I'm aware that there are similar questions, but none of them are really something that I can approach with my current knowledge of groups and representations (which is the first 3 chapters of a book on groups and representations for physicists). As the book introduces representations, it does so through the spherical harmonics $Y_{l,m}$, and specifically the case of a rotation by an angle $\beta$ about the y-axis, for the harmonic $$Y_{1,0} = \sqrt{\frac{3}{4\pi}} \cos \theta =\sqrt\frac{3}{4\pi}\frac{z}r{}$$.
If we identify $D$ with the matrix satisfying $$Y^{'}_{l,m} = \sum_{m'}D^l_{m'm}Y_{l,m'}$$ for this rotation, then the book shows (and I follow) that in this case $D^1_{00} = \cos \beta$, $D^1_{-10} = -D^1_{10} = \frac{1}{\sqrt{2}}\sin\beta$. It then states:
This representation is actually nothing more than the defining representation in disguise: it acts on the spherical harmonics $Y_{1,m}$, which are just simple linear combinations of $x/r$, $y/r$ and $z/r$. However, if we consider any value of $l$ other than $1$, we will encounter genuinely different representations.
My question then is what is a defining representation? If one could answer without going into too much detail beyond finite groups (the book has only touched on continuous groups so far, and I can't find a definition of defining representation in the shortly ensuing pages) that would be appreciated.
I will add that the book then proceeds to state that $Y_{00}$ has a genuinely different representation, as it does not transform at all due to spherical symmetry, and that the $l = 2$ case is also not a defining representation - it gives the explicit example that for a rotation of about the z-axis, each of the harmonics in the new representation is related to the harmonics in the old representation by $Y^{'}_{lm} = exp(-im\alpha)Y_{lm}$. So if one could also explain why specifically the $l = 1$ case is a defining representation, but the other two are not, I think that would help.