The Unapologetic Mathematician

Mathematics for the interested outsider

Admissible Character Tables for Real Reductive Algebraic Groups

I’ve posted my notes for the first of Zuckerman’s lectures. Hopefully my handwriting isn’t too awful for you. I’ve never been very good with that pen-and-paper stuff.

I’m trying to explain this pretty comprehensibly, but I do have to use some terms most mathematicians know without defining them. I’ve got plans to get to them eventually in the main stream of my writings, but for now the exegesis sits at a middle level. Anyhow, there’s a lot to unpack here, so I’ll put it behind the jump.

The diagram on the first page is sort of an idea of what we’re looking at overall. “Admissible” characters are what we’re really interested in, and within that we have “irreducible”, “standard”, and “unitary” characters. A character is a kind of function on a group that encodes information about a representation. Irreducible characters correspond to representations that can’t be broken into pieces, unitary characters correspond to representations with a certain extra structure, and “standard” characters he’ll define on Thursday.

Next we review a character table for a finite group. This is pretty classical stuff that most mathematicians have some handle on. We view it as a function of two variables. One runs over the conjugacy classes of the group, while the other runs over a list of (equivalence classes of) irreducible representations. It turns out that these two sets are the same size, and it’s common to pick an ordering of each to arrange the function into a table. The Atlas of Finite Simple Groups classified all of these groups and calculated their character tables, as I’ve mentioned.

Starting in 2002, the Atlas of Lie Groups project started, with the intent to do much the same thing for Lie groups. Its founding membership included Bill Casselman, Jeff Adams, David Vogan, Dan Barbasch, Siddhartha Sahi, and Fokko DuCloux. Adams was Zuckerman’s first graduate student, Sahi was his third, and (somewhat unrelatedly) I’m his latest to complete.

The project really studies infinite groups G of the form \underline{G}(\mathbb{R}), where \underline{G} is a “connected real algebraic group of reductive type”. This is a lot of technical assumptions I’ll try to avoid explicitly mentioning, but here are some examples:

  • {\rm GL}(n,\mathbb{R}) — the group of invertible n\times n matrices with real entries.
  • {\rm SO}(p,q) — the group of (p+q)\times(p+q) matrices preserving a bilinear form of signature (p,q).
  • {\rm Sp}(2n,\mathbb{R}) — the group of 2n\times2n symplectic matrices.
  • E_{8,{\rm split}}(\mathbb{R}) — the real form of E_8 from the now-famous calculation.

We write G' for the subgroup G'(\mathbb{R}) of “regular semisimple elements” of G — those contained in a unique maximal torus, and we write \hat{G} for the (deep breath…) set of infinitesimal equivalence classes of irreducible (typically) infinite-dimensional representations by bounded linear operators in a Hilbert space \mathcal{H}. A lot of this definition is what the rest of the lecture unpacks.

First of all, for a representation \pi\in\hat{G} and a vector v\in\mathcal{H}, the function on G sending g to \pi(g)v is continuous on G in the “classical” topology — considering it as a real submanifold of {\rm GL}(n,\mathbb{R}). This means that we’re really using the topology of the Hilbert space to pick out “nice” representations, rather than any further structure.

Now a bit of a historical digression. The root system of the Lie algebra E_8 was discovered by Killing, and Cartan constructed the Lie algebra from that. Freudenthal gave explicit constructions of the other sporadic Lie algebras G_2, F_4, E_6, and E_7 in the 1940s.

Chevalley, Serre, and Harish-Chandra gave a universal treatment of constructing a Lie algebra from a root system, and this has led into the modern theory of (infinite-dimensional) Kac-Moody algebras. By-the-by, “Kac” is pronounced like “cats” and (if I haven’t mentioned it) “Lie” is pronounced “lee”.

Anyhow, Lie algebras are important to particle physicists, but they tend to use techniques more like Freudenthal’s rather than the more modern approach.

This \hat{G} was introduced as a class of representations by Harish-Chandra, and it subsumes work on particular groups G by Gelfand, and Naimark, among others.

