Spectral zeta function on pseudo H-type nilmanifolds

Wolfram Bauer (), Kenro Furutani () and Chisato Iwasaki ()
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Wolfram Bauer: Leibniz Universität
Kenro Furutani: Tokyo University of Science
Chisato Iwasaki: University of Hyogo

Indian Journal of Pure and Applied Mathematics, 2015, vol. 46, issue 4, 539-582

Abstract: Abstract We explain the explicit integral form of the heat kernel for the sub-Laplacian on two step nilpotent Lie groups G based on the work of Beals, Gaveau and Greiner. Using such an integral form we study the heat trace of the sub-Laplacian on nilmanifolds L\G where L is a lattice. As an application a common property of the spectral zeta function for the sub-Laplacian on L\G is observed. In particular, we introduce a special class of nilpotent Lie groups, called pseudo H-type groups which are generalizations of groups previously considered by Kaplan. As is known such groups always admit lattices. Here we aim to explicitly calculate the heat trace and the spectrum of the (sub)-Laplacian on various low dimensional compact nilmanifolds including several pseudo H-type nilmanifolds L\G, i.e. where G is a pseudo H-type group. In an appendix we discuss a zeta function which typically appears as the Mellin transform for these heat traces.

Keywords: Sub-Laplacian; sub-Riemannian manifold; nilmanifold; heat kernel; spectral zeta function; pseudo H-type group; admissible module (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s13226-015-0151-6

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