Lie Theory Through Examples

John Baez

Lecture 3

1 Representations of Lie Groups

We’ve been having some fun getting lattices from simply-connected complex simple Lie groups. If someone hands us one of these, say G, we first choose a maximal compact subgroup K ⊆ G. They’re all conjugate inside G, so it doesn’t matter which one we choose. Then, we choose a maximal torus T ⊆ K. Again, they’re all conjugate inside K, so it doesn’t matter which one we choose. Then we can work out the Lie algebra 𝔱 of T and find a lattice

L = ker e =⊆ 𝔱,

where

\begin{matrix}e: 𝔱 & → & T \\ x & ↦ & exp(2πx). \end{matrix}

So far we’ve done this for G = SL(3, ℂ) and K = SL(4, ℂ). The other SL(n, ℂ)’s work similarly, and soon we’ll do even more examples. It’s a lot of fun.

But, what’s it all good for?

Among other things, it’s good for classifying the complex-analytic representations of G, and the unitary representations of K. So, we need a word or two about these.

Remember that a representation of any Lie group H is a smooth homomorphism

ρ: H → GL(V)

where V is a vector space and GL(V) is the group of invertible linear transformations of V. In what follows we’ll always assume V is finite-dimensional. When V = ℂ^{n} we also call GL(V) the general linear group GL(n, ℂ).

Now, the group GL(V) is always a complex manifold: we can cover it with coordinate charts that look like ℂ^{n}, with complex-analytic transition functions. It makes sense to talk about complexanalytic maps between complex manifolds. And indeed, GL(V) is a complex Lie group: a complex manifold where the functions describing multiplication and inverses are complex analytic. To see this, just use the usual formulas for multiplying and taking inverses of matrices.

If H is a complex Lie group, we say a representation ρ: H → GL(V) is complex-analytic if it is complex-analytic as a map between complex manifolds. Such representations are easy to come by:

Exercise 1 Show that SL(n, ℂ) is a complex Lie group, and the obvious representation of SL(n, ℂ) on ℂ^{n} is complex analytic.

Exercise 2 Show that if ρ, σ: G → GL(V) are complex-analytic representations, so are ρ ⊕ σ and ρ ⊗ σ.

Exercise 3 Show that any subrepresentation of a complex-analytic representation is complex-analytic.

On the other hand, unitary representations are also nice. Given a Lie group K and a finitedimensional Hilbert space H, we define a unitary representation to be a smooth homomorphism:

ρ: K → U(H)

where U(H) is the group of unitary operators on a Hilbert space H. When H = ℂ^{n} we also call U(H) the unitary group U(n).

Compact Lie groups have lots of unitary representations:

Exercise 4 Suppose ρ: K → GL(V) is a (finite-dimensional) representation of a compact Lie group. Show that there is an inner product on V that is invariant under ρ, so that letting H denote V made into a Hilbert space with this inner product, we have

ρ: K → U(H).

The exercises we saw for complex-analytic representations all have analogues for unitary representations:

Exercise 5 Show that SU(n) is a compact Lie group, and the obvious representation of SU(n) on ℂ^{n} is unitary.

Exercise 6 Show that if ρ, σ: G → GL(V) are unitary representations, so are ρ ⊕ σ and ρ ⊗ σ.

Exercise 7 Show that any subrepresentation of a unitary representation is unitary.

But, the really cool part is that when G is a complex simple Lie group and K is its maximal compact subgroup, the complex-analytic representations of G correspond in a one-to-way to unitary representations of K. This fact was called the unitarian trick by Hermann Weyl, who used it to do great things. Let’s state it a bit more precisely:

Theorem 1 Suppose G is a complex simple Lie group and K is its maximal compact subgroup. Given a (finite-dimensional) complex-analytic representation

ρ: G → GL(V),

there exists an inner product on V making ρ|_{K} into a unitary representation. Conversely, given a (finite-dimensional) unitary representation

ρ: K → U(H),

there exists a unique extension of ρ to a complex-analytic representation of G on the vector space H.

