# The Unapologetic Mathematician

## Character Tables for Exceptional Real Reductive Groups

I’ve posted my notes for Zuckerman’s third and final talk. This should get us up to “what exactly does the Atlas Project do?”

Maybe we have enough pull around here to get Jeff Adams to come up and answer that question in a talk before I’m banished from the academy, but even if not I may be able to get time to sit down with him one-on-one once I head back to the Maryland area. Anyhow, my explanations are behind the jump, as usual.

Again we’re looking at a connected real algebraic group $\underline{G}$. This time we restrict our attention to those so that $\underline{\mathfrak{g}}(\mathbb{C})$ (which we’ll usually just call $\mathfrak{g}$ is an exceptional simple Lie algebra.

It turns out, to Zuckerman’s delight, that there are 26 such groups. He loves it when 26 shows up, and it does a lot. It’s in string theory, for one, where he used to do some work. He also notes (with a strong note of ha ha only serious) that it’s the most common gematric value of the Hebrew names for the deity.

Anyhow, the list breaks down like this: the group $E_6$ has five real forms in each of two different types, and $E_7$ has four in each of those two types. For $E_8$, $F_4$, and $G_2$ there’s only one type, and they have three, three, and two real forms respectively. The one that’s been in the news is the “split real form” of $E_8$. The Atlas project is very intent on understanding these 26 groups.

Again we consider the real points $G=\underline{G}(\mathbb{R}$, and note that $G$ is connected if and only if $\underline{G}(\mathbb{C})$ is simply-connected. We recall the notation $\mathcal{U}\mathfrak{g}$ for the universal enveloping algebra of $\mathfrak{g}$, $Z$ for its center, and $\hat{Z}$ for the algebra homomorphisms from $Z$ to $\mathbb{C}$. Also remember that $\hat{Z}$ is isomorphic to the affine space $\mathbb{C}^l$ — though not canonically — where $l$ is the rank of the group $\underline{G}$. In the above list it’s the subscript on the name of the complex group, so $E_8$ has rank $8$.

Okay, so we pick a group $\underline{G}$ and a “central character” $\chi\in\hat{Z}$. The set of “virtual characters” ${\rm Ch}(G,\chi)$ is sort of like how we got integers from natural numbers. Start with all the admissible $G$-modules $M$ whose central character is $\chi$. The global character $\theta_M$ is a function on $G$ completely determined by its values on the subgroup of regular semisimple elements $G'\subseteq G$, and invariant under conjugation in $G$. We can add the characters of two modules to get the character of their direct sum, but we also want to subtract. So we just throw in all the differences as well. The result might not be the character of any actual $G$-module, but it will be a “virtual” character.

Now, ${\rm Ch}(G,\chi)$ has two canonical bases: irreducible characters and standard characters. The irreducible characters are just the functions that come from irreducible modules, while the standard characters are those that come from the standard modules defined last time. Last time we talked about the collection of all characters, which had uncountably many generators, but here we’re separating out the characters by how they behave on $Z$. As a result, it turns out that ${\rm Ch}(G,\chi)$ is always a finitely-generated free abelian group, and the rank is the number of distinct standard characters associated to $\chi$.

As an example, assume that $\underline{G}$ is split over $\mathbb{R}$. That is, there is a Borrel subgroup $\underline{B}$ defined over $\mathbb{R}$ and we say $B=\underline{B}(\mathbb{R})$. Now give a one-dimensional continuous representation of $B$, $\xi:B\rightarrow\mathbb{C}^\times$, we can form the induced representation $I_B^G\xi$, which will be standard. For typical $\chi$, then every standard character in ${\rm Ch}(G,\chi)$ is of this form, so the rank of ${\rm Ch}(G,\chi)$ is $2^l$ for such $\chi$. In fact, in this case “standard” and “irreducible” are the same thing.

At the other extreme, $1$ denotes the trivial representation of $G$, which has central character $\chi_1$. One problem is to compute the rank of ${\rm Ch}(G,\chi_1)$. There is no general formula for this, but we have some special cases.

For $\underline{G}_{2,2}$, the split real form of type $G_2$ this rank is $1+12=13$. Compare this to the result for typical characters: $2^2=4$. For $\underline{E}_{8,8}$, the split real form of type $E_8$, the Atlas project tells us that the rank is $1+73410+453060=526471$. Where do these numbers — and particularly this way of writing them as sums — come from?

Let’s go back a bit and try to see where the group of virtual characters really came from. We consider the category $\mathcal{A}(G,\chi)$ of admissible $(\mathfrak{g},K)$-modules with central character $\chi$, whose morphisms preserve both the $\mathfrak{g}$ and $K$ actions. Then the Grothendieck group of this category is just ${\rm Ch}(G,\chi)$! In general this category decomposes into the direct sum of finitely many subcategories, called blocks.

