Regularized determinants of the Rumin complex in irreducible unitary representations of the (2,3,5) nilpotent Lie group

TL;DR

This paper investigates the spectral properties and determinants of Rumin differentials in irreducible unitary representations of a specific 5-dimensional nilpotent Lie group, revealing connections to quantum harmonic oscillators and analytic torsion.

Contribution

It provides explicit spectral computations and determinant evaluations of Rumin differentials in various representations of the (2,3,5) nilpotent Lie group, extending classical analysis.

Findings

Spectrum and zeta regularized determinants computed for Schrodinger representations.

Analytic torsion evaluated for generic representations.

Connections established between Rumin differentials and quantum harmonic oscillators.

Abstract

We study the Rumin differentials of the 5-dimensional graded nilpotent Lie group that appears as osculating group of generic rank two distributions in dimension five. In irreducible unitary representations of this group, the Rumin differentials provide intriguing generalizations of the classical quantum harmonic oscillator. For the Schrodinger representations, we compute the spectrum and the zeta regularized determinant of each Rumin differential. In the generic representations, we evaluate their alternating product, i.e., the analytic torsion of the Rumin complex.