Harish-Chandra Plancherel theorem

Remarks

The existence and uniqueness of $\mu$ come from the abstract Plancherel theorem for groups, since Harish-Chandra proved that $G$ is a type I group. Harish-Chandra's work determines $\mu$ completely: the discrete series of $G$ appear as atoms with mass $d_\sigma$, and each series $\pi_{P, \sigma, \nu}$ contributes a measure in $\nu$ with density $p_P(\sigma, \nu)$. The proof uses Harish-Chandra's Schwartz space, the classification of the discrete series by their characters, and the asymptotics of Eisenstein integrals governed by the Harish-Chandra c-function. The group $G$ has discrete series if and only if $G$ and a maximal compact subgroup $K$ have equal rank; otherwise the term for $P = G$ is absent.

Example: for $G = \mathrm{SL}(2, \R)$ there are two cuspidal parabolic classes, $G$ itself and the minimal parabolic subgroup. Up to normalizing constants, the discrete series of lowest $K$-type $n \ge 2$ have formal degree proportional to $n - 1$. Induce the even and odd principal series from the characters $\mathrm{diag}(a, a^{-1}) \mapsto \abs{a}^{i \nu}$ and $\mathrm{diag}(a, a^{-1}) \mapsto \sgn(a) \abs{a}^{i \nu}$. Then they carry densities proportional to $\nu \tanh(\pi \nu / 2)$ and $\nu \coth(\pi \nu / 2)$.

Going deeper

For a student

The theorem is the nonabelian analogue of Fourier inversion $f(0) = \int \hat f(\xi) \, d\xi$ on $\R^n$: the value $f(e)$ is recovered from the characters of $f$. Its restriction to bi-$K$-invariant functions is the Plancherel theorem for spherical functions, where only the spherical principal series appear and the density is $\abs{c(\lambda)}^{-2}$. The densities $p_P(\sigma, \nu)$ are built from Harish-Chandra's $\mu$-functions, which generalize $\abs{c}^{-2}$. The result is a cornerstone of the trace formula and of the Langlands program for real groups.