What question did this study set out to answer?

The research aims to explore the Rumin differentials in irreducible unitary representations of nilpotent Lie groups.

March 6, 2026Journal of Fourier Analysis and Applications0 citationsOpen Access

Regularized Determinants of the Rumin Complex in Irreducible Unitary Representations of the (2,3,5) Nilpotent Lie Group

Key Points

The research aims to explore the Rumin differentials in irreducible unitary representations of nilpotent Lie groups.
Analyzed the Rumin differentials of a 5-dimensional graded nilpotent Lie group.
Computed the spectrum of the Schrödinger representations.
Evaluated the zeta regularized determinant of each Rumin differential.
Assessed the analytic torsion of the Rumin complex in generic representations.
The spectrum of the Rumin differentials was successfully computed.
Zeta regularized determinants for each Rumin differential were obtained.
The analytic torsion related to the Rumin complex was evaluated, providing significant insights.

Abstract

Abstract We study the Rumin differentials of the 5-dimensional graded nilpotent Lie group that appears as the 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 quantum harmonic oscillator. For the Schrödinger 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.

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