Now back to the main stream: pick a separable complex Hilbert space \mathcal{H}. We write \mathcal{L}(\mathcal{H}) for the algebra of bounded (i.e.: continuous) linear operators on \mathcal{H}. A representation of G on \mathcal{H} is a function \sigma:G\rightarrow\mathcal{L}(\mathcal{H}) with the properties that

  • \sigma(g_1g_2)=\sigma(g_1)\sigma(g_2)
  • \sigma(e)=I
  • for all v\in\mathcal{H}, g\mapsto\sigma(g)v is a continuous map from the Lie group G to the topological Hilbert space \mathcal{H}

The first two conditions just say that \sigma defines a group action, and the last is this continuity requirement. Eventually we’ll manage to do away with explicit consideration of \mathcal{H}.

Zuckerman doesn’t mention it here, but as mathematicians know a Hilbert space comes with an “inner product” sort of like the dot product people first run into in multivariable calculus or in physics. Writing the inner product of v and w as \langle v,w\rangle, there is a special kind of representation people were classically interested in. We say that a representation \sigma is “unitary” if \langle\sigma(g)v,\sigma(g)w\rangle=\langle v,w\rangle for all elements g of the group and vectors v,w of the Hilbert space. Those were the ones people were really interested in that launched a lot of this later work.

Now, choose a maximal compact subgroup K\subseteq G. Here are some examples for our example groups:
\begin{array}{cc}G&K\\{\rm GL}(n,\mathbb{R})&{\rm O}(n)\\{\rm SO}(p,q)&{\rm S}({\rm O}(p)\times{\rm O}(q))\\{\rm Sp}(2n,\mathbb{R})&{\rm U}(n)\\E_{8,{\rm split}}&{\rm Spin}(16)/\mathbb{Z}_2\\\end{array}

Such a maximal compact K always exists, and is unique up to conjugation in G. This may have been shown by Cartan, but Mostow seems to have had something to do with it. When Zuckerman asked him (sitting in the audience), though, he begged off the credit.

Anyhow, finally we can say what “admissible” means. We say a representation \sigma of the group G is admissible if its restriction to a representation of K decomposes as the topological direct sum of finite-dimensional representations of K. Remember that the original Hilbert space is often infinite-dimensional, but we’re insisting that if we forget everything but the K action the space breaks into an infinite number of pieces (as a topological vector space), each of which is finite-dimensional. We write:

\mathcal{H}=\widetilde{\bigoplus\limits_{\delta\in\hat{K}}}{\rm Hom}_K(V_\delta,\mathcal{H})\otimes V_\delta

where \hat{K} again is the set of equivalence classes of irreducible representations of K, there are countably-infinitely many of them, and for each \delta we pick a V_\delta as a representative of the equivalence class. We require that for each \delta the dimension of the space {\rm Hom}(V_\delta,\mathcal{H}) is finite.

Now here’s the Big Theorem, due to Harish-Chandra around 1950: Any irreducible unitary representation is admissible! Even better, we have an explicit bound on the dimension of {\rm Hom}(V_\delta,\mathcal{H}). In fact, it’s always less than the dimension of V_\delta!

Now let \sigma be an admissible representation of G in a Hilbert space \mathcal{H}. Despite the fact that \mathcal{H} breaks up into finite-dimensional pieces, a given vector may touch enough of them so its K-orbit is infinite-dimensional. We call the ones whose K-orbits are finite-dimensional “K-finite”, and write \mathcal{H}^{(K)} for the subspace of all K-finite vectors. This subspace is the algebraic direct sum \bigoplus\limits_{\delta\in\hat{K}}{\rm mult}(V_\delta)V_\delta — the direct sum as just plain vector spaces, as opposed to the direct sum as topological vector spaces from before.

This subspace has some important properties:

  • \mathcal{H}^{(K)} is dense in \mathcal{H}.
  • Any K-finite vector v\in\mathcal{H}^{(K)} defines a real analytic function g\mapsto\sigma(g)v.
  • The Lie algebra \mathfrak{g}(\mathbb{R}) sends K-finite vectors to other K-finite vectors.

Finally we can define “infinitesimal equivalence”. We say that admissible representations \sigma_1 and \sigma_2 are infinitesimally equivalent if \mathcal{H}_1^{(K)} and \mathcal{H}_2^{(K)} are equivalent as (\mathfrak{g},K) modules. It’s important to note here that even though the Lie algebra \mathfrak{g} acts on \mathcal{H}^{(K)}, the Lie group G doesn’t!

