I’ve been at a couple of interesting conferences lately and so I have a lot to talk about. I’ll start by summarizing an excellent expository talk by Jonathan Brundan which he gave at an MSRI introductory workshop last week.
Let G be a reductive group over an algebraically closed field k of characteristic p. The topic of this post is the algebraic representations of G. In other works, we want to study algebraic maps where V is a finite dimensional vector space over k. Over the years, a few people (Soroosh, Carl, Alex Ghitza) have asked me what I knew about this theory and I’m afraid that I always gave them very incomplete or inaccurate answers. Now, that I’ve been to Brundan’s talk I think that I understand what is going on much better and I’d like to summarize it. Of course there will be nothing “new” in this post — I think that all the theory was worked out 20 years ago.
First, let us consider a construction of the group G. We start with our usual reductive Lie algebra over the complex numbers. Consider its universal envelopping algebra . It has a Kostant- form, , which is generated by etc. Then tensor with k to get and dualize to get the Hopf algebra of functions on G.
Now we can build Verma modules (ie modules over ) for (the weight lattice), just as we do over . By general nonsense, has a unique irreducible quotient .
is finite-dimensional if and only if is dominant.
The proof of this is quite interesting. The hard part is to show that is finite dimensional. It suffices to find at least one finite-dimensional module with highest weight . One way to do this is to start with the usual irrep over the complex numbers. Then we have a form by acting on the highest weight vector with . Then we tensor with k to get which is a finite dimensional highest weight module, called the Weyl module.
So for any dominant we have two representations of G, and with highest weight . In fact is universal among such representations, since we have the following Borel-Weil theorem.
Here as over , the higher cohomology of these line bundles vanishes. Moreover, the character of is given by the Weyl character formula. The dimension of the weight spaces of will be smaller than that of .
Sometimes and coincide. For example, . These are called the rth Steinberg modules.
I don’t know if the characters of are known in general (is there an expert out there who can answer this question?), but there is the following remarkable theorem which reduces their study to that of a finite number of characters.
Since our group is defined over (it is defined over ), we have a Frobenius map which is a group homomorphism. We can use this Frobenius map to twist representations. It has the result of multiplying all the weights by p, and leaving representations irreducible, so we have . This has a remarkable generalization.
Let and suppose . Then .
In particular, note that we can make such a decomposition of with all living in the region defined by the inequalities . So to “know” all the irreducible representations, it is enough just to know those ones for in this region.
What is remarkable about this theorem is that its proof involves considering the kernel of the Frobenius — a highly non-reductive group scheme which has just one k point.
Aside from thinking about these irreducibles, there is much more going on. The category is not semisimple, so it is not enough to understand just the irreducible objects. In fact there is a well-developed block theory which is related to the action of affine Weyl group on generated by reflections in the fundamental alcove.