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Using representation theory, we compute the spectrum of the Dirac operator onthe universal covering group of $SL_2(\mathbb R)$, exhibiting it as thegenerator 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 thelimit-of-discrete-series representation, which is the frontier between thediscrete and the principal series of the group. We provide a detailed analysisof the localised spectra of the Dirac operator and compute the Diraccohomology.

The local spectrum of the Dirac operator for the universal cover of $SL_2(\mathbb R)$