Now another Big Theorem of Harish-Chandra: if representations \pi_1 and \pi_2 are unitary and admissible, then they are unitarily equivalent (isomorphic as unitary representations) if and only if they are infinitesimally equivalent (isomorphic as admissible representations)!

What we’ve covered here is enough machinery to completely classify the unitary representations of {\rm SL}(2,\mathbb{R}) — the split real form of {\rm SL}(2,\mathbb{C}). In fact Harish-Chandra did just that.

Finally we come back to characters. An admissible character is a kind of function on the conjugacy classes of G, just like finite groups. And just like for finite groups we calculate its value at g\in G by taking the trace of the image of g. At this point I asked, “what’s a trace?” And well I should have. Since these representations we’re looking at are usually infinite-dimensional, the usual definition of “trace” is complete nonsense!

So we have a notion of “regulated trace”. If we take a representation \pi and a smooth, compactly-supported “test function” f\in C_c^\infty(G) we define
\pi(f)=\int\limits_G f(g)\pi(g)\,dg
where dg is Haar measure on the group G.

Harish-Chandra (him again) showed in general what Gelfand and Naimark had done for special cases: if \pi is an irreducible admissible representation then this operator \pi(f) is “trace class”. That is, if we pick a basis \lbrace v_i\rbrace of \mathcal{H} we find \sum\limits_i\lvert\langle\pi(f)v_i,v_i\rangle\rvert<\infty. We can then define its trace: {\rm tr}(\pi(f))=\sum\limits_i\langle\pi(f)v_i,v_i\rangle. And now we can define the character \theta_\pi(f)={\rm tr}(\pi(f)) as a linear functional on test functions. This turns out to be a distribution, but we can avoid all that messy distribution theory!

Harish-Chandra’s Regularity Theorem of 1963 states that if we restrict our attention to that subgroup G' of “regular semisimple” elements, the function \theta_\pi is actually real analytic! Even better, we can get the “trace” we defined above by this integral
{\rm tr}(\pi(f))=\int\limits_{G'}f(g)\theta_\pi(g)\,dg
And finally, admissible representations \pi_1 and \pi_2 are infinitesimally equivalent if and only if their characters \theta_{\pi_1}(g) and \theta_{\pi_2}(g) are equal for all g\in G'. At long last we get a character table!

But can we compute it?

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April 9, 2007 - Posted by | Atlas of Lie Groups


  1. I thought the representations of compact groups always broke up into direct sums of finite-dimensional representations. (Or is the key point that it’s not topological? What’s a topological direct sum?)

    Comment by Walt | April 11, 2007 | Reply

  2. Walt, topological direct sum is a direct sum in the category of topological vector spaces, not the category of all vector spaces. See, category theorists and functional analysts have some common ground after all. :D

    Anyhow, your memory isn’t exactly wrong, but the topological direct sum has somthing to do with it.

    I also think you’re importing some unspoken assumptions without meaning to. One common one in introductory courses is to always work with unitary representations. Despite the fact that we’ve got a Hilbert space, we’re actually trying to only use its topology, not its norm. We’re not requiring an admissible character be unitary, even when restricted to K. I think that’s a condition on the theorem you have in mind.

    Anyhow, the other side is something I may not have been clear on: not only does the G-module decompose into finite-dimensional K-modules, but those K-modules occur with finite multiplicities. That is, each of them shows up only finitely-many times in the decomposition.

    Comment by John Armstrong | April 11, 2007 | Reply

  3. I can’t quite remember what theorem I’m thinking of. (Peter-Weyl? When you have a theorem wrong, it’s hard to be sure what theorem you’re thinking of. :-))

    So do compact groups have infinite dimensional representations that are not algebraic direct sums of finite dimensional representations?

    Comment by Walt | April 11, 2007 | Reply

  4. Peter-Weyl is the wellspring of most compact Lie group representation theory, yes, and it talks a lot about unitarity as I recall.

    As for representations, surely there are infinite-dimensional K-modules in which a given finite-dimensional module shows up with infinite multiplicity.

    I don’t have a good example of the finer distinction between algebraic and topological direct sum at the ready, but it’s sort of the difference between a basis of a Hilbert space in the usual sense and a basis of a Hilbert space in the plain linear algebra sense.

