David Vogan was visiting Columbia last week, giving the Ritt Lectures, on the topic of Geometry and Representations of Reductive Groups. He has made available the slides from his lectures here.

Vogan’s talks concentrated on describing the so-called “orbit method” or “orbit philosophy”, which posits a bijection for Lie groups G between

and

^{*}

This is described as a “method” or “philosophy” rather than a theorem because it doesn’t always work, and remains poorly understood in some cases, while at the same time having shown itself to be a powerful source of inspiration in representation theory.

It is probably best understood as an expression of the deep relationship between quantum mechanics and representation theory, and the surprising power of the notion of “quantization” of a classical mechanical system. In the Hamiltonian formalism, a classical mechanical system with symmetry group G corresponds to what a mathematician would call a symplectic manifold with an action of the group G preserving the symplectic structure. “Geometric quantization” is supposed to associate in some natural way a quantum mechanical system with symmetry group G to this symplectic manifold with G-action, with the Hilbert space of the quantum system providing a unitary representation of the group G. The representation is expected to be irreducible just when the group G acts transitively on the symplectic manifold. One can show that symplectic manifolds with transitive G action correspond to orbits of G on (Lie G)^{*}, the dual space to the Lie algebra of G, with G acting by the dual of the adjoint action. So it is these “co-adjoint orbits” that provide geometrical versions of classical mechanical systems with G symmetry, and the orbit philosophy says that we should be able to quantize them to get irreducible unitary representations, and any irreducible unitary representation should come from this construction.

That such a “quantization” exists is perhaps surprising. To a quantum system one expects to be able to associate a classical system by taking Planck’s constant to zero, but there is no good reason to expect that there should be a natural way of “quantizing” a classical system and getting a unique quantum system. Remarkably, we are able to do this for many classes of symplectic manifolds. For nilpotent groups like the Heisenberg group, that the orbit method works is a theorem, and this can be extended to solvable groups. What remains to be understood is what happens for reductive groups.

Already for the simplest case here, compact Lie groups, the situation is very interesting. Here co-adjoint orbits are things like flag manifolds, and the Borel-Weil-Bott theorem says that if an integrality condition is satisfied one gets the expected irreducible representations, sometimes in higher cohomology spaces. One can take “geometric quantization” here to be essentially “integration in K-theory”, realizing representations using solutions to the Dirac equation. Recently Freed-Hopkins-Teleman gave a beautiful construction that gives the inverse map, associating an orbit to a given representation.

For non-compact real forms of complex reductive groups, like SL(2,**R**), the situation is much trickier, with the unitary representations infinite dimensional. Vogan’s lectures were designed to lead up to and explain the still poorly understood problem of how to associate representations to nilpotent orbits of such groups. At the end of his slides, he gives two references one can consult to find out more about this.

Finally, there is a good graduate level textbook about the orbit method, Kirillov’s Lectures on the Orbit Method. For more about the orbit method philosophy, its history and current state, a good source to consult is Vogan’s review of this book in the Bulletin of the AMS.

This orbit method of Kirillov-Kostant-Souriau is a very interesting tool

discovered by mathematicians in the 70’s. Should be read in conjunction with the work of Souriau (starting in the 60’s).

It is a typical examble of being shallow that only one author (Kirillov) of this method is mentioned, at the end of the message, moreover as

an author of a graduate level introduction into the method.

What about Souriau? Kostant? geometric quantization? second

geometric quantization?

See this, for example:

http://books.google.com/books?id=PuhpI98ZkDgC&dq=%22geometric+quantization%22+Souriau&lr=

Marius,

I made no attempt in this post to discuss the history of any of this, which is why only Kirillov appeared, and only in a reference to his expository text. Geometric quantization of course has a long history. The book you mention by Woodhouse doesn’t discuss at all the orbit method or the connection to representation theory.

Peter,

Maybe this helps (Kirillov, A. A.

Geometric quantization [MR0842909 (87k:58104)]. Dynamical

systems, IV, 139–176, Encyclopaedia Math. Sci., 4, Springer,

Berlin, 2001)

http://books.google.com/books?hl=en&lr=&id=CCGCFCj-QNsC&oi=fnd&pg=PA139&dq=%22orbit+method%22+Woodhouse+Kirillov&ots=o4YtWHGqQ1&sig=SUE7nUR3c_1U74GJZr4DGPm-MbI

especially section 1.3 “The statement of the quantization problem. The

connection with the method of orbits in representation theory”.

