On holomorphic representations of discrete Heisenberg groups. (English. Russian original) Zbl 1232.43005

Funct. Anal. Appl. 44, No. 2, 156-159 (2010); translation from Funkts. Anal. Prilozh. 44, No. 2, 92-96 (2010).
For two finitely generated abelian groups \(H\) and \(H'\), and a bilinear map \(H\times H'\to C\), the author considers a so-callled discrete Heisenberg group \(G\), which as a set is \(H\times H'\times C\). For a character \(\chi\in \operatorname{Hom}(G,C^*)\) he constructs a space \(V_\chi\) and a representation of \(\pi_\chi\) in the space \(V_\chi\). The space \(V_\chi\) is defined as a suitable subspace of the space of complex functions on \(G\).
To define a character of the representation \(\pi_\chi\), the author considers an extension of \(G\), more precisely, he takes a semidirect product \(\widehat{G}\) of \(G\) with a certain subgroup \(A\subset \operatorname{Hom}(H,H')\). Then he extends the representation \(\pi_\chi\) to a representation of \(\widehat{G}\) by adding a twist.
Finally, he considers the moduli space of this extended representation and shows that for \(g\in\widehat{G}\), the trace of the operator \(\pi_\chi(g)\) is a holomorphic function on this moduli space.


43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
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