Embeddings of unitary highest weight representations and generalized Dirac operators
type: Item
instance of: scholarly article
title: Embeddings of unitary highest weight representations and generalized Dirac operators
zbMATH Open document ID: 0882.22014
author: Thomas J. Enright
author: Nolan Wallach
published in: Mathematische Annalen
publication date: 1998-03-02T00:00:00Z
review text: Let \(G\) be a simple connected finite-dimensional real Lie group with center \(Z\), let \(G_0= G/Z\) and \(K=p^{-1}(K_0)\), where \(p\) is the canonical projection of \(G\) onto \(G_0\) and \(K_0\) is a maximal compact subgroup of \(G_0\). Let \(\mathfrak k\) be the complexified Lie algebra of \(K\) looked upon as a Lie subalgebra of the complexified Lie algebra \(\mathfrak g\) of \(G\) and assume that \(G/K\) is a Hermitian symmetric space. Then \(K\) has a one-dimensional center and we can choose a decomposition \({\mathfrak k}= CH\oplus[{\mathfrak k},{\mathfrak k}]\) with \(\text{ad }H\) having eigenvalues \(-1\), \(0\), \(1\). If \({\mathfrak u}= \{X|X\in{\mathfrak g}\) and \([H,X]= X\}\) and \(\overline{{\mathfrak u}}= \{X|X\in {\mathfrak g}\) and \([H,X]=-X\}\), then \({\mathfrak q}={\mathfrak k}\oplus{\mathfrak u}\) is a \(\theta\)-stable parabolic subalgebra of \({\mathfrak g}\), where \(\theta\) is the Cartan involution of \(\mathfrak g\) with respect to \(\mathfrak k\). Fix a Cartan subalgebra \(\mathfrak h\) of \(\mathfrak g\) with \({\mathfrak h}\subset{\mathfrak k}\) and let \(\mathfrak b\) be a \(\theta\)-stable Borel subalgebra of \(\mathfrak g\) containing \(\mathfrak h\) and contained in \(\mathfrak q\). There exists a parabolic subalgebra \({\mathfrak q}'\) of \(\mathfrak g\), and a corresponding parabolic subgroup \(Q'\) of \(G\), such that \({\mathfrak q}'\cap{\mathfrak g}_0\) is a real parabolic subalgebra of the (real) Lie algebra \({\mathfrak g}_0\) of \(G\) and \({\mathfrak q}\cap{\mathfrak q}'\) is a Levi factor of both \(\mathfrak q\) and \({\mathfrak q}'\). The Hermitian symmetric pair \((G,K)\) is assumed to be of tube type, i.e., the largest root of the system \(\Phi^+\) of positive roots corresponding to \(\mathfrak b\) of the root system of \(\mathfrak g\) with respect to \(\mathfrak h\) is the biggest element \(\gamma_l\) of the Harish-Chandra's system \(\gamma_1<\gamma_2<\cdots<\gamma_l\) of strongly orthogonal roots for \(\Phi^+_n=…
Mathematics Subject Classification ID: 22E45
Mathematics Subject Classification ID: 22E47
Mathematics Subject Classification ID: 22E70
zbMATH DE Number: 1031510
zbMATH Keywords: real Lie group
zbMATH Keywords: Lie algebra
zbMATH Keywords: Hermitian symmetric space
zbMATH Keywords: degenerate principal series
zbMATH Keywords: intertwining operator
zbMATH Keywords: generalized Dirac operator
MaRDI profile type: Publication
full work available at URL: s002080050053
OpenAlex ID: W1991278757
DOI: 10.1007/S002080050053
description: scientific article; zbMATH DE number 1031510