Indeed, maybe Woodhouse is not the best reference in order to quickly understand that all is about the fact that the “orbit method”,

“geometric quantization”, and some topics in reprezentation

theory are in fact one.

This has been discovered by Souriau, in fact. In his book (in french, then translated in english) he defines (elementary) particles as

(some) coadjoint orbits of the Poincare group, after introducing what

is now known as (first) geometric quantization. Each such orbit has

some invariants associated with (and identifying it), like the charge,

and so on. As far as I understand, Souriau ideea is just this: take all

hamiltonian systems with symmetry group G, then you can classify

(or decompose?) the dynamics into a family of dynamics of “particles”,

each “particle” being a canonical dynamical system on a coadjoint

orbit of G. The “canonical” word is related to the “orbit method”, which

relates coadjoint orbits with “simplest” representations of G.

I think Peter gave a very nice overview of a topic that’s much too vast to really fit into a blog entry. One comment:

This is true, but it makes the situation sound a bit more “scary” or “bad” than it is. It’s actually as beautiful as you could hope. Every finite-dimensional irrep of a compact simple group comes from geometrically quantizing an integral coadjoint orbit. And every integral coadjoint orbit gives you a finite-dimensional irrep of the compact simple group. There’s a 1-1 correspondence!

And, if you happen to know the other famous way to get your paws on these finite-dimensional irreps – namely, from integral weights in the Weyl chamber – you’ll be pleased to know that an “integral weight in the Weyl chamber” is just what you get when intersect an integral coadjoint orbit with the Cartan, and look at the point that lies in the Weyl chamber.

Well, I’m sure that sounds scary too to most people. But it’s really great.

On the other hand, I’m still terrified of the noncompact case, which is where Vogan comes into his own.

Thanks John,

Didn’t mean to make the compact case sound scary (yes, the non-compact reductive one is…). It’s an incredibly beautiful story, one that needs more like a few weeks of a graduate course than a blog posting to do justice to it.

There is one subtlety that you don’t mention which fascinates me: there are two possible choices of which orbit to associate to which representation (the “rho-shift”). Does the trivial representation get associated to the trivial orbit, or to the orbit of half the sum of the positive roots? Opinions differ. And half the sum of the positive roots is the highest weight of the spinor representation, for spinors on the flag manifold. The Dirac equation is everywhere…

Peter writes:

You too, eh?

This happens to be something James Dolan has been obsessed with over the last month… and my guess as to the solution was confirmed in Dieudonne’s book on Universal Enveloping Algebras. I don’t want to give it all away, since I plan to write about this in This Week’s Finds someday, but there are two different ways to get a representation of a complex simple Lie algebra G starting from a representation of its Borel B: induction, and coinduction. These are dual in some sense, as the names hint – but there’s a kind of compromise halfway in between, which involves taking the induced representation and tensoring with a special representation whose highest weight is “half the sum of the positive roots”.

But here’s what finally convinced me this stuff make sense. The line bundle on the flag manifold (G/B) whose sections give this special representation is just the bundle of

half-forms!Strictly speaking I should say “a” bundle of half-forms, not “the”. Given an oriented n-manifold, a bundle of half-forms is any line bundle whose square is the bundle of n-forms. An n-form is something you can integrate over your manifold, so a half-form is something whose

squareyou can integrate. This is why “tensoring with a bundle of half-forms” is important in geometric quantization: it gives you a bundle whose sections naturally form a Hilbert space!There are typically lots of square roots of the bundle of n-forms. But in the case of a flag manifold G/B there’s a god-given choice: the bundle induced from a certain special 1-dimensional representation of B, namely the one corresponding to “half the sum of positive roots”!

So, it’s all starting to make sense. It’s sad how hard it’s been to squeeze this explanation out of the literature. Lots of books on Lie algebras talk about “half the sum of positive roots”. But few seem to give the geometrical explanation in terms of geometric quantization. I guess the authors are trying to stay pure, and hide the geometry behind a veil of algebra.

By the way, my remark about something “seeming more scary than it really is” concerned the phrase “sometimes in higher cohomology spaces” in your description of the Bott-Borel-Weil theorem.

