The Laplacian on $p$-forms on the Heisenberg group

Details The Laplacian on $p$-forms on the Heisenberg group The Laplacian on $p$-forms on the Heisenberg group

1998-07-27

arxiv.org/abs/math/9807148v1

Mathematics

Differential Geometry

AMS-LaTeX2e 34 pages

Scientific paper

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.

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