Factoring the Shapovalov determinant. II (Q1355626)

[Part I, cf. Queen's Pap. Pure Appl. Math. 94, 29-34 (1994; Zbl 0818.17031.)] Let \({\mathfrak g}\) be a Kac-Moody algebra (not necessarily symmetrizable) and let \(M(\lambda)\) be the Verma module for \({\mathfrak g}\) with highest weight \(\lambda\). Then \(M(\lambda)\) admits a unique (up to scalar multiple) contravariant form, whose determinant is called the Shapovalov determinant. Kac and Kazhdan showed how to express this determinant as a product of linear factors when \({\mathfrak g}\) is symmetrizable, but this formula no longer holds if \({\mathfrak g}\) is not symmetrizable. The present paper shows, however, that when the Kac-Kazhdan formula is suitably reinterpreted, it holds provided \({\mathfrak g}\) is `generic' in a suitable sense (which induces non-degeneracy, but not symmetrizability).