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.