Beloshapka, I. V. On irreducible representations with finite weight of a certain discrete nilpotent group. (English. Russian original) Zbl 1344.43003 Russ. Math. Surv. 70, No. 4, 777-778 (2015); translation from Usp. Mat. Nauk 70, No. 4, 207-208 (2015). From the paper: Definition 1. Let \(\pi:G\to\operatorname{End}(V)\) be a representation of a group \(G\). The representation \(\pi\) has finite weight, or satisfies the finite-weight condition, if there exist a subgroup \(H\subset G\) and a character \(\chi\) of \(H\) such that the weight subspace \(V(H,\chi)=\{v \in V \mid \pi(h)v= \chi(h)v \;\forall\,h \in H\}\) is non-zero and finite-dimensional; \(H\) is then called a weight subgroup of the representation \(\pi\). Let \(G\) be the group of unipotent \(4 \times 4 \) matrices with integer coefficients and let \(\pi : G \to \operatorname{End}(V)\) be an irreducible representation of it on a complex at most countably dimensional vector space. Theorem 1. The irreducible representation \(\pi\) is monomial, that is, is induced from a character of a subgroup, if and only if it has finite weight. MSC: 43A65 Representations of groups, semigroups, etc. (aspects of abstract harmonic analysis) 20C15 Ordinary representations and characters 20F18 Nilpotent groups PDFBibTeX XMLCite \textit{I. V. Beloshapka}, Russ. Math. Surv. 70, No. 4, 777--778 (2015; Zbl 1344.43003); translation from Usp. Mat. Nauk 70, No. 4, 207--208 (2015) Full Text: DOI References: [1] С. А. Арналь, А. Н. Паршин 2012 Матем. заметки92 3 323–330 · Zbl 1132.03001 [2] English transl. S. A. Arnal’ and A. N. Parshin 2012 Math. Notes92 3 295–301 · Zbl 1264.43007 [3] I. D. Brown 1973 Pacific J. Math.45 1 13–26 · Zbl 0243.22007 [4] P. C. Kutzko 1977 Proc. Amer. Math. Soc.64 173–175 · Zbl 0375.22005 [5] M.-F. Vigne\acute{}ras 1996 Repre\acute{}sentations l-modulaires d’un groupe re\acute{}ductif p-adique avec l\ne p Progr. Math. 137 Birkha\ddot{}user Boston, Inc., Boston, MA xviii+233 pp. 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.