On the Quantum KPRV Determinants for Semisimple and Affine Lie Algebras

Abstract

Let g be a semisimple or affine Lie algebra and U q (g) its quantized enveloping algebra. Extending earlier work, the KPRV determinant for an admissible integrable U q (g) module V relative to a parabolic subalgebra p⊂g is defined and shown to be nonzero. These determinants had previously been evaluated for g semisimple and p a Borel subalgebra. The present results can be used to extend this to g affine as will be shown in a subsequent publication. For a parabolic subalgebra the evaluation of these determinants is much more difficult. For appropriate overalgebras of the primitive quotients of the enveloping algebra U(g) defined by one-dimensional representations of p, these determinants had been calculated for g semisimple. However the quantum case is interesting because it is unnecessary to pass to overalgebras and besides for U(g):g affine, it is not even clear how these determinants should be defined. Here for g semisimple, the degrees of the determinants are computed and shown to depend on being the same type of functions as in the enveloping algebra case; yet in a different fashion. Some special cases (in type A 4) are computed explicity. Here, as in the Borel case, the determinants take a remarkably simple form and notably can be expressed as a product of ‘linear’ factors. However compared to the enveloping algebra case one finds additional factors corresponding to what are called quantum zeros and whose origin remains unknown.