On the Kazhdan-Lusztig conjecture for Kac-Moody algebras (Q5903162)

Let \({\mathfrak g}\) be a complex semisimple Lie algebra with Cartan subalgebra \({\mathfrak h}\) and Weyl group W. Then, for any \(y\leq w\in W\), the multiplicity mtp(y,w) of the irreducible \({\mathfrak g}\)-module \(L(w(\lambda +\rho)-\rho)\) in the Verma module \(M(y(\lambda +\rho)-\rho)\) (with highest weight \(y(\lambda +\rho)-\rho)\) is known to be independent of the dominant integral weight \(\lambda\in {\mathfrak h}^*\), (where \(\rho\in {\mathfrak h}^*\) is, as usual, the element satisfying \(\rho (h_ i)=1\), for all simple co-roots \(h_ i)\) and further Kazhdan-Lusztig conjectured (and later proved by Beilinson-Bernstein and Brylinski-Kashiwara) that \(mtp(y,w)=p_{y,w}(1)\), where the \(p_{y,w}\) are, now well-known as, Kazhdan-Lusztig polynomials, which are combinatorially constructed from the Coxeter group W. Now for arbitrary symmetrizable Kac-Moody algebras, Deodhar-Gabber-Kac conjectured a similar result. Let \({\mathfrak g}\) be an arbitrary Kac-Moody algebra (with Cartan subalgebra \({\mathfrak h}\) and Weyl group W). The author proves, by a very straightforward reduction to the finite case, that the multiplicity of L(w\(\rho\)-\(\rho)\) in M(y\(\rho\)-\(\rho)\) is the same as the predicted one, i.e., equal to \(P_{y,w}(1)\), but for very special Weyl group elements \(y\leq w\). More precisely for those \(y\leq w\) which belong to a finite sub Weyl group \(\subset W\).