Orthogonal basis for the Shapovalov form on $A_n$

Abstract

Let $U$ be either classical or quantized universal enveloping algebra of $\s\l(n+1)$ extended over the field of fractions of the Cartan subalgebra. We suggest a PBW basis in $U$ over the extended Cartan subalgebra diagonalizing the contravariant Shapovalov form on generic Verma module. The matrix coefficients of the form are calculated and the inverse of the form is explicitly constructed.