Orthogonal basis for the Shapovalov form on $U_q(\mathfrak{sl}(n + 1))$

Abstract

Let U be either the classical or quantized universal enveloping algebra of the Lie algebra [Formula: see text] 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 form is explicitly constructed.