Let G be a connected reductive algebraic group defined over an algebraically closed field of positive characteristic p and suppose that the derived subgroup of G is simply connected, p is a good prime for the root system of G and the Lie algebra 𝔤=Lie(G) admits a non-degenerate Ad G-invariant symmetric bilinear form. If G is a simple algebraic group of type other than A, the above assumptions mean that p is a good prime for G, i.e. p ≥ 3 if G is of type B, C or D, p ≥ 5 if G is of type G2, F4, E6 or E7, and p ≥ 7 if G is of type E8. If all components of G have type A, B, C, D we set R=ℤ[1/2]. If G has a component of exceptional type but has no components of type E8 we set R=ℤ[1/6]. If G has a component of type E8 we set R=ℤ[1/30]. Given a linear function χ on 𝔤 we denote by Uχ(𝔤) the reduced enveloping algebra of 𝔤 associated with χ. By the Kac-Weisfeiler conjecture (now a theorem), any Uχ(𝔤)-module has dimension divisible by pd(χ) where 2d(χ) is the dimension of the coadjoint G-orbit of χ. In my talk, based on a joint work with Lewis Topley, I’ll discuss a natural question raised in the 1990s by Kac, Humphreys and myself and explain that for any χ ∈ 𝔤* the reduced enveloping algebra Uχ(𝔤) has an irreducible module of dimension pd(χ). Forms of finite W-algebras over the ring R and their reductions modulo good primes play a crucial role in our arguments. We also use some recent results on multiplicty-free primitive ideals of U(𝔤𝒞) associated with the rigid nilpotent orbits in complex simple Lie algebras 𝔤𝒞.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
