On the quantum KPRV determinants for semisimple and affine Lie algebras
\textit{K. R. Parthasarathy, R. Ranga Rao}, and \textit{V. S. Varadarajan} [Ann. Math. (2) 85, 383--429 (1967; Zbl 0177.18004)] defined the now so-called PRV determinants relative to a standard Borel subalgebra of a finite-dimensional complex simple Lie algebra. Later, \textit{B. Kostant} [Lie groups and their representations (ed. I. Gelfand), Summer School Conference, Budapest, 1971, Halsted/Wiley, New York, 231--331 (1975; Zbl 0327.22010)] introduced similar determinants relative to certain parabolic subalgebras while having in mind applications to principal series representations. The goal of the paper under review is to define and study analogs of KPRV determinants for the quantized enveloping algebra \(U_q(\mathfrak g)\) of a split finite-dimensional semisimple or affine Lie algebra \(\mathfrak g\) over a field of characteristic zero. (Note that it does not seem to be not clear how to define KPRV determinants for the universal enveloping algebra \(U(\mathfrak g)\) of an affine Lie algebra \(\mathfrak g\).) The first section contains an overview of the paper and some of the motivation for introducing quantum KPRV determinants. The second section is a review of the necessary background and notation for quantized enveloping algebras and their representation theory. In the third section the authors consider simple integrable admissible \(U_q(\mathfrak g)\)-modules \(V\) with non-vanishing zero weight space \(V_0\). They define a filtration of \(V_0\) with a corresponding basis which they call Baumann filtration and Baumann basis, respectively. The latter is then used to define the quantum KPRV determinant \(\mathfrak p_{\pi^\prime,V}\) associated to \(V\) and a set \(\pi^\prime\) of simple roots corresponding to a parabolic subalgebra of \(\mathfrak g\). The authors show that \(\mathfrak p_{\pi^\prime,V}\) is non-zero and up to a constant multiple independent of the choice of the Baumann basis. The rest of the paper almost exclusively is concerned with the case of a split finite-dimensional semisimple Lie algebra. In the fourth section the authors compute the degree of the quantum KPRV determinants for simple highest weight modules. The fifth section discusses applications to the study of annihilators of quantized generalized Verma modules. In the last two sections the authors specialize further to the case that \(\mathfrak g\) is simple of type \(A\) and \(V\) is the adjoint module. Compared to KPRV determinants for the universal enveloping algebra \(U(\mathfrak g)\) quantum KPRV determinants have additional factors corresponding to so-called quantum zeros whose origin is not known. Finally, it should be mentioned that due to the work of the first author and \textit{G. Letzter} quantum KPRV determinants relative to a standard Borel subalgebra (i.e., \(\pi^\prime=\emptyset\)) have been computed for split finite-dimensional semisimple Lie algebras [Ann. Sci. École Norm. Supér. (4) 28, 493--526 (1995; Zbl 0838.17011)] and for affine Lie algebras [Math. Res. Lett. 9, 307--322 (2002; Zbl 1019.17003)].
scientific article; zbMATH DE number 1758597