# The Unapologetic Mathematician

## Character Tables and the Atlas of Lie Groups

I’ve posted my notes from Zuckerman’s second lecture. Again, there’s 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. This is a big, thick, technical book, not for the faint of heart, but if you really want to know more of the details they’re in there.

The basic thing we’re studying here is (again) a real reductive algebraic group $\underline{G}$. For now we’re assuming that $\underline{G}$ is connected — the group $\underline{G}(\mathbb{C})$ of complex points is a connected complex Lie group, and the group $\underline{G}(\mathbb{R})$ of real points has finitely many components — but in the long run we hope to drop that requirement. As an example, the group ${\rm GL}(n,\mathbb{C})$ is connected, while ${\rm GL}(n,\mathbb{R})$ splits into two pieces. We count it as “connected”. The same goes for ${\rm SO}(p,q)$.

We’re mostly interested in the group $G=\underline{G}(\mathbb{R})$ as a real Lie group. By the reductivity of $\underline{G}$ we get a “Haar measure” $dg$ on $G$, which allows us to bring in analytic tools to study its representations.

The “dual” object $\hat{G}$ is again the set of equivalence classes of irreducible admissible representations, as covered last time. It has a topological structure, but a rather weird one — it’s not even Hausdorff in general. Luckily we can work with something else that is Hausdorff.

Okay, so let’s consider a representation $\pi\in\hat{G}$ acting in some Hilbert space $\mathcal{H}_\pi$. Again we let $K$ denote a maximal compact subgroup of $G$ and $\mathcal{H}_\pi^{(K)}$ the space of $K$-finite vectors, which carries an action of the complex Lie algebra $\mathfrak{g}=\underline{\mathfrak{g}}(\mathbb{C})$. I know it’s weird that we let $G$ be the Lie group of real points of $\underline{G}$ and $\mathfrak{g}$ be the Lie algebra of complex points of $\underline{\mathfrak{g}}$, but the notation is pretty much standard so we’ll just have to deal.

Now we let $Z$ denote the center of $\mathcal{U}\mathfrak{g}$ — the universal enveloping algebra of $\mathfrak{g}$. We know this is ((non-canonically) isomorphic to) a polynomial algebra in $l$ variables, where $l$ is the rank of the Lie algebra $\mathfrak{g}$. We also know we have a Schur lemma stating that $Z$ acts by scalars in $\mathcal{H}_\pi^{(K)}$. Now we define a function $\chi_\pi:Z\rightarrow\mathbb{C}$ taking every element $z$ of the center to the scalar $\chi_\pi(z)$ by which it acts on $\mathcal{H}_\pi^{(K)}$. We call this function $\chi_\pi$ the “central character” of the representation $\pi$; the older literature says “infinitesimal character”.

Now if $\hat{Z}$ is the set of all algebra homomorphisms from $Z$ to $\mathbb{C}$, we know it’s isomorphic to $\mathbb{C}^l$ as an affine algebraic variety — again, not canonically. Now we have the fundamental mapping $\chi:\hat{G}\rightarrow\hat{Z}\cong\mathbb{C}^l$ taking each representation to its central character. If we use the isomorphism with $\mathbb{C}^l$ to induce a topology on $\hat{Z}$ this map is continuous.

We have a theorem of (wait for it…) Harish-Chandra telling us that the fibers of this map are finite sets, which is basically a consequence of the Regularity Theorem from the end of last time. This means that for any given point in $\hat{Z}$ there is just a finite set of representations having that point as their central character. That’s nice and all, but we really need more information about the fibers.

An example here is helpful. We say that $\underline{G}$ is “split” if there is a real Cartan subgroup $\underline{H}$ so that all the roots of $\underline{H}$ are defined over the reals (no complex numbers needed). For instance, $\underline{GL}(n)$ is split, $\underline{SO}(p,q)$ is split for $|p-q|\leq1$, and (our favorite) $\underline{E}_{8,{\rm split}}$ is split. A theorem of Harish-Chandra (with an assist by Gelfand and Naimark) tells us that if $\underline{G}$ is split then $\chi$ is surjective — every point in $\hat{Z}$ is actually the central character of some representation. In fact, the implication goes the other way: if every point is actually a central character, then $\underline{G}$ is split.

