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I can't find any online references on these harish chandra modules and I have a hard time starting this question. Does anyone have any good references or some examples I can see.

Let the group be $\mathrm{SU}(1,1)$, choose maximal compact subgroup $$ K_{\mathbb{R}}=\left\{ \left(\begin{array}{cc} e^{i\theta} & 0\\ 0 & e^{i\theta} \end{array}\right),\,\theta\in \mathbb{R}\right\} \simeq \mathrm{SO}(2)\simeq \mathrm{U}(1), $$ and let $g = \mathfrak{sl}(2,\mathbb{C})$. Why is it that for any $(g,K)$-module $V$, all eigenvalues of $$ H=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right) $$ are integers. Here $\mathfrak{sl}(2,\mathbb{C})$ is the usual set of $2$ by $2$ with trace $0$.

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math.utah.edu/vigre/minicourses/sl2/schedule.html Also there is a book by Serge Lang. I vote to close because this is a rather elementary question. –  Marc Palm Nov 23 '12 at 11:28
The book by Serge lang is SL(2,R). –  Marc Palm Nov 23 '12 at 11:46
Your question amounts to "why do representations of $SO(2)$ split into one dimensional characters?". –  S. Carnahan Nov 23 '12 at 14:43
what is the connection with SO(2) splitting into one dim characters –  Eugene lee Nov 23 '12 at 21:27

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