Given two simple objects $X_1$ and $X_2$ in $\mathcal{A}(G,\chi)$ we say that $X_1\sim X_2$ if ${\rm Ext}^1_{\mathcal{A}(G,\chi)}(X_1,X_2)\neq0$. That is, if there is a nonsplit short exact sequence
$0\rightarrow X_1\rightarrow M\rightarrow X_2\rightarrow0$
We extend $\sim$ to an equivalence relation on the set of simple modules, and the blocks are the full subcategories generated by the equivalence classes.

For example, if we consider the split real form $\underline{G}=\underline{SL}(3)$ — so $G=SL(3,\mathbb{R})$ — then $\mathcal{A}(G,\chi_1)$ has seven simple objects in two blocks. One contains the trivial module and five more simples, and the other contains exactly one simple module. The category $\mathcal{A}(G,\chi)$ is the direct sum of these blocks.

If $\underline{G}$ is a compact real form, then $\mathcal{A}(G,\chi)$ is always semisimple with either one or zero simple modules, and the theory of compact Lie groups tells us that only a discrete set of central characters have any representations at all. Contrast this with the split case where every central character shows up in at least one representation.

This brings us to four big papers and a book, all by David Vogan. The papers are all titled, “Irreducible Characters of Semisimple Lie Groups”, and they appeared in 1979, 1979, 1981, and 1982, respectively. The book is “Representations of Real Reductive Lie Groups”, from 1981. They laid the groundwork for all that follows.

For each group we want to reduce the computation of all the irreducible characters to a finite calculation. We break them up by their central characters into finite-rank free abelian groups ${\rm Ch}(G,\chi)$. In each of those, we will get the irreducible characters by finding the standard characters and applying a “change-of-basis matrix”. For typical $\chi$ and split $\underline{G}$, everything’s pretty straightforward: each block has one simple object and there are $2^l$ blocks. The problem is all in the non-typical characters.

Now if $\chi$ is the central character of a finite-dimensional simple module, it’s non-typical. For all the other non-typical $\chi$, it turns out that $\mathcal{A}(G,\chi)$ is equivalent to $\mathcal{A}(G_1,\chi_1)$, where $G_1$ is a group of dimension less than or equal to that of $G$, and $\chi_1$ is the central character of a finite-dimensional simple $G_1$-module. This was done in Vogan IV, and the proof uses some very heavy machinery involving the Kazhdan-Lusztig-Vogan polynomials (there they are!).

And now we have to deal with the finite-dimensional simple modules. If $F$ is a finite-dimensional simple $G$-module, then we can relate the theory at $\chi_F$ back to that at $\chi_1$! In fact, if $\underline{G}$ is simply-connected then we have a theorem proved in various ways by Zuckerman, Bernstein and Gelfand, and Bernstein and Beilinson. There is a canonical equivalence of categories $\mathcal{A}(G,\chi_1)\cong\mathcal{A}(G,\chi_F)$. It takes a module $M$ with central character $\chi_1$ to the “$\chi_f$-primary component” $(M\otimes F)_{\chi_F}$, which sends standard modules to standard modules and even preserves the blocks.

So, we start with a nontypical $\chi$. If it doesn’t come from a finite-dimensional simple, reduce to a lower-dimensional group and a finite-dimensional simple character. If is is a finite-dimensional already, translate it to $\chi_1$. Thus understanding $\mathcal{A}(G,\chi_1)$ gives us everything!

Now there’s a set ${\rm Irr}(G,\chi_1)$ of irreducible modules with central character $\chi_1$. The problem is that it’s hard to give an explicit ordered list of its elements. We use Langlands classification theory to give an explicit bijection between irreducible and standard modules, so now we need an explicit list of the standard characters.

A standard character will have the form $I_P^G\theta^M$, where $\theta^M$ is a “relative discrete series character”. In the case of simply-connected $G$, we let $\{H_1,...,H_s\}$ be a complete set of non-conjugate real Cartan subgroups of $G$, where we know $s$ is finite. We can parametrize the standard characters by $(H_i,w,\tau)$, where $H_i$ is one of these Cartans, $w$ is in the Weyl group of $\underline{G}$, and $\tau$ is a homomorphism from $H_i$ to $\mathbb{Z}_2$.

Unfortunately, this repeats a bunch of characters. The parameters of a standard representation are defined up to conjugation of $w$ by $W(H_i)={\rm Norm}(H_i)/H_i$, where ${\rm Norm} (H_i)$ is the normalizer of $H_i$ in $G$. Further, there’s no obvious order on $W(\underline{G})/W(H_i)$, or of such $\tau$. The Atlas project has figured out these technical details, given such an explicit listing of the standard characters, and (as reported) calculated the change-of-basis matrix for ${\rm Ch}(\underline{E}_{8,8},\chi_1)$, with the help of a supercomputer.

More specifically what they computed was the matrix of Kazhdan-Lusztig-Vogan polynomials, and when we evaluate these polynomials at $q=1$ we get the change-of-basis matrix we want.