Suppose that we have (to fix ideas) a unitary irreducible representation of a semi-simple Lie group $G$ (such as the non-compact Lie groups that you write down) on a Hilbert space. (Here I mean irreducible in the Hilbert space sense, i.e. there are no proper invariant closed subspaces.) Let's call the Hilbert space $V$ (just to give it a name).
A theorem of Harish-Chandra then says that $V$ is admissible, which means the following: fix a maximal compact subgroup $K$ of $G$. Then each irreducible representation $W$ of $K$ appears with finite multiplicity as a subrepresentation
of $V$. If we call this multiplicity $m_W$, then we may write
$V = \hat{\oplus}_W W^{m_W}$, i.e. as the Hilbert space direct sum (i.e. the completed direct sum) of the various $W$, each appearing with multiplicity $m_W$. (This is a consequence of the Peter--Weyl theorem, and is true
for any unitary representation of a compact group in which each irrep. appears
with finite multiplicity.)
Now inside $\hat{\oplus} W^{m_W}$ we have the actual algebraic direct sum
$\oplus_W W^{m_W}$, and this has an intrinsic characterization as a subspace
of $V$, as the $K$-finite vectors. (A vector $v$ is called $K$-finite if the
linear span of all its translates by elements of $K$ is finite-dimensional.)
Let's denote it by $V_K \subset V$.
It turns out that $V_K$, although it is not invariant under the action of $G$
(typically, unless $V$ happens to be finite-dimensional, which it usually won't be), is invariant under $\mathfrak g$, the Lie algebra of $G$.
One calls $V_K$ a $(\mathfrak{g},K)$-module, or also a Harish-Chandra module
(because it has actions of $\mathfrak{g}$ and $K$). It turns out that
$V_K$ determines $V$, and the basis of Harish-Chandra's approach to the study
of unitary reps. of $G$ is to work instead with the underlying $(\mathfrak g,K)$-modules.
Now in principle, to recover $V$, one really needs $V_K$ as a $(\mathfrak g, K)$-module; i.e. forgetting $\mathfrak g$ and just remembering the $K$-action is throwing away a lot of information.
But in practice (at least in the examples that I know) non-isomorphic irreducible $V$ have different list of multiplicities $m_W$, and so just knowing
$V_K$ as a $K$-rep. may well already pin down $V$.
In fact, often one doesn't even have to know all the $m_W$, but just
the first non-zero vanishing value. (If I think of the reps. $W$ as being
labelled by their highest weights lying in some choice of dominant Weyl chamber for $K$.)
A good place to read about this (it is not short, but I found it very good for dipping into) is Knapp's book Representation theory of semisimple groups: an
overview based on examples. He gives the basic definitions, a lot of examples, and goes on to develop various aspects of the theory (e.g. the theory of
the relationship between $V$ and the multiplicities $m_W$: this is known as
the theory of $K$-types).
Incidentally, Harish-Chandra was a student of Dirac, and (as far as I know) his study of unitary reps. of semisimple groups was inspired by Bargmann's treatment of the special case of $SL_2(\mathbb R)$, which was in turn inspired in part by the role of this group in physics.
On the other hand, I don't know of a treatment of the theory which directly relates it to the physics literature, and I can't parse the physics argument that you wrote down in detail.
