×

On irreducible representations of discrete Heisenberg groups. (English. Russian original) Zbl 1264.43007

Math. Notes 92, No. 3, 295-301 (2012); translation from Mat. Zametki 92, No. 3, 323-330 (2012).
A representation of a discrete group \(G\) is said to be monomial if it is induced from a character (not necessarily unitary) of a subgroup \(H.\) Let \(\pi:G \to {\operatorname{End}}(V_\pi)\) be a representation of a discrete group \(G\). A subspace \(V(H, \chi)\) is called a weight space if \[ V(H, \chi) = \{ v \in V_\pi:~\pi(h)v = \chi(h)v\;\forall\,h \in H\} \] for a subgroup \(H\) of \(G\). The representation \(\pi\) is said to have finite weight if there is \(H\) and \(\chi\) such that \(V(H, \chi) \neq 0\) and \(\dim_{\mathbb C} V (H, \chi) < \infty\). The authors prove that, if \(G\) is a finitely generated nilpotent group ,then unitary irreducible representations are monomial if and only if they have finite weight. Applications are given for representations of discrete Heisenberg groups.

MSC:

43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis)
43A40 Character groups and dual objects
22A25 Representations of general topological groups and semigroups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] M.-F. Vignéras, Représentations l-modulaires d’un groupe réductif p-adique avec l p, in Progr. Math. (Birkhäuser Boston, Boston,MA, 1996), Vol. 137.
[2] A. N. Parshin, ”Representations of higher adelic groups and arithmetic,” in Proceedings of the International Congress of Mathematicians. Vol. 1. Plenary Lectures and Ceremonies, Hyderabad, India, 19–27 August 2010 (Hindustan Book Agency, New Delhi, 2010), pp. 362–392.
[3] J.-P. Serre, Représentations linéaires des groupes finis (3rd revised ed., Hermann, Paris, 1978; English translation of the 2nd ed. Springer-Verlag, New York-Heidelberg, 1977; Russian translation of the 1st ed. Mir, Moscow, 1970).
[4] A. A. Kirillov, ”Unitary representations of nilpotent Lie groups,” Uspekhi Mat. Nauk 17(4), 57–110 (1962) [Russ. Math. Surv. 17 (4), 53–104 (1962)].
[5] G. van Dijk, ”Smooth and admissible representations of p-adic unipotent groups,” Compositio Math. 37(1), 77–101 (1978). · Zbl 0499.22010
[6] I. D. Brown, ”Representations of finitely generated nilpotent groups,” Pacific J. Math. 45, 13–26 (1973). · Zbl 0243.22007
[7] G.W. Mackey, ”On induced representations of groups,” Amer. J. Math. 73(3), 576–592 (1951). · Zbl 0045.30305
[8] A. N. Parshin, ”On holomorphic representations of discrete Heisenberg groups,” Funktsional. Anal. i Prilozhen. 44(2), 92–96 (2010) [Funct. Anal. Appl. 44 (2), 156–159 (2010)]. · Zbl 1232.43005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.