The finite dimensional irreducible unitary representations of $SU(2)$ are labelled by $j$ which needs to be half-integer, the dimension of the representation is $2j+1$. This is well-known, all is good.

If we do not require finite dimension for the representation, is it possible to make sense of representations with an arbitrary real number $j$? They will, presumably, be infinite dimensional but hopefully still unitary.

In the half-integer case when represented on at most $2j$ degree holomorphic polynomials, the 3 basis elements of the Lie-algebra in representation $j$ act as

$e_1 = \frac{1-z^2}{2}\frac{d}{dz} + jz$

$e_2 = \frac{1+z^2}{2i}\frac{d}{dz} + ijz$

$e_3 = -z\frac{d}{dz} + j$

Clearly, if $j$ is not half-integer and we start from $f(z) = z$ and start acting on it with $e_i$, it will generate an infinite dimensional space.

This kinda gives me the feeling that perhaps non-half-integer $j$ representations are still meaningful and are infinite dimensional. But I'm not sure, is this really the case or something will go wrong?

Basically what I'm asking is whether analytic continuation in $j$ makes any sense.