Sugawara operators and Kac-Kazhdan conjecture
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.
highest weight irreducible representation
scientific article; zbMATH DE number 4103310