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 $0$.
$(g,k)$ modules for $\mathrm{SU}(1,1)$
2
$\begingroup$
group-theory
-
0Please check that the notation is still correct after my edit. – 2012-11-20