Now for $l$ the rank of $\underline{G}$ we have that “typical” points $\omega:Z\rightarrow\mathbb{C}$ the fiber $\chi^{-1}(\omega)$ has $2^l$ points — most points are the central characters of this many representations. “Typical” is a pretty loaded term here, though. The “non-typical” set is pretty badly behaved. For one thing, it has infinitely many components in general.

Let’s consider just about the simplest case: $\underline{G}={\rm SL}(2)$ with rank $l=1$. In this case the center of the enveloping algebra is $\mathbb{C}[\Omega]$, where $\Omega$ is the Casimir operator. We normalize the Casimir to have eigenvalue $n^2-1$ in the $n$-dimensional simple $\mathfrak{sl}(2,\mathbb{C})$ module. Now we have the fundamental map $\chi:\widehat{{\rm SL}(2,\mathbb{R})}\rightarrow\mathbb{C}$. The “typical” eigenvalues here are all the complex numbers other than those of the form $n^2-1$ for positive natural numbers $n$. These “typical” eigenvalues correspond to principal series representations.

In general the set of typical points is open, but not “Zariski open”, if we want to consider the algebraic geometry.

Anyhow, for typical $\omega$ and representations with central character $\chi_\pi=\omega$ the full character $\theta_\pi$ is actually pretty easy to calculate, and the formula is given in Knapp’s book. All the difficulty comes in calculating the character table for irreducible representations whose central characters are not typical, and of course the nontypical oness aren’t distributed quite as nicely as we might wish they were among $\hat{Z}$.

This brings us to the theory of irreducible “relatively tempered” representations.

First, fix a representation $\pi\in\hat{G}$, and let $A(G)$ be the biggest subgroup of the center of $G$ isomorphic to a real vector space. We see that $G$ has a canonical decomposition, “factoring out” this vector subgroup: $G=G_1\times A(G)$. The remaining group $G_1$ is the intersection of the kernels of all homomorphisms from $G$ to the real numbers $\mathbb{R}$. In general it’s disconnected, but with only finitely many components.

If we restrict $\pi$ to the vector subgroup Schur’s lemma again tells us it acts by scalars: $\pi\vert_{A(G)}=\xi_\pi I_{\mathcal{H}_\pi}$. This action is related to the character — if $H$ is in the Lie algebra of $A(G)$ we have $\xi_\pi(\exp H)=e^{\chi_\pi(H)}$. This part of the action of $\pi$ is so simple we may as well just restrict to the remaining subgroup $G_1$.

Now we need a bit of terminology. We say an analytic function $\phi$ on $G$ is a “matrix coefficient” of $\pi$ if there are $K$-finite vectors $v,w\in\mathcal{H}_\pi^{(K)}$ so that $\phi(g)=\langle\pi(g)v,w\rangle$. First of all, given two vectors this function actually is real-analytic. Second of all, though the specific vectors representing a given $\phi$ depend on the inner product of $\mathcal{H}$, the class of which functions have such a representation at all doesn’t. If we move to an equivalent representation we get the same collection of matrix coefficient functions.

Now we define a “tempered” representation as one for which every matrix coefficient $\phi$ is in $L^{2+\epsilon}(G)$ for all positive $\epsilon$, and a “relatively tempered” representation as one for which every $\phi$ restricted to $G_1$ is in $L^{2+\epsilon}(G_1)$ for all positive $\epsilon$ — they’re tempered “relative to the subgroup $G_1$“. We don’t have to be able to integrate the square of a matrix coefficient over the subgroup $G_1$, but we have to almost be able to. This seems to be a completely arbitrary analytic requirement, but it’s actually related to the Plancherel theorem for the group $G_1$.

We get a theorem: if $\pi$ is relatively tempered, then $\pi\vert_{G_1}$ is unitarizable. That is, the action might not be unitary as it stands, but we can pick another inner product on the representation space so that it is.

