Tohoku Mathematical Journal

Heat kernel transform on nilmanifolds associated to H-type groups

Aparajita Dasgupta and Sundaram Thangavelu

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We study the heat kernel transform on a nilmanifold $ \Gamma \backslash N $ associated to an H-type group. Using a reduction technique we reduce the problem to the case of Heisenberg groups. The image of $ L^2(\Gamma \backslash N) $ under the heat kernel transform is shown to be a direct sum of weighted Bergman spaces.

Article information

Tohoku Math. J. (2), Volume 64, Number 3 (2012), 439-451.

First available in Project Euclid: 11 September 2012

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Zentralblatt MATH identifier

Primary: 22E30: Analysis on real and complex Lie groups [See also 33C80, 43-XX]
Secondary: 35H20: Subelliptic equations 35K05: Heat equation 58J35: Heat and other parabolic equation methods

H-type groups representations Laplacians heat kernels Weil-Brezin transforms Bergman spaces


Dasgupta, Aparajita; Thangavelu, Sundaram. Heat kernel transform on nilmanifolds associated to H-type groups. Tohoku Math. J. (2) 64 (2012), no. 3, 439--451. doi:10.2748/tmj/1347369372.

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  • L. Auslander and J. Brezin, Translation invariant subspace in $L^{2}$ of a compact nilmanifold I, Invent. Math. 20 (1973), 1–14.
  • J. Brezin, Harmonic analysis on nilmanifolds, Trans. Amer. Math. Soc. 150 (1970), 611–618.
  • J. Cygan, Heat kernels for class 2 nilpotent groups, Studia Math. 64 (1979), 227–238.
  • A. Kaplan and F. Ricci, Harmonic analysis on groups of Heisenberg type, Harmonic Analysis (Cartona, 1982), 416–435, in Lecture Notes in Mathematics, 992, Springer, Berlin, 1983.
  • B. Krötz, S. Thangavelu and Y. Xu, The Heat kernel transform for the Heisenberg group, J. Funct. Anal. 225 (2005), 301–336.
  • B. Krötz, S. Thangavelu and Y. Xu, Heat kernel transform for nilmanifolds associated to the Heisenberg group, Rev. Mat. Iberoamericana. 24 (2008), 243–266.
  • C. C. Moore, Decomposition of unitary representations defined by discrete subgroups of nilpotent groups, Ann. of Math. (2) 82 (1965), 146–182.
  • D. Müller, Sharp $L^{p}$ estimates for the wave equation on Heisenberg type groups, in Lecture Notes, Orleans, April, 2008.
  • A. L. Onishchik and E. B. Vinberg (Eds.), Lie groups and Lie algebras II, Encyclopaedia Math. Sci. 21, Springer-Verlag, Berlin, 2000.
  • J. Randall, The heat kernel for generalized heisenberg groups, J. Geom. Anal. 6 (1996), 287–316.
  • F. Ricci, Harmonic analysis on generalized Heisenberg groups, Preprint.
  • S. Thangavelu, Harmonic analysis on the Heisenberg nilmanifolds, Rev. Un. Mat. Argentina 50 (2009), 75–93.
  • S. Thangavelu, Gutzmer's formula and the Segal-Bargmann transform, Perspectives in mathematical sciences. II, 209–221, Stat. Sci. Interdiscip. Res., 8, World Sci. Publ., Hackensack, NJ, 2009.
  • R. Tolimieri, Heisenberg manifolds and theta functions, Trans. Amer. Math. Soc. 239 (1978), 293–319.