## 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)

### Keywords:

discrete Heisenberg group; character; trace; representation
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### References:

 [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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