaning; for a number field, it raises the question of which absolute value you intend to use.) Regards, –  Emerton Nov 24 '12 at 18:06
    
Dear Emerton. Thank you for your kind reply. I am very sorry for not asking the question clearly. There was a little hasting in posting this. Here, K is maximal compact subgroup as you indicated and I am asking it in local fields case.(especially real and p-adic field) This is the link that connect to the post in mathstackexchange regarding this. math.stackexchange.com/questions/243291/… Best, –  Jude Nov 25 '12 at 11:08
    
1. Please edit your question such that it will mansion that $F$ is a local field . 2. please take care about the (2) subscript in the definition of $M$. Probably you just need to replace (2) by {2}. I'll try to answer your question soon –  Rami Nov 25 '12 at 18:13
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1 Answer

up vote 2 down vote accepted

Let me formulate the Cartan decomposition for a reductive group over a local field (for more information see e.g. http://www.math.tau.ac.il/~bernstei/Unpublished_texts/unpublished_texts/Bernstein93new-harv.lect.from-chic.pdf section 2.1):

Let $G$ be a reductive group defined over over a local field $F$ of characteristic $0$. Let $A$ be the maximal split torus. That is, the maximal algebraic subgroup defined over $F$ of $G$ which is isomorphic to $(F^{\times})^k$ for some $k$. Note that it is unique up to a conjugation. Let $K$ be the maximal compact subgroup of $G$. Note that in the Archemedian case it is an algebraic subgroup and it is unique up to conjugation. In the non-Archemedian it is not algebraic and often it is not unique up to a conjugation. If your group is defined over the ring of integers $O$ of $F$, you can take $K=G(O)$.

The week version of Cartan decomposition says: $$G=KAK.$$

For the stronger version, let $\Lambda=Mor(F^\times, A)$ be the group of co-characters of $A$. Let $\Lambda^{++}$ be the Wile chamber of $\Lambda$. ruffly spicing it is the fundamental domain for $W:=N_G(A)/Z_G(A)$. We have an obvious map $\Lambda \times F^\times \to A$. Let $A^+$ be:\ 1. In the Archemedian case the image of $\Lambda^{++} \times \mathbb{R}_{>0}$. 2. In the non-Archemedian case the image of $\Lambda^{++} \times \pi$, where $\pi$ is the uniformizer of $F$.

Than $$G=KA^+K.$$ Moreover, any $K\times K$ double co-set have a unique element from $A^+$.

The case of $U(1)$, this is not that interesting. First of all the group is commutative, so we do not need to write $K$ twice. In the non-split case, the group is compact, then $A$ is trivial and the Cartan decomposition says $G=K$. In the split case, $A$ is one dimensional and $A^+$ is either $\mathbb{R}_{>0}$ or $\pi^\mathbb{Z}$.

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Dear Rami, I have no word to thank you enough. It helped me very much. Though I am not so sure that how your answer is in well-harmony in the above U(2) case, but I think that this comes from my ignorances. I will study the material you recommended and then I will meditate it again. Best regards, –  Jude Nov 27 '12 at 3:52
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