Marty's answer discusses the Plancherel formula, and in a comment on his answer, I mentioned Harish-Chandra's work on the Plancherel formula in the case of reductive Lie groups. Yemon Choi's answer also mentions the case of semisimple Lie groups as being easier than the general case. The point of this answer is to elaborate slightly on my comment, and to point out that, while the semi-simple case might be easier, it is a very substantial piece of mathematics; indeed, it is essentially Harish-Chandra's life's work.
When Harish-Chandra began his work, it was known (thanks to Mautner?) that semisimple Lie groups were type I, and hence that for such a group $G$, the space $L^2(G)$ admits a
well-defined direct integral decomposition into irreducibles (typically infinite
dimensional, if $G$ is not compact). However, this is a far cry from knowing the precise
form of the decomposition.
The most fundamental, and difficult, question, turns out to
be whether there are any atoms in the Plancherel measure, i.e. whether $L^2(G)$ contains
any non-zero irreducible subspaces, i.e. whether the group $G$ admits discrete series.
This was solved by Harish-Chandra, who established his famous criterion: $G$ admits discrete series if and only if it contains a compact Cartan subgroup. He also gave a complete enumeration of the discrete series representations up to isomorphism, and described
their characters via formulas analogous to the Weyl character formula.
Harish-Chandra then want on to describe
the Plancherel measure on $L^2(G)$ inductively in terms of direct integrals
of parabolic inductions of discrete series represenations of Levi subgroups of $G$.
(It is the appearance of Levi subgroups, which are always reductive but typically
never semisimple, that also forces one to generalize from the semisimple to the reductive case.)
After completing the theory of the Plancherel measure for reductive Lie groups, he then went on to develop the analogous theory for $p$-adic reductive groups. However, in this case, one still doesn't have a complete enumeration of the discrete series representations in general: there are certain "atomic" discrete series representations, called "supercuspidal", which have no analog for Lie groups, and which aren't yet classified in general (i.e. for all $p$-adic reductive groups).
Harish-Chandra's work, as well as standing on its own as an amazing edifice, was a central
inspiration for Langlands in his development of the Langlands program, and remains
at the core of the Langlands proram today.
For a very nice introduction to Harish-Chandra's work, and the surrounding circle of ideas,
one can read this article by Rebecca Herb.
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