ABSTRACT

The State of KPRV Determinants.

Antony Joseph

Let U(g) be the enveloping algebra of a semisimple Lie algebra g. A classical result of Kostant asserts that U(g) is a free module over its centre Z(g). Moreover one can choose the free basis to span a g module in which the multiplicity of a given simple module V is just the dimension d of its zero weight space. Through the triangular decomposition of U(g) one obtains a d by d determinant depending on V with coefficents in the symmetric algebra S(h) of a Cartan subalgebra h. These determinants were evaluated by Parthasarathy, Ranga-Rao and Varadarajan in 1968. They are called KPRV determinants.

More generally KPRV determinants are defined whenever one can similarly reunite the three ingredients, freeness, the multiplicity equality and triangular decomposition. Thus these determinants have been evaluated in the super reductive case and for quantized enveloping algebras. It is also possible to define and calculate them with respect to parabolic subalgebras and for affine Lie algebras.

Several open problems remain. The first is to evalute the KPRV determinants in the quantum parabolic case, in particular to interpret their zeros. Secondly for g affine it is as yet unknown if every simple integrable module with a zero weight space is admissible and this both in the classical and quantum theories. Thirdly for g affine one should like to show that the quantum trace of a weight zero vector in such a module converges in the Krull topology to an element of k((q)).