    Comment by John Armstrong | April 11, 2007 | Reply

  5. I see the distinction now. I should have known what you meant, but my mind drew a blank.

    I think I’ve figured out what’s true, based on the always reliable Google search. For a compact group acting on a Hilbert space, you can use Haar measure to define an inner product so that the representation is unitary. Then you prove that the irreducible unitary representations of a compact group are finite-dimensional. This allows the possibility that the group has irreducible infinite dimensional representations that cannot be realized as a Hilbert space representation.

    Comment by Walt | April 12, 2007 | Reply

  6. Well, we do start with a Hilbert space representation. As I’m looking deeper I think that basically the only thing that can go wrong is the finite multiplicity condition. Some G-modules restrict to K-modules where a given irrep shows up with infinite multiplicity. Those are the ones that admissibility rules out.

    Wasn’t there someone on your functional analysis thread complaining about “arbitrarily” throwing out things you just don’t like? :D

    Comment by John Armstrong | April 12, 2007 | Reply

  7. That must be right.

    Comment by Walt | April 12, 2007 | Reply

  8. Great overview. Just want to point out one thing: Harish-Chandra’s regularity theorem actually says that the character of an admissible representation, as a distribution, is equal to a locally L^1 function on G.

    It is an easy fact that the character is a real-analytic function when restricted to the open dense set G’. (Essentially because if a distribution on {R^n/a discrete subgroup} is an eigendistribution for all the constant-coefficient differential operators, then it is a sum of exponentials and hence a real analytic function. Apply this to the character restricted to a Cartan subgroup of G, and the invariant differential operators of G. This restriction is defined on the open set of regular points, because the character is “constant along conjugacy orbits”, and regular conjugacy orbits are “perpendicular” to the Cartan subgroup.)

    It is also not difficult to show that this real analytic function is locally L^1 on G, when extended by 0 outside G’. The meat of Harish-Chandra’s theorem is that if you take the character minus this function, the distribution you get (supported on G – G’) is actually 0. The proof is devilishly difficult, of course. Even Atiyah-Schmid’s simpler proof is unpublished, for being too long.

    Also, a couple of historical points: The unitary representations of SL(2,R) were classified by Bargmann, not Harish-Chandra. Bargmann did this in 1942 (published in 1947) soon after Wigner and Dirac proposed the possible significance of the question for quantum mechanics. Harish-Chandra and Gelfand-Naimark also published some far weaker results on this problem in 1947-48.

    The fact that a semisimple group has a unique maximal compact subgroup upto conjugation was proved by Cartan. This same fact for an arbitrary Lie group was proved by Mostow.

    Comment by Rishi | April 12, 2007 | Reply

  9. I should have said “irreducible admissible representation”, instead of “admissible representation” in the 1st paragraph. The statement may not be true for admissible representations of infinite length.

    Comment by Rishi | April 12, 2007 | Reply

  10. Rishi: you’re right about the regularity theorem, and that’s all hidden in my “moreover” in the notes, or my “even better” in the exegesis. I’m trying here to write the results of the functional analysis as they apply to the algebra rather than the details.

    As for the history, I’m transcribing my lecture notes of Zuckerman’s lecture, so any historical errors can be traced to him.

    Comment by John Armstrong | April 12, 2007 | Reply

  11. [...] a lot to unpack here, so it goes after the jump. The bulk of this lecture, as well as the last one, is covered in Knapp’s Representations of Semisimple Lie Groups, an Overview by Examples. [...]

    Pingback by Character Tables and the Atlas of Lie Groups « The Unapologetic Mathematician | April 14, 2007 | Reply

  12. [...] Admissible Character Tables for Real Reductive Groups [...]

    Pingback by Ars Mathematica » Blog Archive » Real Reductive Groups | May 29, 2007 | Reply

  13. [...] what progress we’ve made! Of course, we’re not up to the point of really understanding Zuckerman’s three lectures, but all in good [...]

    Pingback by Lie Groups in Nature « The Unapologetic Mathematician | January 11, 2010 | Reply

  14. [...] what progress we’ve made! Of course, we’re not up to the point of really understanding Zuckerman’s three lectures, but all in good [...]

    Pingback by Lie Groups in Nature | Drmathochist's Blog | August 28, 2010 | Reply

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