2 The Weight Lattice

Now say we have our favorite kind of Lie group G: a simply-connected complex simple Lie group. Say someone hands us a complex-analytic representation of G. We want to understand it and classify it. By the above theorem, we can think of it as a unitary representation of the maximal compact K without losing any information.

So, let’s do that: say we have unitary representation ρ of K on a finite-dimensional Hilbert space H. What do we do now? Since the maximal torus T is a subgroup of K, we get a unitary representation of T:

ρ|_{T}: T → U(H).

And in fact, the maximal torus is big enough that we can completely recover ρ from ρ|_{T}. This is not supposed to be obvious! But it’s great, because unitary representations of tori are incredibly easy to understand.

Here’s how we understand them. Any unitary representation of a torus

α: T → U(H)

can be composed with the exponential map to give a unitary representation of the vector space 𝔱, thought of as a group:

β = αe.

Clearly β is trivial on the kernel of e:

β(x) = 1 for all x ∈ L.

Conversely, it’s easy to see a representation

β: 𝔱 → U(H)

comes from a representation α as above iff β is trivial on L. Even better, since the exponential map is onto, knowing β tells us α. It’s also easy to see that β is irreducible iff α is.

So, we just need to understand unitary representations

β: 𝔱 → U(H)

that are trivial on L. Let’s start with irreducible ones. By Schur’s Lemma, an irreducible representation of an abelian group is 1-dimensional. In this case, β(x) is just multiplication by some unit complex number; to be a representation we also need

β(x + y) = β(x)β(y).

So, it’s easy to see that we get all irreducible unitary representations of 𝔱 from elements ℓ of the dual vector space 𝔱^{∗}, like this:

β(x) = e^{2πiℓ(x)}.

For this to be trivial on L we need

ℓ(x) ∈ ℤ for all x ∈ L.

This means that ℓ needs to lie in the set

L^{∗} = {ℓ ∈ 𝔱^{∗} : ℓ(x) ∈ ℤ for all x ∈ L}.

In fact L^{∗} is a lattice in 𝔱^{∗} just as L is a lattice in 𝔱! We call L^{∗} the dual lattice of L. Summarizing what we’ve seen so far:

Theorem 2 An irreducible unitary representation α of a torus T is specified by choosing a point ℓ in the dual lattice L^{∗}.

We call the point ℓ ∈ L^{∗} the weight of the representation. When T is the maximal torus of a simply-connected compact simple Lie group K, we call L^{∗} the weight lattice of K.

But what if α fails to be irreducible? Then it’s a direct sum of irreducible representations — and in an essentially unique way. To specify each of these irreducible representations, we pick a point in L^{∗}. We can pick the same point more than once. But we need to pick just finitely many points, since H is finite-dimensional. So:

Theorem 3 Finite-dimensional unitary representation ρ of a torus T are classified up to unitary equivalence by maps

d: L^{∗} → ℕ

such that

∑_{ℓ∈L^{∗}} d(ℓ) < ∞.

The map d says how many times each irreducible unitary representation of T shows up in ρ. The theorem says that given any map d: L^{∗} → ℕ with ∑_{ℓ∈L^{∗}} d(ℓ) < ∞, there exists a unitary representation ρ with this d. Moreover, it says that two unitary representations of T are unitarily equivalent iff they have the same d. Here we say ρ: K → U(H) and ρ′: K → U(H′) are unitarily equivalent if there is unitary operator U: H → H′ such that

ρ′(k)U = Uρ(k)

for all k ∈ K.

For example, say we have a unitary representation of the circle group:

α: U(1) → U(H).

Here 𝔲(1) ≅ ℝ and L ⊆ ℝ is just the set of integers. So, to specify a unitary irreducible representation of U(1) we pick an integer ℓ. Concretely, it goes like this:

α(e^{iθ}) = e^{iℓθ}.

In general, unitary representations of U(1) are classified by maps

d: ℤ → ℕ

as in the theorem.