On differential operators attached to certain representations of classical groups
type: Item
instance of: scholarly article
title: On differential operators attached to certain representations of classical groups
zbMATH Open document ID: 0558.10023
DOI: 10.1007/BF01388834
published in: Inventiones Mathematicae
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143156
review text: Let \(\sigma\) : GL\({}_ m({\mathbb{C}})\to GL_ n({\mathbb{C}})\) be an irreducible representation which occurs in the space of all homogeneous polynomial functions of degree r on the complex symmetric matrices. To each such \(\sigma\) and another representation \(\zeta\) of \(GL_ m({\mathbb{C}})\) the author attaches a certain non-holomorphic differential operator \(\Delta_{\zeta,\sigma}\) which sends a scalar-valued holomorphic Eisenstein series to a non-holomorphic vector valued one. Exact criteria for the non-vanishing of the latter can be obtained by studying the operator \(\Delta_{\zeta,\sigma}\). This problem is solved in this paper not only for \(GL_ m({\mathbb{C}})\), but for the automorphism groups of the classical Hermitian symmetric spaces.
reviewed by: Birgit Speh
Mathematics Subject Classification ID: 11F27
Mathematics Subject Classification ID: 32M15
Mathematics Subject Classification ID: 22E40
zbMATH DE Number: 3889638
zbMATH Keywords: non-holomorphic differential operator
zbMATH Keywords: holomorphic Eisenstein series
zbMATH Keywords: automorphism groups
zbMATH Keywords: Hermitian symmetric spaces
MaRDI profile type: Publication
cites work: Extension of a Formula by Cayley to Symmetric Determinants
cites work: Q5520117
cites work: Q3849170
cites work: On a ring of invariant polynomials on a Hermitian symmetric space
cites work: Die Randwerte holomorpher Funktionen auf hermitesch symmetrischen Räumen
cites work: The arithmetic of certain zeta functions and automorphic forms on orthogonal groups
cites work: Arithmetic of differential operators on symmetric domains
cites work: Confluent hypergeometric functions on tube domains
cites work: On Eisenstein series
cites work: Differential operators and the singular values of Eisenstein series
cites work: Über die Zetafunktionen indefiniter quadratischer Formen
author: Gorō Shimura
description: scientific article; zbMATH DE number 3889638