On the Kazhdan-Lusztig conjecture for Kac-Moody algebras (Q5903491)

Let \(A=(a_{ij})_{1\leq i,j\leq n}\) be a not necessarily symmetrizable GCM and let \({\mathfrak g}(A)\) be the corresponding Kac-Moody algebra, \({\mathfrak h}(A)\) the Cartan subalgebra. Let \(\rho\in {\mathfrak h}(A)^*\) which takes the value one on each simple coroot. Take a subset I of \(\{\) 1,...,n\(\}\) such that \(A_ I=(a_{ij})_{i,j\in I}\) is the Cartan matrix of a complex semisimple Lie algebra \({\mathfrak g}_ I\). Denote by \(W_ I\) the subgroup of the Weyl group \(W=W(A)\) generated by simple reflections \(s_ i\) (i\(\in I)\). Let \(\leq\) be the standard partial order on the Coxeter group (W,S). The main result of this paper is to prove the following Kazhdan-Lusztig type theorem: \[ [M(y\rho -\rho) : L(w\rho - \rho)=P_{y,w}(1) \] for all \(y,w\in W_ I\) with \(y\leq w\), where \(P_{y,w}\) is the Kazhdan-Lusztig polynomial for (W,S).