Geodesic
The Mellin Transform and the Plancherel Theorem for the Discrete Heisenberg Group
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 193-211.

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In the classical representation theory of locally compact groups, there are well-known constructions of a unitary dual space of irreducible representations, the Fourier transform, and the Plancherel theorem. In this paper, we present analogs of these constructions for the discrete Heisenberg group and its irreducible infinite-dimensional representations in a vector space without topology. 
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A. N. Parshin. The Mellin Transform and the Plancherel Theorem for the Discrete Heisenberg Group. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Algebra, number theory, and algebraic geometry, Tome 307 (2019), pp. 193-211. http://geodesic.mathdoc.fr/item/TM_2019_307_a9/

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