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My aim is to study some special representations of $SO(3)$ using characters, and to do this I need an explicit expression of the Haar measure on $SO(3)$. I've found some "versions" of the Haar measure on $SO(3)$, and the most common one seem to be the acquired by parameterizing $SO(3)$ using Euler angles (can be found here).

Since the characters of $SO(3)$ only depend on the angle from the axis-angle representation, it seems as if it would be suitable to find a Haar measure based on this parameterization. However, I don't really know where to start, and I've searched the web like a maniac.

Could someone provide some help?

Thanks in advance!

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The Haar measure can be explicitly calculated as follows in terms of a given parametrization $R(\phi,\theta,\psi)$ of $SO(3)$.

The first step is to define a metric thereon, which can be done by means of the Killing form: denoting by $X$, $Y$ and $Z$ the generators of the $so(3)$ algebra, $$ [X,Y]=Z\,,\qquad [Y,Z]=X\,,\qquad [Z,X]=Y\,, $$ the Killing form on $so(3)$ is defined as the symmetric, bilinear form $(\cdot\,,\cdot)$ on $so(3)$ satisfying $$ (X,X)=(Y,Y)=(Z,Z)=1\,,\qquad (X,Y)=(Y,Z)=(Z,X)=0\,. $$ In fact, one can check that this Killing form is indeed invariant under the Lie product, $([A,B],C)+(B,[A,C])=0$.

Now that we have a metric on the tangent space to the identity, we can define a metric at any point as follows: let $\dot R=\frac{d}{dt}R(t)$ be some vector, defined at the point $R=R(t)$; we may transport it back to the tangent space to the identity by means of the map induced by left multiplication. In this case, this simply means taking $\Omega=R^T\dot R$, which is indeed an element of the Lie algebra. Then, we may define a metric $ds^2$ by means of $ds^2=(\Omega, \Omega)\,dt^2$, the so-called Killing metric.

More explicitly, in the fundamental representation, we have $$ (A,B)=\frac{1}{2}\mathrm{tr}(A^TB)\, $$ and $$ ds^2=\frac{1}{2}\mathrm{tr}(\Omega^T\Omega)\,dt^2\,. $$ For instance, employing the Euler parametrization $$ R(\phi,\theta,\psi)=R_z(\phi)R_y(\theta)R_z(\psi) $$ with $\phi,\psi\in[0,2\pi)$ and $\theta\in[0,\pi)$, $$ R_z(\phi)=\left( \begin{matrix} \cos\phi & -\sin\phi & 0\\ \sin\phi & \cos\phi &0\\ 0 & 0 & 1 \end{matrix} \right) \,,\qquad R_y(\theta)=\left( \begin{matrix} \cos\theta & 0 & \sin\theta\\ 0 & 1 & 0\\ -\sin\theta &0 & \cos\theta \\ \end{matrix} \right) $$ you should obtain (after a somewhat lengthy calculation) $$ ds^2= d\phi^2 + 2\cos\theta\, d\phi\, d\psi + d\theta^2 + d\psi^2\,. $$ The determinant of this metric, $(\sin\theta)^2$, provides the Haar measure/volume form by means of $$ \omega = \sqrt{\mathrm{det}g}\,d\phi\wedge d\theta\wedge d\psi= \sin\theta \,d\phi \wedge d\theta\wedge d\psi. $$ Note that this measure is automatically right- and left-invariant, since so is the Killing metric.

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