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

##### 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)