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 of the form , where 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:

• — the group of invertible matrices with real entries.
• — the group of matrices preserving a bilinear form of signature .
• — the group of symplectic matrices.
• — the real form of from the now-famous calculation.

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

First of all, for a representation and a vector , the function on sending to is continuous on in the “classical” topology — considering it as a real submanifold of . This means that we’re really using the topology of the Hilbert space to pick out “nice” representations, rather than any further structure.

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Now a bit of a historical digression. The root system of the Lie algebra was discovered by Killing, and Cartan constructed the Lie algebra from that. Freudenthal gave explicit constructions of the other sporadic Lie algebras , , , and 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 was introduced as a class of representations by Harish-Chandra, and it subsumes work on particular groups by Gelfand, and Naimark, among others.

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Now back to the main stream: pick a separable complex Hilbert space . We write for the algebra of bounded (i.e.: continuous) linear operators on . A representation of on is a function with the properties that

• 
• 
• for all , is a continuous map from the Lie group to the topological Hilbert space

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

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 and as , there is a special kind of representation people were classically interested in. We say that a representation is “unitary” if for all elements of the group and vectors 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 . Here are some examples for our example groups:

Such a maximal compact always exists, and is unique up to conjugation in . 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 of the group is admissible if its restriction to a representation of decomposes as the topological direct sum of finite-dimensional representations of . Remember that the original Hilbert space is often infinite-dimensional, but we’re insisting that if we forget everything but the action the space breaks into an infinite number of pieces (as a topological vector space), each of which is finite-dimensional. We write:

where again is the set of equivalence classes of irreducible representations of , there are countably-infinitely many of them, and for each we pick a as a representative of the equivalence class. We require that for each the dimension of the space 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 . In fact, it’s always less than the dimension of !

Now let be an admissible representation of in a Hilbert space . Despite the fact that breaks up into finite-dimensional pieces, a given vector may touch enough of them so its -orbit is infinite-dimensional. We call the ones whose -orbits are finite-dimensional “-finite”, and write for the subspace of all -finite vectors. This subspace is the algebraic direct sum — 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:

• is dense in .
• Any -finite vector defines a real analytic function .
• The Lie algebra sends -finite vectors to other -finite vectors.

Finally we can define “infinitesimal equivalence”. We say that admissible representations and are infinitesimally equivalent if and are equivalent as modules. It’s important to note here that even though the Lie algebra acts on , the Lie group doesn’t!

Now another Big Theorem of Harish-Chandra: if representations and 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 — the split real form of . In fact Harish-Chandra did just that.

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Finally we come back to characters. An admissible character is a kind of function on the conjugacy classes of , just like finite groups. And just like for finite groups we calculate its value at by taking the trace of the image of . 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 and a smooth, compactly-supported “test function” we define

where is Haar measure on the group .

Harish-Chandra (him again) showed in general what Gelfand and Naimark had done for special cases: if is an irreducible admissible representation then this operator is “trace class”. That is, if we pick a basis of we find . We can then define its trace: . And now we can define the character 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 of “regular semisimple” elements, the function is actually real analytic! Even better, we can get the “trace” we defined above by this integral

And finally, admissible representations and are infinitesimally equivalent if and only if their characters and are equal for all . At long last we get a character table!

But can we compute it?