Spectral zeta function on pseudo H-type nilmanifolds

• Published in: Indian journal of pure and applied mathematics Vol. 46; no. 4; pp. 539 - 582
• Main Authors: Bauer, Wolfram, Furutani, Kenro, Iwasaki, Chisato
• Format: Journal Article
• Language: English
• Published: New Delhi Indian National Science Academy 01.08.2015
• Subjects: Applications of Mathematics; Mathematics; Mathematics and Statistics; Numerical Analysis; admissible module; heat kernel; spectral zeta function; Sub-Laplacian; sub-Riemannian manifold; nilmanifold; pseudo; type group
• DOI: 10.1007/s13226-015-0151-6
• ISSN: 0019-5588

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.