Rotation of spherical harmonics lead out of representation space? The spherical harmonics $Y^m_l$ with differing $l$'s are the irreducible representations of the rotation Group SO(3). The representation space should be closed under group transformation. Furthermore the group elements rotate these functions in the usual way.
Visualizations of these functions can be found at wiki: https://en.wikipedia.org/wiki/Spherical_harmonics
If we look at a visualization of the spherical harmonic $Y^0_{l}$ for $l>1$ and rotate it 90° along the x (or y) axis it looks like the resulting function can no longer be expressed as a linear combination of $Y^m_l$. This would mean that the functions $Y^m_l$ are not closed under SO(3) transformations.
Is this problem just an artefact of the visualization and it is very much possible to express the rotated $Y^0_l$ as a linear combination of $Y_l^m$, or am I misunderstanding something about the concept of irreducible representations?
 A: In the Wikipedia article you linked, if you consider the section about rotations, you'll find that they partly answer your question: When applying a rotation to a spherical harmonic of degree $l$, this rotated spherical harmonic can itself be expressed as a linear combination of spherical harmonics of degree $l$, with coefficients given by this fairly complicated Wigner D-matrix.
Interpreting the visualization might indeed be a bit tricky, especially since it may not be clear how you go from the visualizations on these page to the original functions. 
For example, take a look at the real spherical harmonics for $l = 1$. In the visualization on top of the page, you get that their images correspond to the three dumbbell-shaped objects in the second row. But of course, the spherical harmonics take values on the sphere, so their images aren't literally these dumbbells. 
Quoting the subtext of the image: 
"The distance of the surface from the origin indicates the absolute value of $Y^m_l(\theta,\phi)$ in angular direction $(\theta, \phi)$." Thus, what this image actually represents is that for $l = 1$, the three real spherical harmonics are functions which are positive on one half of the sphere and negative on the opposite half, where the three different spherical harmonics correspond to viewing the halves of the sphere on the $x$-axis, the $y$-axis and the $z$-axis, respectively.
Now, if you rotate one of these spherical harmonics, you basically rotate these half spheres where the function takes positive/negative values. Indeed, if you rotate, for example, the $x$-half spheres into the $y$-half spheres, that corresponds to applying a rotation to the $Y_l^{-1}$ which made it into $Y_l^0$. Now, with that thought in mind, personally, I find it more believable that if I rotate my half spheres along some different axis, that I can then represent the result as a weighted sum of my $x$-, $y$- and $z$-half spheres. So, in this crude sense, any decomposition of a sphere into two half-spheres given by a rotated spherical harmonic is just a linear interpolation between the decompositions of the sphere into $x$-, $y$- and $z$-half spheres, corresponding to your original, unrotated spherical harmonics.
If you did not understand that last paragraph, however, I do not blame you, it feels a bit esoterical to me as well, it was just a way to make the images on Wikipedia plausible for my own brain, which might well work different from yours :) It is difficult to make a visual interpretation of these fairly difficult functions intuitive. Maybe it resonates with you regardless! But rest assured that you are not misunderstanding the spherical harmonics as representations of $SO(3)$.
