Sugawara operators and their applications to Kac-Kazhdan conjecture
Let \(M(\lambda)\) be a Verma module for an affine Kac-Moody Lie algebra \({\mathfrak g}\) and let \(L(\lambda)\) be the unique irreducible quotient of \(M(\lambda)\). Assume that \(\lambda (c)+g=0\) where c is the canonical central element in \({\mathfrak g}\) and g is the dual Coxeter number. The author proves that the character of \(L(\lambda)\) is given by the formula \(ch(L(\lambda))=e\prod (1-e^{-\alpha})^{-1},\) where the product is over all positive real roots. The result is a more general version of the Kac-Kazhdan conjecture. To prove the result the author introduces the higher order analogy of the Sugawara operators. If \({\mathfrak g}\) is of type \(A_ n^{(1)}\), \(B_ n^{(1)}\) or \(C_ n^{(1)}\), then the author proves the existence of a `fundamental' set of Sugawara operators. Further, when \({\mathfrak g}\) is of type \(A_ n^{(1)}\) the operators are constructed explicitly as elements in a completion Û(\({\mathfrak g})\) of the universal enveloping algebra of \({\mathfrak g}\).
scientific article; zbMATH DE number 4045972