Representations of \(sl_ q(3)\) at the roots of unity (Q1357128)

In [Prog. Math. 92, 471-506 (1990; Zbl 0738.17008)] \textit{C. De Concini} and \textit{V. G. Kac} started to study the representation theory of quantized universal enveloping algebras \(U_q ({\mathfrak g})\) of a finite-dimensional complex simple Lie algebra \({\mathfrak g}\) at an odd primitive \(\ell\)-th root of unity \(q= \varepsilon\). By analogy with the representation theory of the corresponding Lie algebra over a field of prime characteristic, they exhibit a large central subalgebra of \(U_\varepsilon ({\mathfrak g})\) which implies that every irreducible representation of \(U_\varepsilon ({\mathfrak g})\) is finite-dimensional, and furthermore, they show that the maximal dimension of such a representation is \(\ell^N\), where \(N\) is the number of positive roots of \({\mathfrak g}\). \textit{C. De Concini}, \textit{V. G. Kac} and \textit{C. Procesi} proved in [J. Am. Math. Soc. 5, 151-189 (1992; Zbl 0747.17018)] that the isomorphism classes of the irreducible \(U_\varepsilon ({\mathfrak g})\)-modules are parametrized by the conjugacy classes of the connected complex Lie group \(G\) with Lie algebra \({\mathfrak g}\) and trivial center. Similar to the Kac-Weisfeiler conjecture for the corresponding Lie algebra over an algebraically closed field of prime characteristic (which recently was proved by \textit{A. A. Premet} [Invent. Math. 121, 79-117 (1995; Zbl 0828.17008)]) they conjectured that the dimension of a simple module corresponding to the conjugacy class \({\mathcal O}\) is divisible by \(\ell^{\dim {\mathcal O}/2}\), and they verified this for the regular conjugacy class in [Commun. Math. Phys. 157, 405-427 (1993; Zbl 0795.17006)]. The main result of the paper under review is an explicit description of every irreducible \(U_\varepsilon (sl(3))\)-module corresponding to a subregular unipotent conjugacy class as being induced from a restricted irreducible \(U_\varepsilon (sl(2))\)-module extended to a suitable subalgebra of \(U_\varepsilon (sl(3))\) which by the results cited above completes the verification of the De Concini-Kac-Procesi conjecture for \(U_\varepsilon (sl(3))\).