[Paper Review] The Laplacian on $p$-forms on the Heisenberg group

Luke M. Schubert|ArXiv.org|Jul 27, 1998

Advanced Algebra and Geometry22 references3 citations

TL;DR

This paper computes the eigenvalues of the Laplacian on p-forms over the (2n+1)-dimensional Heisenberg group by decomposing it via generalized Bargmann representations, leading to a complete determination of the Novikov-Shubin invariants. The analysis uses operators commuting with the Laplacian in an anti-Fock space of square-integrable anti-holomorphic functions, yielding precise spectral data for the group's geometry and topology via heat kernel decay rates.

ABSTRACT

The Novikov-Shubin invariants for a non-compact Riemannian manifold M can be defined in terms of the large time decay of the heat operator of the Laplacian on square integrable p-forms on M. For the (2n+1)-dimensional Heisenberg group H, the Laplacian can be decomposed into operators in the conjugate of the generalised Bargmann representations which, when restricted to the centre of H, are characters. The representation space is an anti-Fock space of anti-holomorphic functions on complex n-space which are square integrable with respect to a Gaussian measure. In this paper, the eigenvalues of decomposed operators are calculated, using operators which commute with the Laplacian; this information determines all the Novikov-Shubin invariants of H. Further, some eigenvalues of operators connected with nilpotent Lie groups of Heisenberg type are calculated in the later sections.

Motivation & Objective

• To determine the Novikov-Shubin invariants of the (2n+1)-dimensional Heisenberg group using spectral analysis of the Laplacian on p-forms.
• To decompose the Laplacian on p-forms via generalized Bargmann representations, particularly in the context of the group's center as a character.
• To characterize the spectrum of the Laplacian on p-forms by identifying operators that commute with it in the anti-Fock space of square-integrable anti-holomorphic functions.
• To extend the spectral analysis to nilpotent Lie groups of Heisenberg type, providing eigenvalue computations for broader classes of sub-Riemannian manifolds.
• To link the large-time decay of the heat kernel on p-forms to the Novikov-Shubin invariants through precise eigenvalue data.

Proposed method

• The Laplacian on p-forms over the Heisenberg group is decomposed using the conjugate of generalized Bargmann representations, which diagonalize the operator in the representation space.
• The representation space is identified as an anti-Fock space of anti-holomorphic functions on C^n, equipped with a Gaussian measure, enabling the use of holomorphic functional calculus.
• Operators that commute with the Laplacian are constructed and used to compute eigenvalues, leveraging the symmetry of the Heisenberg group and its center.
• The eigenvalues are computed explicitly by analyzing the action of these commuting operators in the anti-Fock setting, leading to a complete spectral description.
• The method extends to nilpotent Lie groups of Heisenberg type by adapting the representation-theoretic framework to their structure and associated Fock-type spaces.
• The spectral data are then used to compute the Novikov-Shubin invariants via the asymptotic decay rate of the heat trace on p-forms.

Experimental results

Research questions

• RQ1 What are the Novikov-Shubin invariants of the (2n+1)-dimensional Heisenberg group, and how are they determined by the spectrum of the Laplacian on p-forms?
• RQ2 How can the Laplacian on p-forms over the Heisenberg group be decomposed using generalized Bargmann representations?
• RQ3 Which operators commute with the Laplacian in the anti-Fock space setting, and how do they facilitate eigenvalue computation?
• RQ4 What is the spectral structure of the Laplacian on p-forms for nilpotent Lie groups of Heisenberg type, and how does it generalize the Heisenberg group case?
• RQ5 How does the large-time decay of the heat kernel on p-forms relate to the Novikov-Shubin invariants through the computed eigenvalues?

Key findings

• The Novikov-Shubin invariants of the (2n+1)-dimensional Heisenberg group are fully determined by the computed eigenvalues of the Laplacian on p-forms.
• The eigenvalues of the decomposed Laplacian operators are explicitly calculated using commuting operators in the anti-Fock space of square-integrable anti-holomorphic functions.
• The representation space for the generalized Bargmann transform is identified as the anti-Fock space on C^n with a Gaussian measure, providing a concrete realization of the spectral decomposition.
• The method successfully extends to nilpotent Lie groups of Heisenberg type, yielding eigenvalue computations for a broader class of sub-Riemannian manifolds.
• The spectral data obtained directly determine the asymptotic decay rate of the heat trace on p-forms, which governs the Novikov-Shubin invariants.
• The analysis confirms that the Novikov-Shubin invariants are non-trivial and computable for the Heisenberg group, reflecting its non-compact, nilpotent geometry.