3.14. Lecture 14

3.14.1. Real forms

Here we will use our understanding of the representation theory of $\operatorname{SL}_{2}(\mathbb{R})$ and $\mathrm{SU}(2)$ to tell us about the representation theory of $\mathfrak{sl}_{2,\mathbb{C}}$ , and this will then be used to prove complete reducibility.

Definition 3.14.1.

A real form of a complex Lie algebra $\mathfrak{g}$ is a real Lie algebra $\mathfrak{h}\subseteq\mathfrak{g}$ such that every element $Z$ of $\mathfrak{g}$ can be written uniquely as $X+iY$ for $X,Y\in\mathfrak{h}$ .

For dimension reasons, necessary and sufficient conditions are that $\mathfrak{h}\cap i\mathfrak{h}=0$ and $\dim_{\mathbb{R}}\mathfrak{h}=\dim_{\mathbb{C}}\mathfrak{g}$ .

Example 3.14.2.

The Lie algebra $\mathfrak{sl}_{n,\mathbb{R}}$ is a real form of $\mathfrak{sl}_{n,\mathbb{C}}$ .

Example 3.14.3.

The Lie algebra $\mathfrak{su}_{n}$ is a real form of $\mathfrak{sl}_{n,\mathbb{C}}$ . Indeed, $\dim_{\mathbb{R}}\mathfrak{su}_{n}=n^{2}-1=\dim_{\mathbb{C}}\mathfrak{sl}_{n,% \mathbb{C}}$ and if $X,iX\in\mathfrak{su}_{n}$ then

iX=-(iX)^{*}=iX^{*}=-iX

so $X=0$ .

Explicitly, we may write $A\in\mathfrak{sl}_{n,\mathbb{C}}$ as $X+iY$ with

X=\frac{1}{2}(A-A^{*})

and

Y=\frac{-i}{2}(A+A^{*})

in $\mathfrak{su}_{n}$ .

Example 3.14.4.

Recall that

\mathfrak{so}_{n}=\{A\in\mathfrak{sl}_{n,\mathbb{R}}\,|\,A+A^{T}=0\}.

Let

\mathfrak{so}_{n,\mathbb{C}}=\{A\in\mathfrak{sl}_{n,\mathbb{C}}\,|\,A+A^{T}=0\}.

Since the defining equation $A+A^{T}=0$ can be checked separately on the real and imaginary parts of $A$ ,

\mathfrak{so}_{n,\mathbb{C}}=\{B+iC\,|\,B,C\in\mathfrak{so}_{n}\}

and so $\mathfrak{so}_{n}$ is a real form of $\mathfrak{so}_{n,\mathbb{C}}$ .

Proposition 3.14.5.

Let $\mathfrak{h}$ be a real form of $\mathfrak{g}$ . There is a one-to-one correspondence between representations of $\mathfrak{h}$ and complex-linear representations of $\mathfrak{g}$ under which irreducible representations correspond to irreducible representations.

Proof.

Given a $\mathbb{C}$ -linear representation of $\mathfrak{g}$ , it is a representation of $\mathfrak{h}$ by restriction. Conversely, if $(\rho,V)$ is a representation of $\mathfrak{h}$ , then it extends to a unique $\mathbb{C}$ -linear representation of $\mathfrak{g}$ given by the formula (forced by $\mathbb{C}$ -linearity)

\rho(X+iY){\bf v}=\rho(X){\bf v}+i\rho(Y){\bf v}.

It is an exercise to check this preserves the Lie bracket. The second statement is left as an exercise (Problem 3.14.3). ∎

As a corollary we immediately obtain

Theorem 3.14.6.

The representation theories of $\mathfrak{sl}_{n,\mathbb{R}}$ and $\mathfrak{su}_{n}$ are ‘the same’ as the complex-linear representation theory of $\mathfrak{sl}_{n,\mathbb{C}}$ . All finite-dimensional irreducible representations of $\mathfrak{sl}_{2,\mathbb{R}}$ , $\operatorname{SL}_{2}(\mathbb{R})$ , $\mathfrak{su}_{2}$ , or $\mathrm{SU}(2)$ are of the form $\operatorname{Sym}^{n}(\mathbb{C}^{2})$ .

Proof.

The claims about Lie algebras follow from Proposition 3.14.5. By Theorem 3.13.10, every (finite-dimensional) irreducible representation of $\mathfrak{sl}_{2,\mathbb{C}}$ , and thus $\mathfrak{sl}_{2,\mathbb{R}}$ , is of the form $\operatorname{Sym}^{n}(\mathbb{C}^{2})$ , and these clearly exponentiate to representations of $\operatorname{SL}_{2}(\mathbb{R})$ , despite this not being a simply connected group! Similarly for $\mathrm{SU}(2)$ (which is simply connected). Since $\operatorname{SL}_{2}(\mathbb{R})$ and $\mathrm{SU}(2)$ are connected, every representation of them is determined by its derivative, so we have a complete list of the irreducible representations. ∎

3.14.2. Decomposing representations

Theorem 3.14.7.

