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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.

MSC:

43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
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[1] J. N. Bernstein and A. V. Zelevinsky, Uspekhi Mat. Nauk, 31:3 (1976), 5–70; English transl.: Russian Math. Surveys, 31:3 (1976), 1–68.
[2] L. Gerritzen, Math. Ann., 196 (1972), 323–346. · Zbl 0255.14013
[3] V. Kac and D. Peterson, Adv. Math., 53:2 (1984), 125–264. · Zbl 0584.17007
[4] G. Mackey, Amer. J. Math., 73 (1951), 576–592. · Zbl 0045.30305
[5] D. Mumford, Invent. Math., 1 (1966), 287–354. · Zbl 0219.14024
[6] A. Pressley and G. Segal, Loop Groups, The Clarendon Press, Oxford Univ. Press, New York, Oxford, 1986.
[7] T. Pytlik, Monatsh. Math., 93:4 (1982), 309–328. · Zbl 0487.22006
[8] J.-P. Serre, Représentations linéaires des groupes finis, Hermann, Paris, 1967.
[9] M.-F. Vignéras, Représentations l-modulaires d’un groupe réductif p-adique avec l p, Birkhäuser, Boston, MA, 1996.
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