Localised analytic torsion and relative analytic torsion for non compact Lie groups of type I

Let G be a (non compact) connected, simply connected, locally compact, second countable Lie group, either abelian or unimodular of type I, and let ρ be an irreducible unitary representation of G. Then, we define the analytic torsion of G localised at the representation ρ. The idea of considering localised invariants is due to Brodzki, Niblo, Plymen and Wright [5], and was exploited in [31] to define a localised eta function. Next, let Γ be a discrete co compact subgroup of G. We use the localised analytic torsion to define the relative analytic torsion of the pair $(G,\Gamma )$, and we prove that the last coincides with the Lott $L^2$ analytic torsion of a covering space. We illustrate these constructions analysing in some details two examples: the abelian case, and the case $G=H$, the Heisenberg group.