Gradient-type differential operators and unitary highest weight representations of SU(p,q)
type: Item
instance of: scholarly article
title: Gradient-type differential operators and unitary highest weight representations of SU(p,q)
zbMATH Open document ID: 0678.22004
DOI: 10.1016/0022-1236(88)90114-0
published in: Journal of Functional Analysis
publication date: 1988-01-01T00:00:00Z
review text: Let \(G=SU(p,q)\) and K a maximal compact subgroup of G. Let \({\mathfrak g}={\mathfrak k}+{\mathfrak p}\) denote the Cartan decomposition of \({\mathfrak g}=Lie(G)\), \(\{X_ j\}\) an orthonormal basis of \({\mathfrak p}_ C\) and \({\mathfrak p}^{\pm}_ C\) the upper and lower triangular matrices of \({\mathfrak p}_ C\). For \((\tau_{\lambda},H_{\lambda})\in \hat K\) with highest weight \(\lambda\) let \(C^{\infty}(G,\tau_{\lambda})=\{f\in C^{\infty}(G,H_{\lambda})\); \(f(gk)=\tau_{\lambda}(k)^{-1}f(g)\) \((k\in K\), \(g\in G)\}\), \(\nabla\) the gradient operator defined by \(\nabla f(g)=\sum r(X_ j)f(g)\otimes \bar X_ j\) and \({\mathcal P}_{\lambda}: H_{\lambda}\otimes {\mathfrak p}_ C\to W(\lambda)^{\perp}\) a projection onto the orthogonal complement of a K-invariant subspace \(W(\lambda)\) of \(H_{\lambda}\otimes {\mathfrak p}_ C\). Then we call \({\mathcal D}_{\lambda}={\mathcal P}_{\lambda}\nabla\) a differential operator of gradient type. When \(W(\lambda)=H_{\lambda}\otimes {\mathfrak p}^-_ C\), \({\mathcal D}_{\lambda}\) is the Schmid operator and the discrete series arises as the \(L^ 2\)-kernel of \({\mathcal D}_{\lambda}.\) In this paper, noting the Fock realization of unitary highest weight representations \(F_{\lambda}\) (\(\lambda\in \Lambda)\), the authors construct \(W(\lambda)\subset H_{\lambda}\otimes {\mathfrak p}^-_ C\) as the direct sum of K-irreducible subspaces \(H^{\lambda-\beta}\) with highest weights \(\lambda-\beta\) \((\beta \in \Phi^+_ n)\) which satisfy \((\lambda +\delta,2\beta)/(\beta,\beta)<1\). Let \(\Lambda_ 1\) be the subset of \(\Lambda\) for which there exists \(H^{\lambda -\beta}\) with \((\lambda +\delta,2\beta)/(\beta,\beta)=1\). Then \({\mathcal D}_{\lambda}\) \((\lambda \in \Lambda_ 1)\) is not the Schmid operator and the unitary highest weight representation with highest weight \(\lambda\) contained in Ker(\({\mathcal D}_{\lambda})\) corresponds to a point at the far right of the discrete set, opposite the discrete serie…
Mathematics Subject Classification ID: 22E30
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 22E47
Mathematics Subject Classification ID: 43A65
zbMATH DE Number: 4110966
zbMATH Keywords: maximal compact subgroup
zbMATH Keywords: Cartan decomposition
zbMATH Keywords: orthogonal complement
zbMATH Keywords: differential operator of gradient type
zbMATH Keywords: Fock realization
zbMATH Keywords: unitary highest weight representations
zbMATH Keywords: highest weights
zbMATH Keywords: representation spaces
zbMATH Keywords: ladder representations
author: Mark G. Davidson
author: Ronald J. Stanke
reviewed by: Takeshi Kawazoe
MaRDI profile type: Publication
full work available at URL: 0022-1236(88)90114-0
OpenAlex ID: W2014901971
cites work: Compactification of arithmetic quotients of bounded symmetric domains
cites work: Knapp-Wallach Szegö integrals and generalized principal series representations: The parabolic rank one case
cites work: The harmonic representation of U(p,q) and its connection with the generalized unit disk
cites work: Q3954003
cites work: Q3318972
cites work: Q4751164
cites work: Q3223062
cites work: On singular holomorphic representations
cites work: The last possible place of unitarity for certain highest weight modules
cites work: Hermitian symmetric spaces and their unitary highest weight modules
cites work: Wave and Dirac operators, and representations of the conformal group
cites work: On the Segal-Shale-Weil representations and harmonic polynomials
cites work: Szegoe kernels associated with discrete series
cites work: Q3333257
cites work: An integral transform in \(L^ 2\)-cohomology for the ladder representations of U(p,q)
cites work: Dirac operator and the discrete series
cites work: Unitary Structures on Cohomology
cites work: Q5642743
cites work: Q5682518
cites work: Q3215171
description: scientific article; zbMATH DE number 4110966