Some analysis of two generalized Heisenberg groups

Hailong Xu ()
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Hailong Xu: Nanjing University

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 1, 153-167

Abstract: 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.

Keywords: Generalized Heisenberg group; Dual space; Spectral decomposition.; 22D10; 43A85; 57S25. (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-022-00353-3

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