Highest weight modules associated with classical irreducible regular prehomogeneous vector spaces of commutative parabolic type
type: Item
instance of: scholarly article
title: Highest weight modules associated with classical irreducible regular prehomogeneous vector spaces of commutative parabolic type
zbMATH Open document ID: 0758.17005
author: Shuichi Suga
published in: Osaka Journal of Mathematics
publication date: 1992-09-27T00:00:00Z
review text: The purpose of this paper is to give realization of highest weight modules in terms of irreducible relative invariants of prehomogeneous vector spaces. Let \({\mathfrak g}\) be a classical complex simple Lie algebra. We assume that \({\mathfrak g}\) has a \(\mathbb{Z}\)-gradation of the form \({\mathfrak g}={\mathfrak g}(-1)+{\mathfrak g}(0)+{\mathfrak g}(1)\). Then the natural action of \({\mathfrak g}(0)\) on \({\mathfrak g}(\pm 1)\) makes an irreducible prehomogeneous vector space. Let \(d\lambda\) be a 1-dimensional representation of the parabolic subalgebra \({\mathfrak p}={\mathfrak g}(0)+{\mathfrak g}(1)\) and let \(\mathbb{C}_{d\lambda}\) be its representation space. Let \(U({\mathfrak g})\) and \(U({\mathfrak p})\) be the universal enveloping algebras of \({\mathfrak g}\) and \({\mathfrak p}\), respectively. We denote by \(V(d\lambda)\) the generalized Verma module induced from \(d\lambda:V(d\lambda)=U({\mathfrak g})\otimes_{U({\mathfrak p})}\mathbb{C}_{d\lambda}\) and by \(L(d\lambda)\) its irreducible quotient. Then the \(U({\mathfrak g})\)-module \(L(d\lambda)\) is realized using the irreducible relative invariant polynomial of the prehomogeneous vector space \(({\mathfrak g}(0),{\mathfrak g}(-1))\). As an application, we recover the reducibility criterion of \(V(d\lambda)\) and show that it has a natural interpretation in terms of the zeros of the \(b\)-function of the irreducible relative invariant.
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 14M17
zbMATH DE Number: 59005
zbMATH Keywords: highest weight modules
zbMATH Keywords: prehomogeneous vector spaces
zbMATH Keywords: complex simple Lie algebra
zbMATH Keywords: generalized Verma module
reviewed by: Masakazu Muro
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 59005