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 $\rho$ an irreducible unitary representation of $G$. Then, we define the analytic torsion of $G$ localised at the representation $\rho$. Next, let $\mathrm{\Gamma }$ a discrete cocompact subgroup of $G$. We use the localised analytic torsion to define the relative analytic torsion of the pair $(G,\mathrm{\Gamma })$, and we prove that it 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.