An introduction to the Lusztig Conjecture

Summary

In these lectures we discuss the main and central problem of the representation theory of reductive algebraic groups in characteristic p > 0 : the problem of determining the formal characters of the irreducible modules. In particular, we discuss the conjecture of Lusztig, which predicts the characters of certain key modules from which the character of an arbitrary irreducible module may be determined, provided that p is greater than or equal to 2h − 3, where h is the Coxeter number of the reductive group. Thanks to work of Kashiwara–Tanisaki, Kazhdan–Lusztig, Lusztig, and Andersen–Jantzen–Soergel, the conjecture is now known to hold for p ≫ 0, in the sense that there is an integer n(Φ) for each root system Φ such that the conjecture holds for all semisimple, simply connected groups with root system Φ defined over an algebraically closed field of characteristic p > n(Φ). However, no explicit bound for the integer n(Φ) is known at the present time (except in a few cases when the rank of Φ is very small). In Lusztig's conjecture, the characters are given as ℤ-linear combinations of Weyl characters, with coefficients described in terms of the Kazhdan–Lusztig polynomials.

We start in the first section with the general framework. In the second section we go through the example of SL₂. Many of the features of this example are present also in the general set-up and to see this it is convenient to use the Chevalley construction of a semisimple group G over an algebraically closed field K of characteristic p > 0, via an admissible ℤ-form of a finite dimensional module for a complex semisimple Lie algebra.