Once again, I’m preparing to teach a class and needing some advice concerning an important point. I’m teaching a course of representation theory as a followup to an excellent course on compact Lie groups, taught this semester by Eckhard Meinrenken. In my class, I would like to explain transition from compact Lie groups to complex reductive groups, as a first step towards the Borel-Weil theorem.
A priori, compact connected Lie groups and complex reductive groups, seem to have little in common and live in different worlds. However, there is a 1-1 correspondence between these objects — for example and are related by this correspondence. Surprisingly, it is not that easy to realize this correspondence.
Let us imagine that we start with a compact connected Lie group and want to find the corresponding complex algebraic group . I will call this process complexification.
One approach to complexification is to first show that is in fact the real points of a real reductive algebraic group. For any particular this is obvious — for example is described by the equation . But one might wonder how to prove this without invoking the classification of compact Lie groups. I believe that one way to do this is to consider the category of smooth finite-dimensional representation of the group and then applying a Tannakian reconstruction to produce an algebraic group. This is a pretty argument, but perhaps not the best one to explain in a first course. A slightly more explicit version would be to simply define to be where ranges over the irreducible complex representations of (the Hopf algebra structure here is slightly subtle).
In fact, not only is every compact Lie group real algebraic, but every smooth map of compact Lie groups is actually algebraic. So the
the category of compact Lie groups embeds into the category of real algebraic groups. For a precise statement along these lines, see this very well written
MO answer by BCnrd.
A different approach to complexification is pursued in
Allen Knutson’s notes and in Sepanski’s book. Here the complexification of is defined to be any such that there is an embedding , such that on Lie algebras . (Actually, this is Knutson’s definition, in Sepanski’s definition we first embed into .) This definition is more hands-on, but it is not very obvious why such is unique, without some structural theorems describing the different groups with Lie algebra .
At the moment, I don’t have any definite opinion on which approach is more mathematically/pedagogically sound. I just wanted to point out something which I have accepted all my mathematical life, but which is still somewhat mysterious to me. Can anyone suggest any more a priori reasons for complexification?