(g,k) modules examples - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T05:52:36Z http://mathoverflow.net/feeds/question/114214 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114214/g-k-modules-examples (g,k) modules examples Eugene lee 2012-11-23T06:34:15Z 2012-11-23T06:34:15Z <p>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. </p> <p>Let the group be $\mathrm{SU}(1,1)$, choose maximal compact subgroup $$K_{\mathbb{R}}=\left{ \left(\begin{array}{cc} e^{i\theta} &amp; 0\ 0 &amp; e^{i\theta} \end{array}\right),\,\theta\in \mathbb{R}\right} \simeq \mathrm{SO}(2)\simeq \mathrm{U}(1),$$ and let $g = \mathfrak{sl}(2,\mathbb{C})$. Why is it that for any $(g,K)$-module $V$, all eigenvalues of $$H=\left(\begin{array}{cc} 1 &amp; 0\ 0 &amp; 1 \end{array}\right)$$ are integers. Here $\mathfrak{sl}(2,\mathbb{C})$ is the usual set of $2$ by $2$ with trace $0$.</p>