Contravariant forms on generalized Verma modules and \(b\)-functions
type: Item
instance of: scholarly article
title: Contravariant forms on generalized Verma modules and \(b\)-functions
zbMATH Open document ID: 0976.17005
author: Akihito Wachi
published in: Hiroshima Mathematical Journal
publication date: 2002-01-13T00:00:00Z
review text: Let \(G\) be a complex Lie group, \({\mathfrak g}\) its Lie algebra, \({\mathfrak p}\) a parabolic subalgebra such that \(({\mathfrak g},{\mathfrak p})\) is of ``commutative parabolic type'', i.e. correponds to an Hermitian symmetric space (most results of the paper are given for the special case where \(({\mathfrak g},{\mathfrak p})\) is of ``regular type'', which corresponds to symmetric spaces of tube type). For \(\lambda\in\Hom({\mathfrak p},\mathbb{C})\), let \(M(\lambda)\) be the generalized \(U({\mathfrak g})\)-module with heighest weight \(\lambda\). Two bilinear forms on \(M(\lambda)\) are defined and their ratios on the irreducible components of the Schmid decomposition are computed in terms of \(b\)-functions. The unitarizability of the irreducible quotient of \(M(\lambda)\) is studied as an application.
reviewed by: Guy Roos
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 22E47
zbMATH DE Number: 1310266
zbMATH Keywords: Verma modules
zbMATH Keywords: Hermitian symmetric space
zbMATH Keywords: Schmid decomposition
zbMATH Keywords: unitarizability
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.32917/hmj/1206125161
OpenAlex ID: W1497500732
DOI: 10.32917/HMJ/1206125161
description: scientific article; zbMATH DE number 1310266