When you have a

positiveweight, it gives you a line bundle on the flag manifold G/B whose space of holomorphic sections form an irrep of G. Only when you want to terrify people who haven’t fallen in love with sheaves should you call this space of holomorphic sections the “0th cohomology” of the line bundle – or really, its corresponding sheaf of holomorphic sections.The scary jargon becomes more forgivable when we consider

nonpositiveweights. These again give line bundles on G/B, which again give irreps – but only when we take somehighercohomology group.However, even in this case, the jargon is a bit more scary than necessary. It came as a great relief when I discovered that a “higher cohomology group” of a holomorphic line bundle was just a fancy way of talking about the space of holomorphic sections of this bundle

tensored with another bundle– I guess a bundle of p-forms.Another digression:

The fact that I’m talking about half-forms when you’re talking about spinors suggests that I, at least, haven’t gotten to the very bottom of this “half the sum of positive roots” business. Both half-forms and spinors are “square roots” of something – but different somethings, and in different ways! And, they both show up as correction fudge factors in geometric quantization, but I don’t quite understand how they’re related.

I bet Eckhard Meinrenken does.

John,

I’m looking forward to seeing your version of why the so-called “rho-shift”.

One indication that spinors are the right way to think about this comes from the following: Borel-Weil and Borel-Weil-Bott both crucially depend on the choice of invariant complex structure, which is the same as the choice of positive roots. Acting by the Weyl group changes the complex structure, makes dominant weights no longer dominant, and moves the representation from holomorphic sections to higher cohomology (or, OK, higher order differential forms). If you use the Dirac operator and spinors instead, none of this happens, the whole set up is completely Weyl-invariant.

..at least as far as symplectic spinors are concerned the latter is possibly reflected by the fact that any compatible complex structure on a symplectic manifold (i.e. on a coadjoint orbit) gives rise to a reduction of the metaplectic structure of the manifold to a two-fold-covering of the unitary group, contained in the centralisator of the latter is a one-parameter-subgroup induced by the element in the Lie-algebra whose image under the exponential map projects to the canonical complex structure of R^{2n}. This subgroup factors to a familiy of representations of the circle over the eigenspaces of the harmonic oscillator, the corresponding splitting of the spinor bundle reveals as the ‘lowest eigenmode’ of the harmonic oscillator a line-subbundle, this is then a ‘canonical’ choice for a half-form on M (depending on the complex structure). Associated to a Lagrangian polarization is a O(n)-reduction of the spinor-bundle resp. of the above line-subbundle, this is the ‘half-form’ widely used in geometric quantization, as it seems.

Hi Peter,

Good pedagogical explanations of the orbit method seem to be rare.

I looked at the brief section in your course notes (the section at

the end on the momentum map and the orbit method), but it was

a bit too brief for me to get a thorough understanding. I’ve tried

Kirillov’s course notes, but they require too much from (this) reader.

Is there any chance you could expand your course notes? E.g.,

talk in more detail about the H1 and SU(2) examples you mention,

and perhaps some others? There’s not quite enough detail on the

H1 case to follow it properly, and the SU(2) case is only couple

of paragraphs?

strangerep,

I’ll be teaching that course this semester again, maybe I’ll get a chance to expand the notes, we’ll see.

I probably also discuss the SU(2) case when I discuss Borel-Weil theory, it’s really the same thing, which is one reason for not going on much about the orbit method in the compact group case. In the case of SU(2) orbits are just spheres, weights correspond to line bundles, for weights of the right sign quantization is just taking holomorphic sections of the line bundle. Quite concretely, such sections correspond to homogeneous polynomials of degree given by the weight.

The Heisenberg group case probably does deserve a lot more detail….

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Hi Peter and John,

I’ve been trying to understand the basics, namely this geometric way of thinking about representations of compact Lie groups for a while now; various entries in this blog have helped me in many ways -thanks! However, can you point me to an explicit “equivalence of categories” statement?

For instance, my favourite way of conceptualizing it is the way of Guillemin and Sternberg in their book “Symplectic techniques in physics” (see the section on Kaehler manifolds). This is the approach which doesn’t bother with coadjoint orbits in the dual of the lie algebra, but rather works directly with the representations themselves. The correspondence is roughly:

representations of a compact Lie group G complex G-equivariant line bundles having a finite number of orbitsNote I didn’t say “irreducible representations”; I’m looking for an equivalence of categories here. A morphism between equivariant line bundles category on the right hand side is an

equivariant kernel, i.e an equivariant collection of maps f_{x,y} : L_x –> L’_x.From left to right, given a representation V of a compact Lie group G, one simply takes the G-orbits in the projective space of the representation which are complex submanifolds. (This is a slightly more intrinsic procedure than finding these orbits in the dual of the lie algebra). From right to left, one just takes holomorphic sections.

Does this make sense, and do you know of a precise statemen to this effect, so as to obtain an equivalence of categories?

