Representations of quantum groups at roots of 1. (English) Zbl 0738.17008

De Concini, Corrado; Kac, Victor G.

Operator algebras, unitary representations, enveloping algebras, and invariant theory, Proc. Colloq. in Honour of J. Dixmier, Paris/Fr. 1989, Prog. Math. 92, 471-506 (1990).

[For the entire collection see Zbl 0719.00018.]
Let \(U_ q\) denote the quantized enveloping algebra associated to a complex simple finite dimensional Lie algebra \({\mathfrak g}\). The quantum parameter \(q\) is in this paper a complex number. When \(q\) is generic, i.e. not a root of unity, then the theory of finite dimensional representations of \(U_ q\) resembles the classical theory of finite dimensional representations of \({\mathfrak g}\) (see G. Lusztig [Adv. Math. 70, 237–249 (1989; Zbl 0651.17007)]). The present authors give a formula for the determinant of the contravariant form on the Verma modules for \(U_ q\) (using the classical Shapovalov formula for \(q=1)\). Then they study the center \(Z_ q\) of \(U_ q\). When \(q\) is generic, they obtain a Harish-Chandra-type theorem (compare also the Joseph-Letzter preprint, “Local finiteness of the adjoint action for quantized enveloping algebras”, published in [J. Algebra 153, No. 2, 289–318 (1992; Zbl 0779.17012)]) whereas the situation is quite different when \(q\) is a root of unity. In the latter case they exhibit a nice large subalgebra \(Z_ 0\) of \(Z_ q\), and in the spirit of V. Kac and B. Weisfeiler [Indagationes Math. 38, 136–151 (1976; Zbl 0324.17001)] they study the various fibers of the map \(\text{Rep}(U_ q)\to\text{Spec} Z_ q\to\text{Spec} Z_ 0\). They raise several natural questions concerning the associated “quantized coadjoint orbits” (most of which they have recently answered in their subsequent joint work with C. Procesi [J. Am. Math. Soc. 5, No. 1, 151–189 (1992; Zbl 0747.17018)]).

Reviewer: Henning Haahr Andersen (Aarhus)

MSC:

17B37	Quantum groups (quantized enveloping algebras) and related deformations
16T20	Ring-theoretic aspects of quantum groups
20G05	Representation theory for linear algebraic groups
17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

Keywords:

quantum groups; coadjoint orbit; quantized enveloping algebra; Shapovalov formula