The eta function of the localised Dirac operator for the universal cover of SL2(R)

Recently Brodzki, Niblo, Plymen, and Wright determined a closed explicit description of the spectrum of the Dirac operator D for the universal cover of SL2(R) localised at a representations π in the principal series and in the discrete series [4]. Using this description, we introduce an eta function ηloc(s;D,π) for the localised Dirac operator and we study its analytic properties. We show that the localised eta function has an analytic extension to the complex plane with simple poles at negative integers and at s=1, and it is regular at s=0, and we give a formula for ηloc(0;D,π). We discuss a possible eta invariant for the Dirac operator.