ABSTRACT

SERGE SKRYABIN

Let 𝔤 be a finite dimensional restricted Lie algebra over an algebraically closed field of characteristic p > 0. In the representation theory of 𝔤 one has to deal with the family of reduced enveloping algebras U_{ξ}(𝔤) parametrized by linear functions ξ ∈ 𝔤^{∗}, the p-characters of representations. As the full determination of irreducible 𝔤-modules is a very hard problem, one could ask what are the irreducible modules corresponding to p-characters in a sufficiently general position? There is a nonempty Zariski open subset F ⊂ 𝔤^{∗} (called the fully Azumaya locus in [1]) such that for ξ ∈ F all irreducible U_{ξ}(𝔤)-modules have the maximal possible dimension, and U_{ξ}(𝔤) is isomorphic to a full matrix ring over a commutative ring. If 𝔤 is the Lie algebra of a reductive algebraic group then, under some restrictions, F consists precisely of regular linear functions, and for ξ ∈ F the irreducible U_{ξ}(𝔤)-modules are induced from onedimensional modules of a suitable Borel subalgebra. As such these modules are well understood.

There is another big class of simple Lie algebras, called the Lie algebras of Cartan type, for which not much is known about the generic irreducible representations until now. Even the Kac-Weisfeiler conjecture on the maximum dimension of irreducible modules remains open for those Lie algebras. In this talk I am going to discuss several techniques in a possible approach to the solution of this problem. A description of generic irreducible modules for certain Poisson algebras was obtained recently in [2]. An essential ingredient here are Morita equivalences arising from projective generators introduced by Premet in [3].

References

[1] K.A. Brown, I. Gordon, The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras, Math. Z. 238 (2001) 733-779.

[2] S. Skryabin, Representations of the Poisson algebra in prime characteristic, manuscript.

[3] A. Premet, Special transverse slices and enveloping algebras, to appear in Adv. Math.