Sugawara operators and Kac-Kazhdan conjecture
type: Item
instance of: scholarly article
title: Sugawara operators and Kac-Kazhdan conjecture
zbMATH Open document ID: 0674.17005
DOI: 10.1007/BF01394343
author: Takahiro Hayashi
published in: Inventiones Mathematicae
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143616
review text: Let \({\mathfrak g}\) be an affine Kac-Moody Lie algebra and let L(\(\lambda)\) be a highest weight irreducible representation of \({\mathfrak g}\). If \(\lambda\) is a dominant integral weight then the character of L(\(\lambda)\) is given by the well known Weyl-Kac character formula. For general \(\lambda\), it is more difficult to determine the character formula. In this paper the author proves that if the level of \(\lambda\) is equal to the negative of the dual Coxeter number, and if \(\lambda +\rho (\alpha^ v)\not\in {\mathbb{Z}}_{>0}\) for all positive co-roots, then, \[ ch(\lambda)=e^{\lambda}\prod_{\alpha \in \Delta_+^{re}}(1-e^{-\alpha})^{-1}. \] To prove this formula he introduces higher order Sugawara operators; these are defined in a suitable completion of U(\({\mathfrak g})\). These operators act on the Verma modules M(\(\lambda)\) (\(\lambda\) as before) and in fact intertwine the action of \({\mathfrak g}\). The theorem is obtained as a consequence of the complete description of \({\mathfrak g}\)-intertwining operators.
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 4103310
zbMATH Keywords: Kac-Kazhdan conjecture
zbMATH Keywords: affine Kac-Moody Lie algebra
zbMATH Keywords: highest weight irreducible representation
zbMATH Keywords: dominant integral weight
zbMATH Keywords: character formula
zbMATH Keywords: higher order Sugawara operators
zbMATH Keywords: Verma modules
zbMATH Keywords: intertwining operators
MaRDI profile type: Publication
Wikidata QID: Q122924691
cites work: Introduction to Lie Algebras and Representation Theory
cites work: Moduln mit einem höchsten Gewicht
cites work: Q3693672
cites work: Structure of representations with highest weight of infinite-dimensional Lie algebras
cites work: Q5611879
OpenAlex ID: W2017649081
description: scientific article; zbMATH DE number 4103310