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.



Infinite dimensional unitary representations of SU(2) for non-half-integer j?
I just wrote this answer on your last question. To summarize, you can't have an irreducible representation of a compact group that is infinite dimensional, unless the representation space is very exotic.


By the Peter-Weyl Theorem, all irreducible Hilbert space representations of a compact group (e.g. SU(2)) are finite dimensional. Thus, any infinite dimensional Hilbert space representation will be reducible.


What can be said for non-Hilbert space representations? Given a compact group G acting irreducibly (and continuously) on a locally convex topological vector space V, you can inject V into L2(G) by sending v in V to the function cv(g)=〈g⋅v,v'〉 where v' is basically any nonzero element of the continuous dual of V (this is where local convexity is used). (Note that the irreducibility of V implies the injectivity of the map.)


Thus V is finite dimensional: its image is not necessarily closed in L2(G), but, since V is irreducible, its image has to lie within a single irreducible component of L2(G), which are all finite dimensional.


So to find a representation that is infinite dimensional and irreducible, you'd have to look at non-locally convex vector spaces (actually, you need the dual space to not separate points). Like Lp with p<1.