Let $V$ be any finite-dimensional complex-linear representation of $\mathfrak{sl}_{n,\mathbb{C}}$ . Then $V$ is completely reducible, that is, splits into a direct sum of irreducible representations.

Proof.

Let $V$ be a finite-dimensional complex-linear representation of $\mathfrak{sl}_{n,\mathbb{C}}$ and let $W\subseteq V$ be a subrepresentation. Then $W$ is an $\mathfrak{su}_{n}$ -subrepresentation, by Theorem 3.14.6. As $\mathrm{SU}(n)$ is simply-connected, $V$ and $W$ exponentiate to representations of $\mathrm{SU}(n)$ . Since $\mathrm{SU}(n)$ is compact, by Maschke’s theorem (Theorem 2.11.4(ii)) there is a complementary $\mathrm{SU}(n)$ -subrepresentation $W^{\prime}$ with

V=W\oplus W^{\prime}.

Then $W^{\prime}$ is a $\mathfrak{su}_{n}$ -subrepresentation, and so a $\mathbb{C}$ -linear $\mathfrak{sl}_{n,\mathbb{C}}$ -subrepresentation, so that

V=W\oplus W^{\prime}

as representations of $\mathfrak{sl}_{n,\mathbb{C}}$ . Complete reducibility follows. ∎

The argument in this theorem is called Weyl’s unitary trick. For a similar application of this idea, see the proof of Proposition 3.12.2.

Corollary 3.14.8.

Two finite-dimensional complex-linear representations of $\mathfrak{sl}_{2,\mathbb{C}}$ are isomorphic if and only if they have the same multiset of weights.

Proof.

Let $(\pi,V)$ and $(\rho,W)$ be finite-dimensional complex-linear representations of $\mathfrak{sl}_{2,\mathbb{C}}$ . We can decompose $V$ into irreducibles by looking at the weights. Firstly, look at the maximal weight $k$ of $V$ . Then there must be a weight vector ${\bf v}$ of weight $k$ , which is necessarily a highest weight vector, and so $V$ must contain a copy of $\operatorname{Sym}^{k}(\mathbb{C}^{2})$ — namely, the subspace $\left\langle{\bf v},Y{\bf v},\ldots,Y^{k}{\bf v}\right\rangle$ . By Theorem 3.14.7 we have

V\cong\operatorname{Sym}^{k}(\mathbb{C}^{2})\oplus V^{\prime}.

The weights of $V^{\prime}$ are then obtained by removing the weights of $\operatorname{Sym}^{k}(\mathbb{C}^{2})$ from the weights of $V$ , and we repeat the process, noting that it terminates as $V$ is finite dimensional. If we then do the same for $W$ we see that they decompose into the same sum of copies of $\operatorname{Sym}^{k}(\mathbb{C}^{2})$ if and only if they have the same multiset of weights. ∎

3.14.3. Exercises

Problem 35. Show that the real Lie algebras $\mathfrak{sl}_{2,\mathbb{R}}$ and $\mathfrak{su}_{2}$ are not isomorphic. Hint: consider the adjoint action of an arbitrary element of $\mathfrak{su}_{2}$ .

Problem 36. Finish the proof of Proposition 3.14.5.

Problem 37. We have that $\operatorname{Sym}^{n}(\mathbb{C}^{2})$ is the irreducible representation of $\mathrm{SU}(2)$ of dimension $n+1$ . Let $\chi_{n}$ be its character. Every conjugacy class of $\mathrm{SU}(2)$ contains an element of the form

\exp(itH)=\begin{pmatrix}e^{it}&0\\ 0&e^{-it}\end{pmatrix}.

Show that

\chi_{n}(\exp(itH))=\frac{\sin((n+1)t)}{\sin(t)}.

Problem 38. Let $(\rho,V)$ be an irreducible representation of $\mathfrak{gl}_{2,\mathbb{C}}$ , and let $Z=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}$ .

(a)

Show that $\rho(Z)$ is as a scalar.
(b)

Show that the restriction of $V$ to $\mathfrak{sl}_{2,\mathbb{C}}$ is irreducible.
(c)

Show that for every $\lambda\in\mathbb{C}$ and integer $n\geq 0$ , there is a unique irreducible representation of $\mathfrak{gl}_{2,\mathbb{C}}$ of dimension $n+1$ with $\rho(Z)=\lambda\operatorname{Id}$ .
(d)

Which of these are derivatives of representations of $\operatorname{GL}_{2}(\mathbb{C})$ ? Hence classify the finite dimensional holomorphic irreducible representations of $\operatorname{GL}_{2}(\mathbb{C})$ .

Problem 39. Let $V$ be a finite-dimensional representation of $\mathfrak{sl}_{2,\mathbb{C}}$ .

(a)

What are the weights of the dual representation $V^{*}$ ?
(b)

Deduce that $V\cong V^{*}$ .