A proof of Kirillov's conjecture
type: Item
instance of: scholarly article
title: A proof of Kirillov's conjecture
zbMATH Open document ID: 1034.22010
author: Ehud Moshe Baruch
published in: Annals of Mathematics. Second Series
publication date: 2003-10-27T00:00:00Z
review text: Let \(K=\mathbb R\) or \(\mathbb C\) and let \(P_n(K)\) be the subgroup of the matrices in GL\(_n(K)\) consisting of those whose last row is \((0,0,\dots,0,1)\). In the paper under review the author proves that the restriction of an irreducible unitary representation of GL\(_n(K)\) to \(P_n(K)\) is irreducible. This claim was called Kirillov's conjecture. The history of this claim is long and interesting. \textit{I. M. Gelfand} and \textit{M. A. Naimark} in their book ``Unitäre Darstellungen der klassischen Gruppen'' [Akademie Verlag, Berlin (1957) (which is the German translation of the Russian publication from 1950; Zbl 0077.03405)] proved that a series of parabolically induced unitary representations of SL\(_n(\mathbb C)\) are irreducible by proving that their restriction to a group similar to \(P_n(\mathbb C)\) is irreducible (recall that GL\(_n(\mathbb C)\) is the product of SL\(_n(\mathbb C)\) and the center of GL\(_n(\mathbb C)\)). I. M. Gelfand and M. A. Naimark claimed that their representations exhaust all the irreducible unitary representations of SL\(_n(\mathbb C)\), what would imply that Kirillov's conjecture holds for GL\(_n(\mathbb C)\). In his paper [Ann. Math. (2) 86, 461--490 (1967; Zbl 0188.45303)], \textit{E. M. Stein} has shown that I. M. Gelfand and M. A. Naimark's list of representations is incomplete. Nevertheless, in their book they showed how Kirillov's conjecture implies the irreducibility of unitary parabolic induction. In his paper [Dokl. Akad. Nauk SSSR 114, 37--39 (1962; Zbl 0119.26804)] \textit{A. A. Kirillov} proposed a proof of the claim about the irreducibility of the restriction based on a nice geometric idea. He claimed that each \(P_n(K)\)-invariant distribution on GL\(_n(K)\) is invariant for the whole group. He observed that this claim implies the irreducibility of the restrictions to \(P_n(K)\) of the irreducible unitary representations of GL\(_n(K)\). To show this, for an irreducible unitary representation \(\pi\) of GL\(_n(…
reviewed by: Marko Tadić
Mathematics Subject Classification ID: 22E45
zbMATH DE Number: 1997276
zbMATH Keywords: general linear groups over real and complex fields
zbMATH Keywords: irreducible unitary representations
zbMATH Keywords: restrictions of irreducible unitary representations
zbMATH Keywords: invariant eigendistributions
MaRDI profile type: Publication
full work available at URL: annals.2003.158.207
OpenAlex ID: W2106869452
Wikidata QID: Q123203537
DOI: 10.4007/ANNALS.2003.158.207
description: scientific article; zbMATH DE number 1997276