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I don't have much of any awareness about the representation theory of non-compact Lie groups but I bumped into it for my work.

Is there some idea that the representations of a non-compact group are labeled by those of its maximal compact subgroup? If yes then I would like to know of explanations for the above and of references from where I can pick this up.

  • The conformal group of $3+1$ dimensional space-time is $SO(4,2)$ and apparently any representation of it can be written as a direct sum over representations of $SO(4) \times SO(2)$.

  • The N=2 superconformal group for $2+1$ dimensions is $SO(3,2)\times SO(2)$ and its maximal compact subgroup $SO(2) \times SO(3) \times SO(2)$ ? If yes then I would like to know how can this be proven. (In physics contexts these two $SO(2)$ factors are distinguished by physical generators of different meanings.)

I would like to know of the general framework in which the above fits in.

To quote from a paper a typical argument where such a thing seems to get used,

"Any irreducible representation of the superconformal algebra may be decomposed into a finite number of distinct irreducible representations of the conformal algebra...which are in turn labeled by their own primary states...hence the state content of an irreducible representation of the superconformal algebra is completely specified by the quantum numbers of its conformal primaries"

I would be very glad if someone can also give expository references or explanations specific to the above argument.

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Your title (and a sentence in the body) is quite difficult to parse, as it is not grammatically correct. A non compact group has usually only one maximal compact subgroup (up to conjugation) How can it label the many, many representations of the non-compact group? –  Mariano Suárez-Alvarez Feb 15 '11 at 19:00
@Mariano I have added some further clarifications into the statement of the question. I guess I am now communicating what I want to know. –  Anirbit Feb 16 '11 at 13:00
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1 Answer 1

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Suppose that we have (to fix ideas) a unitary irreducible representation of a semi-simple Lie group $G$ (such as the non-compact Lie groups that you write down) on a Hilbert space. (Here I mean irreducible in the Hilbert space sense, i.e. there are no proper invariant closed subspaces.) Let's call the Hilbert space $V$ (just to give it a name).

A theorem of Harish-Chandra then says that $V$ is admissible, which means the following: fix a maximal compact subgroup $K$ of $G$. Then each irreducible representation $W$ of $K$ appears with finite multiplicity as a subrepresentation of $V$. If we call this multiplicity $m_W$, then we may write $V = \hat{\oplus}_W W^{m_W}$, i.e. as the Hilbert space direct sum (i.e. the completed direct sum) of the various $W$, each appearing with multiplicity $m_W$. (This is a consequence of the Peter--Weyl theorem, and is true for any unitary representation of a compact group in which each irrep. appears with finite multiplicity.)

Now inside $\hat{\oplus} W^{m_W}$ we have the actual algebraic direct sum $\oplus_W W^{m_W}$, and this has an intrinsic characterization as a subspace of $V$, as the $K$-finite vectors. (A vector $v$ is called $K$-finite if the linear span of all its translates by elements of $K$ is finite-dimensional.) Let's denote it by $V_K \subset V$.

It turns out that $V_K$, although it is not invariant under the action of $G$ (typically, unless $V$ happens to be finite-dimensional, which it usually won't be), is invariant under $\mathfrak g$, the Lie algebra of $G$.

One calls $V_K$ a $(\mathfrak{g},K)$-module, or also a Harish-Chandra module (because it has actions of $\mathfrak{g}$ and $K$). It turns out that $V_K$ determines $V$, and the basis of Harish-Chandra's approach to the study of unitary reps. of $G$ is to work instead with the underlying $(\mathfrak g,K)$-modules.

Now in principle, to recover $V$, one really needs $V_K$ as a $(\mathfrak g, K)$-module; i.e. forgetting $\mathfrak g$ and just remembering the $K$-action is throwing away a lot of information.

But in practice (at least in the examples that I know) non-isomorphic irreducible $V$ have different list of multiplicities $m_W$, and so just knowing $V_K$ as a $K$-rep. may well already pin down $V$.

In fact, often one doesn't even have to know all the $m_W$, but just the first non-zero vanishing value. (If I think of the reps. $W$ as being labelled by their highest weights lying in some choice of dominant Weyl chamber for $K$.)

A good place to read about this (it is not short, but I found it very good for dipping into) is Knapp's book Representation theory of semisimple groups: an overview based on examples. He gives the basic definitions, a lot of examples, and goes on to develop various aspects of the theory (e.g. the theory of the relationship between $V$ and the multiplicities $m_W$: this is known as the theory of $K$-types).

Incidentally, Harish-Chandra was a student of Dirac, and (as far as I know) his study of unitary reps. of semisimple groups was inspired by Bargmann's treatment of the special case of $SL_2(\mathbb R)$, which was in turn inspired in part by the role of this group in physics.

On the other hand, I don't know of a treatment of the theory which directly relates it to the physics literature, and I can't parse the physics argument that you wrote down in detail.

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Great answer! I took the liberty of editing out two trivial typos. @Anirbit: Many of my physicist friends swear on Barut and Raczka, Theory of Group representations and applications, but I don't know if this particular topic is covered there. –  t.b. Feb 16 '11 at 14:13
@Theo: Dear Theo, Thanks for the kind words, and the editing. Best wishes, –  Matt E Feb 16 '11 at 14:30
@Matt Thanks for your beautiful overview. Can you kindly explain what is this "Hilbert space direct sum (i.e. the completed direct sum)"? I have access to the other book by Knapp called "Lie Groups beyond an introduction". Will that have further details? Like if you could tell me what would be the key-concepts by which I should search? –  Anirbit Feb 17 '11 at 8:07
@Anirbit: Dear Anirbit, By the Hilbert space direct sum of the spaces $W^{m_W}$ I mean the closure (in $V$) of their usual direct sum. This makes sense because the various spaces $W^{m_W}$ naturally sit inside $V$, and are mutually orthogonal, so that their (usual) direct sum also sits inside $V$. I've never studied the other book of Knapp that you mention, and so can't really say anything about it. The key word for learning more about the ideas in my answer is "Harish-Chandra module". If you google this you will find a lot of material, much of it advanced, but hopefully some ... –  Matt E Feb 17 '11 at 14:21
... more introductory material too. Regards, –  Matt E Feb 17 '11 at 14:21
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