Abstract:

Let g = Lie(G) be the Lie algebra of an algebraic group over an algebraically closed field 
of characteristic p > 0. It is known that the simple g-module has a "p-character" and that 
the set of such characters can be naturally identified with the dual space g* of 
the Lie algebra. Thus we find ourselves in the context of the orbit method, and we might 
expect the structure of g-modules to depend on the structure of coadjoint G-orbits. 
Kac and Weisfeiler made two influential conjectures in this spirit: they predicted that 
the dimensions of simple g-modules depend upon coadjoint orbits in a very precise sense. 
I will start by giving an overview of these conjectures and their partial resolutions.

The second conjecture was only stated for G reductive; nonetheless in a joint work with 
Ben Martin and David Stewart we have developed methods prove both conjectures 
in large characteristics for every Lie algebra attached to Z-group scheme. 
To be more precise, we show that if G is a Z-group scheme then there is a number n > 0 
such that for all primes p > n and all algebraically closed fields of characteristic p > n 
both conjectures hold for Lie(G(k)). Our methods combine deformation quantisation and 
model theory. In the second half of the talk I will explain the novel aspects of our proof.