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.
I am finding the study of quantization of co-adjoint orbits very timely since there’s a lot of interest these days on covariant phase methods in order to compute asymptotic charges associated to asymptotic symmetry groups on GR, such as the flat-space/celestial/Carrollian holography, e.g., the Noether charges associated to BMS group. The paper by Witten on Virasoro and coajoint orbits is still very readable, by the way. Since you like BRST cohomology, this paper by Barnich could be of some interest to some of yours students wishing to see a modern application of the methods presented in the class: https://inspirehep.net/literature/567304
Is the following useful for understanding applications of real Lie groups to physics?
KYUNGPOOK Math J. 42 (2002), 199-272 “The Method of Orbits for Real Lie Groups” by Jae-Hyun Yang
https://www.arxiv-vanity.com/papers/math/0602056/
David Brown,
That article has a lot of material I’ll be covering from a somewhat different point of view. By the way, a better link might be
https://arxiv.org/abs/math/0602056
I’ve just started posting notes for the course, in a couple weeks will write something about the orbit method, will also include references, some of which I think are much more helpful than that article.
Soon I’ll be lecturing on the Weil or oscillator representation. In some sense it’s just one example of the general theory of the Jacobi group that the author of that paper writes about, but I think it’s better understood on its own terms as a very special structure.
Anyway, a detailed treatment of the subject should appear in the notes over the next couple weeks.
Class notes?
By the way, the Berkeley Lectures link doesn’t work for me, although I think I was able to find them elsewhere.
Art,
I fixed the link to the Berkeley Lectures. Class notes I’m writing are linked to from the syllabus. Starting to get way behind though on those.
I’ve just finished the first part of the course, now will turn to more conventional material for a while, probably won’t write up notes on this material since there are many very good places to read about it (the Berkeley notes one example).