2.11. Lecture 11

2.11.1. Representations of U(1) and Maschke’s theorem

We now discuss the representation theory of the unitary group $\operatorname{U}(1)=\{e^{it}\,|\,t\in\mathbb{R}\}$ , which is isomorphic to $\mathrm{SO}(2)$ , the circle group. Its Lie algebra is $i\mathbb{R}\subseteq\mathbb{C}$ with trivial Lie bracket, which is isomorphic to $\mathbb{R}$ .

Proposition 2.11.1.

All irreducible finite-dimensional representations of $\operatorname{U}(1)=\{z\in\mathbb{C}\,|\,|z|=1\}$ are one-dimensional. They are given by

z=e^{it}\longmapsto e^{int}=z^{n}

for $n\in\mathbb{Z}$ .

Proof.

By Schur’s lemma and the fact that $\operatorname{U}(1)$ is abelian, all irreducible finite-dimensional representations of $\operatorname{U}(1)$ are one-dimensional, so are homomorphisms $\operatorname{U}(1)\rightarrow\mathbb{C}^{\times}$ . Since $\operatorname{U}(1)$ is connected, such a homomorphism is determined by the derivative $\mathfrak{u}_{1}\rightarrow\mathfrak{gl}_{1}(\mathbb{C})=\mathbb{C}$ , which has the form $it\mapsto\lambda t$ for some $\lambda\in\mathbb{C}$ . As in Example 1.7.1, this exponentiates to a map $\operatorname{U}(1)\rightarrow\mathbb{C}^{\times}$ if and only if $\lambda=in$ for some $n\in\mathbb{Z}$ , giving the homomorphism $z\mapsto z^{n}$ . ∎

Theorem 2.11.2.

(i)

All finite dimensional representations of $\operatorname{U}(1)$ are unitary.
(ii)

All finite dimensional representations of $\operatorname{U}(1)$ are completely reducible, that is, decompose into irreducible representations.

Proof.

(i)

This is Problem 2.11.2.
(ii)

This follows from (i) as in the proof of Maschke’s theorem for finite groups. We will sketch another proof. Let $(\rho,V)$ be a finite dimensional representation of $\operatorname{U}(1)$ . Consider its differential $D\rho:\mathfrak{u}(1)=i\mathbb{R}\to\mathfrak{gl}_{n,\mathbb{C}}$ . Let $A=D\rho(2\pi i)$ . By the proof of Lemma 1.2.2, we may write $A=D+N$ for some strictly upper triangular matrix $N$ and diagonal matrix $D$ such that $D$ and $N$ commute. Then

$\rho(\exp(A))=\rho(e^{2\pi i})=\rho(1)=I.$

On the other hand,

$\exp(A)=\exp(D+N)=\exp(D)\exp(N),$

since $D$ and $N$ commute. As $D$ is diagonal $\exp(D)$ is diagonal, but $\exp(N)=\exp(D)^{-1}$ and as $N$ is nilpotent we must have $N=0$ . It follows that $\exp(N)=\exp(D)=\operatorname{Id}$ and $A$ is diagonal.

∎

Remark 2.11.3.

Complete irreducibility does not hold for representations of a general Lie group. For example, the standard representation of

N=\left\{\begin{pmatrix}1&x\\ 0&1\end{pmatrix}\,\bigg{|}\,x\in\mathbb{R}\right\}

does not decompose into a direct sum of two one-dimensional invariant subspaces (otherwise we would diagonalise $\begin{pmatrix}1&x\\ 0&1\end{pmatrix}$ , which is impossible). Furthermore, it is not unitary (as unitary matrices are diagonalisable).

Some of this actually generalises substantially:

Theorem 2.11.4.

(Maschke’s theorem for compact Lie groups) Let $G$ be a compact Lie group.

(i)

Any finite-dimensional representation of $G$ is unitarisable.
(ii)

(complete reducibility) Any finite-dimensional representation of $G$ is a direct sum of irreducible representations.

Proof.

The proof of the first part uses the same idea as for finite groups. Let $(\rho,V)$ be a finite-dimensional representation and take $(\,,\,)$ to be any Hermitian inner product on $V$ . Then define

\left\langle{\bf v},{\bf w}\right\rangle=\int_{g\in G}(\rho(g){\bf v},\rho(g){% \bf w})dg.

