Abstract

This paper grew from a first attempt to understand the relationship of a new con-jecture of Lusztig [L2] for representations of quantized enveloping algebras to his conjecture [Ll] for representations of algebraic groups in characteristic p. The main question here is: Does the new conjecture imply the old? Using results of [APW], it is easy to see this question has an affirmative answer if appropriate irreducible representations of a quantum enveloping algebra at a pth root of unity remain irreducible upon ‘reduction modp’. (Equivalently, the quantum and characteristic p irreducible modules involved, labelled by the same weight, must have the same dimension.) Indeed, Lusztig has con-jectured this irreducibility in an equivalent context [L3], where the weights involved are restricted, and p is sufficiently large. Another standard hypothesis (equivalent for p 2 2 h- 3) requires that p be at least as large as the Coxeter number h, and that the weights in question belong to the Jantzen region, which is contained in the lowest p2-alcove. The main question above has been made even more interesting by an announcement of Kazhdan and Lusztig [KL] that they can prove the new conjecture, assuming results asserted by Casian, at least in the case of a simply-laced root system.

Citations