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Some analysis of two generalized Heisenberg groups

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  • Hailong Xu

    (Nanjing University)

Abstract

This paper studies two generalized Heisenberg groups: $${\mathcal {H}}^n := {\mathbb {R}}^n \times {\mathbb {R}}^n \times {\mathbb {R}}$$ H n : = R n × R n × R for $$n > 1$$ n > 1 and $$G := {\mathbb {R}} \times {\mathbb {R}} \times {\mathbb {R}}^2$$ G : = R × R × R 2 . More precisely, we first classify uniform lattices and then give an explicit description of spectral decomposition of for any uniform lattice $$\Gamma $$ Γ , in particular for the uniform lattice $$\Gamma _k$$ Γ k , where $$\Gamma _k$$ Γ k is defined to be $${\mathbb {Z}} \times {\mathbb {Z}} \times \frac{{\mathbb {Z}}}{k} \times \frac{{\mathbb {Z}}}{k}$$ Z × Z × Z k × Z k . Finally using the Selberg trace formula and spectral decomposition of , we obtain an identity which involves the Fourier transform.

Suggested Citation

  • Hailong Xu, 2024. "Some analysis of two generalized Heisenberg groups," Indian Journal of Pure and Applied Mathematics, Springer, vol. 55(1), pages 153-167, March.
  • Handle: RePEc:spr:indpam:v:55:y:2024:i:1:d:10.1007_s13226-022-00353-3
    DOI: 10.1007/s13226-022-00353-3
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