This is also a Hermitian inner product, and

$\displaystyle\left\langle h{\bf v},h{\bf w}\right\rangle$	$\displaystyle=\int_{g\in G}(\rho(gh){\bf v},\rho(gh){\bf w})dg$
	$\displaystyle=\int_{k\in G}(\rho(k){\bf v},\rho(k){\bf w})d(kh^{-1})$	(putting $k=gh$ )
	$\displaystyle=\int_{k\in G}(\rho(k){\bf v},\rho(k){\bf w})dk$	(since $dk=d(kh^{-1})$ )
	$\displaystyle=\left\langle{\bf v},{\bf w}\right\rangle.$

The challenge here is to show that there is an appropriate notion of $\int_{g\in G}f(g)dg$ for which the step “ $dk=d(kh^{-1})$ ” is valid — this goes by the name of ‘existence of Haar measure’. For $U(1)$ you can do it by hand, see problem 2.11.2.

The second part is proved exactly as for finite groups in Michaelmas (Theorem 5.12). Let $(\rho,V)$ be a finite-dimensional representation and let $W$ be a subrepresentation. Let $\left\langle\,,\,\right\rangle$ be the $G$ -invariant Hermitian inner product on $V$ guaranteed by part (i). Then the orthogonal complement $W^{\perp}$ is also a subrepresentation, and $V=W\oplus W^{\perp}$ . Iterating, we obtain that $V$ is a direct sum of irreducible representations.

∎

2.11.2. Exercises

Problem 27. Consider $G=\mathrm{U}(1)$ .

(a)

For $\varphi$ a continuous function on $G$ , we define its integral

$\int_{G}\varphi(g)dg=\tfrac{1}{2\pi}\int_{0}^{2\pi}\varphi(e^{it})dt.$

Note that $\int_{g}1dg=1$ . Show that

$\int_{G}\varphi(hg)dg=\int_{G}\varphi(gh)dg=\int_{G}\varphi(g)dg$

for any $h\in G$ .
(b)

Let $(V,\rho)$ be a finite dimensional representation of $G$ and let $(\,,\,)$ be any Hermitian form on $V$ . Define a new Hermitian form by

$({\bf v},{\bf w})_{\rho}=\int_{G}(\rho(g){\bf v},\rho(g){\bf w})dg.$

Show that $(\,,\,)_{\rho}$ is a $G$ -invariant Hermitian form on $V$ .
(c)

Conclude that every finite-dimensional representation of $\mathrm{U}(1)$ is completely reducible. (Compare this to the proof of Theorem 5.12 from Michaelmas).

Problem 28. Consider the orthogonal group $\operatorname{O}(2)$ .

(a)

Show that $\mathrm{SO}(2)$ has index $2$ in $\operatorname{O}(2)$ . Deduce that every element in $\operatorname{O}(2)$ can be uniquely written as $r_{\theta}$ or $r_{\theta}s$ with $s=\begin{pmatrix}0&1\\ 1&0\end{pmatrix}$ and $r_{\theta}$ the matrix for rotation by $\theta$ . Show that

$sr_{\theta}=r_{-\theta}s.$
(b)

Mimic the method we used for dihedral groups to classify all irreducible finite-dimensional representations of $\operatorname{O}(2)$ .

Problem 29. Let $V$ be the space of functions on $\mathbb{C}^{2}$ that are polynomials in the coordinates $x$ and $y$ . Consider the (left) action of $\operatorname{GL}_{2}(\mathbb{C})$ on $V$ given by

(g\varphi)({\bf v})=\varphi(g^{-1}v)

(here, think of ${\bf v}=\begin{pmatrix}x\\ y\end{pmatrix}\in\mathbb{C}^{2}$ as a column vector).

Compute the derived action for the “standard” basis of $\mathfrak{sl}_{2}(\mathbb{C})$ given by $X=\begin{pmatrix}0&1\\ 0&0\end{pmatrix}$ , $Y=\begin{pmatrix}0&0\\ 1&0\end{pmatrix}$ , and $H=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}$ . You should get something involving the partial derivatives $\frac{\partial}{\partial x}$ and $\frac{\partial}{\partial y}$ .