On monomial representations of finitely generated nilpotent groups. (English) Zbl 1454.43007
Summary: A result of Segal states that every complex irreducible representation of a finitely generated nilpotent group \(G\) is monomial if and only if \(G\) is abelian-by-finite. A conjecture of Parshin, recently proved affirmatively by I. V. Beloshapka and S. O. Gorchinskii
[Sb. Math. 207, No. 1, 41–64 (2016; Zbl 1365.43006); translation from Mat. Sb. 207, No. 1, 45–72 (2016)]
characterizes the monomial irreducible representations of finitely generated nilpotent groups. This article gives a slightly shorter proof of the conjecture using ideas of Kutzko and Brown. We also give a characterization of the finite-dimensional irreducible representations of two-step nilpotent groups and describe these completely for two-step groups whose center has rank one.
MSC:
| 43A65 | Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) |
| 20C15 | Ordinary representations and characters |
| 20F18 | Nilpotent groups |
Keywords:
discrete Heisenberg groups; finitely generated nilpotent groups; induced representations; monomial representationsCitations:
Zbl 1365.43006References:
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