9 Problems for Epiphany 9.1 Problems for section 4 9.3 Problems for section 6

9.2 Problems for section 5

Problem 66.

1.

If $(\rho,V)$ is an irreducible finite-dimensional complex representation of $\mathfrak{g}$ and $\mathfrak{z}$ is the centre of $\mathfrak{g}$ (see problem 64), show that there is a linear map $\alpha:\mathfrak{z}\rightarrow\mathbb{C}$ such that $\rho(Z)v=\alpha(Z)v$ for all $Z\in\mathfrak{z}$ .
2.

For $\mathfrak{g}=\operatorname{GL}_{n,\mathbb{C}}$ , find $\mathfrak{z}$ . Find $\alpha$ when $V=\Lambda^{k}\mathbb{C}^{n}$ , where $\mathbb{C}^{n}$ is the standard representation and $1\leq k\leq n$ . These representations are in fact irreducible, though we haven’t proved that yet; you can just directly show that $\alpha$ exists.

Solution

Problem 67.

Prove that, for $X,Y\in\mathfrak{gl}_{n}$ ,

(\operatorname{ad}_{X})^{m}(Y)=[X,[X,\ldots,[X,Y]\ldots]]=\sum_{i=0}^{m}\binom% {m}{k}X^{k}Y(-X)^{m-k}.

Hence give a direct proof that $\exp(\operatorname{ad}_{X})=\operatorname{Ad}_{\exp(X)}$ .

Solution

Problem 68.

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

1.

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 $h\in G$ .
2.

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

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

Show that $(\,,\,)_{\rho}$ is a $G$ -invariant Hermitian form on $V$ .
3.

Conclude that every finite-dimensional representation of $\mathrm{U}(1)$ is completely reducible. (Compare Problem 12).

Solution

Problem 69.

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

1.

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

$sr_{\theta}=r_{-\theta}s.$
2.

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

Solution

Problem 70.

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

(here, think of $v=\begin{pmatrix}x\\ y\end{pmatrix}\in\mathbb{C}^{2}$ 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 $\frac{\partial}{\partial y}$ .

Solution

Problem 71.

Let $V=\mathbb{C}^{2}$ be the standard representation of $\operatorname{GL}_{2}(\mathbb{C})$ .

1.

Show that $\Lambda^{2}(V)\cong\det$ as Lie group representations.
2.

Show that $\Lambda^{2}(V)\cong\operatorname{tr}$ as representations of $\mathfrak{gl}_{2}(\mathbb{C})$ . (You could just ‘take the derivative’ of part (a), but please do it directly instead.)
3.

Find an explicit homomorphism $\rho:\operatorname{GL}_{2}(\mathbb{C})\rightarrow\operatorname{GL}_{3}(\mathbb% {C})$ corresponding to $\operatorname{Sym}^{2}(V)$ .

Solution