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 $0$.