On the uniqueness of embeddings of Verma modules defined by the Shapovalov elements (Q1112952)

Let \({\mathfrak g}\) be a symmetrizable Kac-Moody Lie algebra (over a field of char 0), with Cartan subalgebra \({\mathfrak h}\), and associated Weyl group W. Let \(\Delta^+_ R\) denote the set of positive real roots. Fix a dominant \(\lambda\in {\mathfrak h}^*\) and define \(\Delta_{\lambda}:=\{\alpha \in \Delta^+_ R:\) \(<\lambda +\rho, \alpha^ v>\) is a positive integer\(\}\). Let \(W_{\lambda}\) be the subgroup of W generated by the reflections \(\{s_{\alpha}\}_{\alpha \in \Delta_{\lambda}}\). Then \(W_{\lambda}\) is a Coxeter group. For any \(w\in W_{\lambda}\), there exists an embedding of the Verma module M(w*\(\lambda)\) (with highest weight \(w*\lambda):=w(\lambda +\rho)-\rho)\) inside M(\(\lambda)\). When \(\lambda\) is dominant integral, the embedding is known to be unique (due to Rocha Caridi-Wallach). Given any reduced decomposition (with respect to the Coxeter group \(W_{\lambda})\) \(\omega\) of w, there is defined an embedding \(i_{\omega}\) (using Shapovalov element) of M(w*\(\lambda)\) in M(\(\lambda)\). One of the main results of the paper under review is that \(i_{\omega}\) does not depend upon the choice of the reduced decomposition \(\omega\) of \(w\in W_{\lambda}.\) The author also proves that, in the case when \(W_{\lambda}\) is of rank 2, any irreducible subquotient of M(\(\lambda)\) has multiplicity one in M(\(\lambda)\); again generalizing a result of Rocha Caridi-Wallach for integral \(\lambda\).