I agree with Peter in that I am also fascinated by the spinor side of the story. Basically, my understanding (from the book “Heat kernels and Dirac operators” by Berline,Getzler and Vergne) is that under this geometric correspondence, the character of the representation computes in terms of the equivariant index of the Dirac operator living over these G-sets; in fact it localizes over the fixed points.

Bruce,

In your way of associating a geometrical object to a representation, doesn’t the orbit depend on which vector you pick?

I’m fond of the K-theoretic “quantization=integration” idea, but to turn this into a statement about categories (of equivariant vector bundles on one side, representations on the other) you need to “categorify” it, and I haven’t thought about that. The people who seem to have precise equivalence of categories statements of this kind are those thinking in terms of D-modules. I think the Beilinson-Bernstein localization theorem is the kind of precise statement about equivalence of categories that you want. In the compact Lie group case, maybe it can be restated in various other ways.

Thanks for the reply! Peter wrote:

No… that’s what I learnt from Guillemin and Sternberg. I guess you are saying to me that it is the orbits of the maximal weight vector which are the orbits we need to pick out, and somehow this “maximal weight vector” requires a choice. But the more intrinsic way of looking at it is that, at least for irreducible representations (heh, I am out on a limb for reducible reps), there is only

oneorbit which is acomplexsubmanifold of the projective space P(V).In other words, from an intrinsic point of view, weight vectors don’t come into it at all. One simply needs to “look for the complex orbits, Luke” (insert Obi-Wan voice here).

Well, indeed I

amin the “categorificaion” game; I work with “2-representations” and all that jazz. But here we’re working with ordinary representations…. surely I’m not asking for a “categorification” of anything, I’m just asking for a nice elegant statement of the geometric correspondence between ordinary representations of Lie groups and “equivariant complex line bundles”… if there is such a statement, one would expect it to be phrased as an equivalence of categories.Gulp! Sadly all that stuff is over my head. I’ll crawl back under my bridge now.

..if we are already that far: one should possibly not forget that the actual goal was to quantize classical

observables, an observable in classical mechanics is nothing more than a function (for instance a ‘symbol’ on a coadjoint orbit), to these functions one aimed to associate more or less pseudodifferential operators on appropriate vectorbundles such that this correspondence is equivariant w.r.t. appropriate ‘products’ on both sides. I only want to point out that one advantage of the spinor view is that it makes it possible to associate to a certain class of functions (again, ‘symbols’) on a cotangent bundlesectionsin a (symplectic) spinor bundle, to do this, one needs a ‘half-form’, which is a subbundle of the symplectic spinor bundle, to compensate for an ambigous choice of (local) ‘signs’. On the other hand one can localize functions on the zero-section of the cotangent-bundle and encode them as well as spinors, this is analogous to the ‘micolocal lift’ in deformation quantization. Using a Fouriertransform for spinors one can pair these two sections by the pointwise (spinor)-L^2-product over the manifold and gets the action of a pseudodifferential operator on smooth functions on the zero-section of the cotangent bundle which is exactly that one used in deformation quantization to define the *-product on symbols.I never saw this spinor-quantization construction anywhere else than in my not-quite-recent diploma thesis, which is (funny enough) in german, but one should recall that a correspondence between orbits and sections of vectorbundles is only the very first step in quantization and unless somebody enlightens me and one is not conerned about varieties and algebraic groups, I do not see what <i<exactly the ‘categorification’-theme aims at. The (symplectic) spinor-business seems for no clear reason to be a bit mystified and judged to be oversubtle or impenetrable, nevertheless it arises very naturally in the symplectic context, that is, in the context of quantization.

Bruce,

I was just using the word “categorification” to show off, since one of the few things I know about it is that people use “decategorification” to mean taking K-theory….

Thinking more about it, what I was thinking about clearly can’t work, but in more detail, the idea was that the index map

K_G(G/T) to K_G(pt.)=R(G) relates the K-theory of the category of G-equivariant vector bundles on the flag variety to the K-theory of the category of representations of G.

But, even at the level of K-theory, this isn’t an isomorphism, since

K_G(G/T)=R(T) and R(G)=R(T)^W. The map is “Dirac induction”, maybe there’s some way of thinking about this though that does give an isomorphism.

As for D-modules, in this case Beilinson-Bernstein is just another way of restating Borel-Weil-Bott, but in a way that I guess gives an equivalence of categories. You need to just see what it is saying in this special case. I think the new book of Hotta et. al. explains this, also some expository papers of Schmid. Might or might not be in some lecture notes of David Ben-Zvi, who is someone who certainly understands this stuff.

Thanks, you’ve given me lots of leads to follow up here.