Murray, Scott H. Representations of parabolic and Borel subgroups. (English) Zbl 1120.20050 Commun. Algebra 35, No. 2, 455-459 (2007). Let \(k\) be a locally compact field. Assuming for convenience that the irreducible complex characters of the general linear groups \(\text{GL}_p(k)\) are known, one wants to relate the sets \(\text{Irr}(P)\) of irreducible representations for the various parabolic subgroups \(P\) of all these general linear groups. For a composition \(\lambda=(\lambda_1,\dots,\lambda_s)\) of \(q=\sum_i\lambda_i\) we denote by \(P^\lambda\) the parabolic subgroup of \(\text{GL}_q(k)\) that stabilizes the flag \(k^{\lambda_1}\subseteq k^{\lambda_1+\lambda_2}\cdots\). By repeated application of Mackey’s theorem on representations of semidirect products, the author expresses the set \(\text{Irr}(P^{(m,n)})\) as a disjoint union of sets \(\text{Irr}(P^\lambda\times\text{GL}_p(k))\), where \(\lambda\) is a composition of \(m\) and \(0\leq p=n-\sum_i(i-1)\lambda_i\). This result can be read in both directions. In particular, taking \(\lambda=(1^n)=(1,\dots,1)\), one learns that the irreducible representations of a Borel subgroup in \(\text{GL}_n(k)\) are determined by representations of maximal parabolic subgroups \(P^{(m,n)}\) in \(\text{GL}_{m+n}(k)\) with \(m=n(n-1)/2\). Reviewer: Wilberd van der Kallen (Utrecht) Cited in 1 Document MSC: 20G05 Representation theory for linear algebraic groups 22D10 Unitary representations of locally compact groups 20C15 Ordinary representations and characters Keywords:Borel subgroups; general linear groups; parabolic subgroups; Clifford theory; irreducible complex characters; irreducible representations PDFBibTeX XMLCite \textit{S. H. Murray}, Commun. Algebra 35, No. 2, 455--459 (2007; Zbl 1120.20050) Full Text: DOI References: [1] Curtis C. W., Methods of Representation Theory (1990) [2] Gabriel P., Representations of Finite-Dimensional Algebras (1997) [3] DOI: 10.1112/plms/s3-52.2.236 · Zbl 0587.20022 · doi:10.1112/plms/s3-52.2.236 [4] DOI: 10.2307/1969423 · Zbl 0046.11601 · doi:10.2307/1969423 [5] Zelevinsky A. V., Representations of Finite Classical Groups (1981) · Zbl 0465.20009 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.