The strong linkage principle for quantum groups at roots of 1. (Q1868742)

The strong linkage principle - see [\textit{H. H. Andersen}, J. Reine Angew. Math. 315, 53--59 (1980; Zbl 0439.20026)] gives a condition on the highest weights of composition factors of the cohomology of line bundles on the flag variety \(G/B\), if \(G\) is a semisimple algebraic group over a field of positive characteristic and \(B\) is a Borel subgroup. The analogous result for quantum groups at a parameter \(q\) of finite order was proved by \textit{H. H. Andersen} and \textit{J. Paradowski} [Commun. Math. Phys. 169, 563--588 (1995; Zbl 0827.17010)], \textit{H. H. Andersen}, \textit{P. Polo} and \textit{Wen Kexin} [Invent. Math. 104, 1--59 (1991; Zbl 0724.17012)], \textit{H. H. Andersen} and \textit{Wen Kexin} [J. Reine Angew. Math. 427, 35--50 (1992; Zbl 0771.17010)], and \textit{L. Thams} [J. Reine Angew. Math. 436, 129--153 (1993; Zbl 0760.17015)], under suitable varying hypotheses. In this paper, a unified proof -- for arbitrary characteristic and arbitrary order of \(q\) -- is presented. Besides a splitting of the category of integrable modules, some other consequences are stated; particularly, the Steinberg module is discussed in this context.