  Using representation theory, we compute the spectrum of the Dirac operator on
the universal covering group of $SL_2(\mathbb R)$, exhibiting it as the
generator of $KK^1(\mathbb C, \mathfrak A)$, where $\mathfrak A$ is the reduced
$C^*$-algebra of the group. This yields a new and direct computation of the
$K$-theory of $\mathfrak A$. A fundamental role is played by the
limit-of-discrete-series representation, which is the frontier between the
discrete and the principal series of the group. We provide a detailed analysis
of the localised spectra of the Dirac operator and compute the Dirac
cohomology.
