Adding some further detail to Will Sawin's answer, in some examples that are clearer than the general case:

For semi-simple real Lie groups $G$ obtained by "restricting scalars" of ("classical") _complex_ semi-simple Lie groups, Gelfand-Naimark in 1947 (I think...?) gave a Plancherel theorem for $L^2(G)$, involving _integrals_ of (the simplest) irreducible unitary repns, namely, the unitary principal series. The "Plancherel measure" was/is a uniform sort of real-line measure on the spectral parameters. They conjectured that these were _all_ the irreducible unitaries of such groups, but it was observed later that there were "unitary degenerate principal series" that occurred as proper subrepns of _non_-unitary (and not unitarizable) principal series. 

After work of Wigner and Bargmann in the late 1940s on some small examples of semi-simple groups not obtained by restriction of scalars on complex groups, in the early 1950s Harish-Chandra started considering the more general semi-simple real case. Although the "holomorphic discrete series" repns of classical groups "of hermitian type" had been implicit in Siegel's and other's early work on modular forms in several complex variables since the late 1930s, it was only in the 1950s that Harish-Chandra made explicit the point that, for such (non-compact) groups there are non-trivial "discrete series" repns, meaning, literally ("discretely") appearing in the decomposition of $L^2(G)$, rather than appearing "continuously".

It took Harish-Chandra many years to treat the _other_ discrete series repns of semi-simple real Lie groups, and, even then, he reasoned indirectly by constructing their characters rather than having models of the repns. 

In fact, constructions of models was an on-going process in the 1960s and 1970s, with W. Schmid, R. Langlands, G. Zuckerman, and others adding new ideas of various sorts. The historical notes in A. Knapp's Princeton Press volume on repn theory of semi-simple real Lie groups include more bibliographic pointers.