Cardinalities of set of equivalence classes of Lie groups' and algebras' representations. I would like to ask about the cardinality of the sets of irreducible, inequivalent representations of Lie groups or Lie algebras. I will use the term irreps (of a group/algebra) to refer to finite-dimensional, inequivalent and irreducible representations.
I know that, for example, $SO(3)$ has irreducible representations only in odd dimensions (hence inequivalent). I imagine therefore the set of irreps is countable (correct?).
Then, there are irreps of $SU(2)$ (equivalently, $\mathfrak{su}(2)$, or its complexification $\mathfrak{su}(2)_{\mathbb{C}} = \mathfrak{sl}(2,\mathbb{C}$)), which are indexed, e.x. by physicists, by $s\in \frac{1}{2}\mathbb{Z_{+}}$, acting on a vector space of polynomials in two complex variables. Therefore, there are infinitely countably many irreps.

*

*Are there some easy to list general results about such classification, whether we concern ourselves with:

a) finitely dimensional representations,
or
b) infinitely-dimensional representations?


*In the general case, is the set of irreps of compact/noncompact Lie group or a Lie algebra infinite, and what is its cardinality?

And in the case of negative answer to 2.:


*Can there be finitely many irreps for either a compact or a non-compact Lie group?

I have not found clear and concise statements in the literature and would be grateful for suggestions.
 A: All irreducible representations of a compact Lie group are finite-dimensional, and a compact Lie group has countably-many irreducible representations. This follows for example from my answer here. For a compact Lie group you are not going to have only finitely-many irreducible representations unless $L^2(G)$ is finite-dimensional. This will not happen unless $G$ is finite.
For non-compact Lie groups there will be in general infinite-dimensional irreducible representations, and continuum-many irreducible representations. The more interesting thing to do is sort them into various families each depending, say, on a single real parameter. This is due in greatest part to Harish-Chandra and the philosophy he developed, and is many decades of work starting in the late 1940s, essentially, with Bargmann's classification of unitary irreducible representations of $SL_2(\mathbb{R})$ (and it's rather easy to see using Weyl's unitary trick that there are no finite-dimensional unitary representations at all in this case).
