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

TL;DR

This paper computes the spectrum of the Dirac operator on the universal cover of SL_2(R) using representation theory, revealing its role in K-theory and analyzing the local spectra and Dirac cohomology.

Contribution

It provides a new, direct computation of the K-theory of the reduced C*-algebra of the group via spectral analysis of the Dirac operator.

Findings

Spectrum of the Dirac operator characterized for the universal cover of SL_2(R)

Identification of the Dirac operator as the generator of KK^1 group

Analysis of the local spectra and Dirac cohomology

Abstract

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.