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Representations of quantum groups at a $$p$$-th root of unity and of semisimple groups in characteristic $$p$$: independence of $$p$$. (English) Zbl 0802.17009
Astérisque. 220. Paris: Société Mathématique de France, 321 p. (1994).
Let $$R$$ be an indecomposable finite root system, and let $$p$$ be an odd prime ($$>3$$ if $$R$$ is of type $$G_ 2$$). Let $$U_ p$$ be the quantized universal enveloping algebra associated to $$R$$ at a $$p$$th root of unity [defined using divided powers, as in G. Lusztig, Introduction to quantum groups, Birkhäuser, Boston (1993; Zbl 0788.17010)]. Also, let $$G_ p$$ be the semisimple, connected, simply-connected algebraic group associated to $$R$$ and defined over an algebraically closed field of characteristic $$p$$. Assume that $$p>h$$, the Coxeter number of $$R$$.
It is known that the simple modules in the block of the trivial one- dimensional $$U_ p$$-module (resp. $$G_ p$$-module) are indexed by certain elements in the affine Weyl group $$W_ a$$ of $$R$$. Let $$L_ w$$ be the simple module indexed by $$w\in W_ a$$, and let $$V_ w$$ be the Weyl module with head $$L_ w$$. The authors’ main result is that there exist integers $$d_{w,n}$$ $$(n\in W_ a)$$, independent of $$p$$, such that $$d_{w,n}$$ is equal to the multiplicity of $$L_ n$$ as a composition factor of $$V_ w$$ for all $$p >h$$ in the $$U_ p$$ case (resp. for all $$p\gg 0$$ in the $$G_ p$$ case).
In 1980, Lusztig made a conjecture on the characters of the representations of $$G_ p$$. Recent work by D. Kazhdan and G. Lusztig [see the papers reviewed above] has shown that a similar conjecture about $$U_ p$$ is equivalent to a conjecture about the affine Lie algebra associated to $$R$$, a proof of which has recently been announced by M. Kashiwara and T. Tanisaki. Combining these results with the main theorem of this paper, one has a proof of Lusztig’s conjecture for $$p\gg 0$$ (it is expected that the conjecture should hold for $$p>h$$).

MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 17-02 Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras 20G40 Linear algebraic groups over finite fields 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
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