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.
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Dedicated to Professor Kalyan B. Sinha on the occasion of his 70th birthday.
The first named author has been supported by an Emmy-Noether grant of Deutsche Forschungsgemeinschaft (DFG); the second named author was supported by a fund of Prof. Li Yutian, Hong Kong Baptist University and the National Center for Theoretical Sciences, National Tsing Hua University, Taiwan. The third named author was supported by the Grant-in-aid for Scientific Research (C) No. 24540189, JSPS and the National Center for Theoretical Sciences, National Tsing Hua University, Taiwan.
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Bauer, W., Furutani, K. & Iwasaki, C. Spectral zeta function on pseudo H-type nilmanifolds. Indian J Pure Appl Math 46, 539–582 (2015). https://doi.org/10.1007/s13226-015-0151-6
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DOI: https://doi.org/10.1007/s13226-015-0151-6