This semester I’m teaching the second half of our graduate course on Lie groups and representations, and have started making plans for the course, which will begin next week. There’s a web-page here which I’ll be adding to as time goes on. The plan is to try and write up lecture notes for most of the course, some of which may be rewrites of older notes written when I taught earlier versions of this course. I’ll post these typically after the corresponding lectures. Any corrections or comments will be welcome.
This year I’m hoping to integrate ideas about “quantization” into the course more than in the past, starting off with the mathematics behind what physicists often call “canonical quantization”. This topic is worked out very explicitly and in great detail in this book, but in this course I’ll be giving a more stream-lined presentation from a more advanced point of view. This subject has a somewhat different flavor than usual for math graduate courses, in that instead of proving things about classes of representations, it’s one very specific representation that is the topic.
This topic is also the simplest example of the general philosophy of trying to understand Lie group representations in terms of the geometry of a “co-adjoint orbit”, and I’ll try and say a bit about this “orbit philosophy” and “geometric quantization”.
The next topic of the course will likely be more standard: the classification of finite dimensional representations of semi-simple complex Lie algebras (or, equivalently, compact Lie groups), and their construction using Verma modules. For this topic it’s hard to justify spending a lot of time writing notes, since there already are several places this has been done very well (e.g Kirillov’s book). After doing this algebraically, I’ll go back to the geometric and orbit point of view and explain the Borel-Weil-Bott theorem giving a geometric construction of these representations.
For the last part of the course, I hope to discuss the representations of SL(2,R) and the general classification of real semi-simple Lie algebras and groups. If I ever manage to understand what’s going on with the real Weil group and the Langlands classification of representations of real Lie groups, maybe I’ll say something about that, but that is probably too much to hope for.
Throughout the course, as well as the relation to quantization, I also hope to explain some things about relations to number theory. These would include the theory of theta functions early on, modular forms at the end.