And then a miracle occurs: there is a complete classification of the relatively tempered representations. This was known for ${\rm SL}(2,\mathbb{R})$ back in the 1940s, and the general case is due to work of Harish-Chandra, Langlands, Hecht and Schmid, and Knapp and Zuckerman. Further, for every relatively tempered irreducible representation $\pi$, the character $\theta_\pi$ can be explicitly calculated as a real-analytic function on the subgroup $G'$ of regular semisimple elements of $G$, and the Regularity Theorem tells us this completely determines the character on all of $G$. This is due to most of the above people, and the finishing touches are due to Becky Herb. This gives us for free an understanding of the so-called “standard” characters.

So what’s a standard character? To explain that I’ll need to talk about parabolic induction.

Let $\underline{G}$ be a reductive algebraic group and $\underline{P}$ be a real parabolic subgroup. This is a Zariski-closed subgroup so that the complex points $\underline{P}(\mathbb{C})$ contain a Borel subgroup of $\underline{G}(\mathbb{C})$. The Borel subgroup isn’t unique, but any two are conjugate so we’re justified in picking one and sticking with it. If the group $\underline{G}$ has rank $l$ then there are $2^l$ subgroups of $\underline{G}(\mathbb{C})$ containing a given Borel, so the number of real parabolic subgroups is bounded (up to conjugacy) by $2^l$. For the compact real form $G$ there’s only one parabolic, while the split real form has all $2^l$. A suggested exercise is to work them all out for the split group ${\rm SO}(p,q)$.

Given a parabolic subgroup, we write $P$ for the real points $\underline{P}(\mathbb{R})$. This is a closed Lie subgroup of $G$, and the space $G/P$ is compact. We can also decompose any parabolic $\underline{P}$ into the product (not direct) of groups $\underline{M}\underline{N}$, where $\underline{M}$ is reductive and $\underline{N}$ is unipotent. This decomposition is unique up to conjugation.

Now parabolic induction is a process that takes an admissible representation $\sigma$ of $M=\underline{M}(\mathbb{R})$ and gives an admissible representation $I_P^G\sigma$ of $G$. By a sort of “Frobenius formula” we can explicitly calculate the character $\theta_{I_P^G\sigma}$ from the character $\theta_\sigma^M$. It’s important to note, though, that even if $\sigma$ is an irreducible representation of $M$, the induced representation $I_P^G\sigma$ of $G$ may not be.

We can now define “standard” characters. Start with some parabolic subgroup $P$ of $G$ with reductive part $M$, and pick an irreducible, relatively tempered representation $\sigma$ of $M$. Use it to induce a representation $I_P^G\sigma$ of $G$. The standard characters are those arising from these representations.

At the level of characters, what we have is a homomorphism from the abelian group ${\rm Ch}(M)$ to ${\rm Ch}(G)$, where these groups are “virtual characters”. Every element is a linear combination of irreducible characters, but they might involve minus signs. In that case the function can’t be the character of an actual representation of the group, but we’ll throw it in anyway so we have a nice abelian group to work with. These groups are really stupendously huge, but at least they’re free. We know that the irreducible characters span the group ${\rm Ch}(G)$, and we know that they’re independent, so they form an uncountably infinite basis.

Here’s the amazing thing: the standard characters also form a basis! Take all the parabolics — there are only a finite number of them — and for each one get the irreducible basis of ${\rm Ch}(M)$. Each of these bases gets sent to some collection of characters in ${\rm Ch}(G)$, and the set of all such characters is a new basis of this abelian group! This is the culmination of work of Harish-Chandra, Langlands, Speh, Borel and Wallach, and (according to Vogan) was formulated like this by Zuckerman.

The character table collects together the characters of all the irreducible representations, and the irreducible representations are now linear combinations of the new basis of standard characters. The standard characters can be calculated explicitly (in terms of the character tables of lower-rank reductive groups $\underline{M}$, so all we need now is a “change-of-basis matrix”. And that’s where the Atlas comes in.