I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for mathematics and mathematical physics? Is there any property which connects this theory to several areas of math?
More precisely, I want to know about applications of metaplectic quantization on coadjoint orbit .
This question is quite general, I'll write just my own point of view, and hope others add more to get a complete picture.
0) Let me quote A. Kirillov himself:
"In conclusion I want to express the hope that the orbit method becomes for my readers a source of thoughts and inspirations as it has been for me during the last thirty- ve years"
Sorry I cannot find another much more colorful quote from him, where he says something like orbit method not only produced results on many principal questions of representation theory, but gives informal guidance how to invent new and new results.
1) What is the context of orbit method and why it is related to mathematical physics. Orbit method as a particular case of quantization ideology.
I think orbit method should be seen in the context of quantization and roughly speaking its relation to the mathematical physics is that orbit method is particular example of "quantization program". (Well, there some other relations with integrable systems, but they are not so central, imho).
Let me try to explain this in more details. Consider universal enveloping $U(g)$, it is an algebra with generators $e_k$ and relations $[e_{k}, e_{l} ] = c^{i}_{kl} e_l $, let us insert parameter $h$ here: $[e_{k}, e_{l} ] = h c^{i}_{kl} e_l$. For $h=0$ we have just the commutative algebra - denote it $S(g)$, for any non-zero $h$ this algebra is isomorphic to $U(g)$.
Now let us look at $h$ "very small", i.e. we can formally take $h^2=0$, what will see from the structure of non-commutative algebra $U(g)$ reduces to commutative algebra $S(g)$ "plus" a Poisson bracket on it.
So the moral is that Non-commutative algebra, when non-commutativity tends to zero is commutative algebra with Poisson bracket. (It is called classical limit).
The big goal of quantization program
is try to express everything about non-commutative algebra in terms of
Poisson algebra. The reason is that Poisson algebra is something more simple than non-commutative algebra.
In particular we can be interested in description of irreps of non-commutative algebra. What are the corresponding objects for the Poisson algebra ? Answer - symplectic leaves. Observation: the symplectic leaves of $S(g)$ (classical limit of $U(g)$) are exactly the coadjoint orbits.
So this puts the orbit method - in more general framework of quantization. Where one may hope to describe the irreps of quantized algebras via symplectic leaves in Poisson algebra. The problem is that such construction does not always exists neither for $U(g)$, nor for general quantum algebras, at least it does not exist in some simple sense, and big activity is to understand what are the borders between true statements from fakes, it is neverending activity.
It is worth to remark that such point of view on orbit method is not the original one, but emerged later when quantization theory begin to develop.
The basic question of representation theory are calculating characters, induction-restriction, tensor products. The natural question: what are the parallel constructions in the Poisson world (i.e. symplectic geometry) ? how representation theory questions can be answered with the help of symplectic geometry ? There are ideological answers to these questions and again it is neverending game to make "ideology" to theorems or to counter-examples.
Some MO-questions with more details on quantization: Q1, Q2.
2) What is naive quantization without metaplectic correction. (It is better to call it correction (imho), but not quantization, but people use both).
So, our basic wish is to construct a irreducible representation of $U(g)$ (or more generally of some quantized algebra $\hat A$). The "naive recipe" is the following:
1) take symplectic leaf
2) Consider algebra of functions on it and split it into two halves "P-part" and "Q-part
3) The representation space is ----- all functions of "Q"-variable, and representation is constructed as: Q-variables act as multiplication operators, while "P" acts as $\partial_Q$.
Now, what I mean "split in two parts", informally you should think as follows Darboux theorem says that symplectic form is $dp\wedge dq$ is appropriate coordinates, so you have these "p" and "q" as my splitting. The problem is that Darboux is local result, and you need something more complicated to make it work, look at the word "polarization" for more information on that.
3) Towards metaplectic correction.
In the previous item I wrote that naively we should take "half of functions" on the orbit(symplectic leaf) as a Hilbert space.
This actually a point to be corrected.
We should not take "functions", but should take "half-forms".
The simplest motivation is that we want to have a Hilbert space, so we need to have an inner product, but there is no canonical one on the functions. But if we take half-forms: $f(q) \sqrt{dq}$ we have the canonical inner product: $\int fg \sqrt{dq}^2 = \int fg dq $.
So the metaplectic correction is story is how to introduce these "half-forms" into a business in an appropriate way. Some time ago we exercised with quantization of sphere $S^2$ and argued that in general this should be consistent with the Duflo isomorphism.
4) Finally to your main question: "quantization of coadjoint orbits".
Well, sorry, I cannot say much. The general ideology here is that we should take a coadjoint orbit and try to construct irrep. Kirillov done it in 1962 for nilpotent groups, for solvable much progress achieved later. For semisimples - generic orbits - no problems, but for many orbits it is impossible (or at least in some naive sense impossib). I do not know what is the current state of art. It might be there are some particular classes of orbits where some people think that one can indeed construct irrep and it is in reach and good topic for a paper of a PhD. It might be that ideas by Ranee Brylinski Geometric Quantization of Real Minimal Nilpotent Orbits can be somehow developed... But I do not know much about it, my impression that all left open problems are quite difficult and technical, I would not start this as a PhD. Any case I would ask David Vogan or Jeffrey Adams (he is sometimes on MO). By the way have a look at D. Vogan's REVIEW OF “LECTURES ON THE ORBIT METHOD,”.
Any way good luck !
Coadjoint orbits, the symplectic structure on them and polarizations give rise to representations, and you all of them for nilpotent and even solvable groups. For other groups you have to enrich the coadjoint orbit to a bundle over them to get more representations.
The method was carried over to geometric quantization where you want to deform the pointwise multiplication of functions with the Poisson bracket to a noncommutative product with the Poisson bracket as a first order approximation. Ultimately this lead to star product quantization.
Let $\hat{G}$ be the set of irreducible representations of a group $G$. Kirillov's method provides a geometric method for understanding $\hat{G}$ as the orbits of $G$ in the dual of the Lie algebra. It is also a means to introducing non-commutative harmonic analysis to engineering.
Tony Dooley from the university of Bath in the UK gave a talk on it and the Kirillov character formula as part of a seminar at my university so it may be productive to check out his homepage.
