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 rho be an irreducible unitary representation of G. Then, we define the analytic torsion of G localised at the representation rho. 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 Gamma 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. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).