Discrete series representation

Remarks

For $u, v \in H$, the matrix coefficient of $\pi$ is the function $g \mapsto \ip{\pi(g) u}{v}$. Equivalently, $\pi$ is in the discrete series if and only if some nonzero matrix coefficient lies in $L^2(G)$. Then every matrix coefficient lies in $L^2(G)$.

Example: let $G$ be a compact group, with Haar measure of total mass $1$. Each matrix coefficient of a unitary representation is continuous and bounded by $\norm{u} \norm{v}$, so it lies in $L^2(G)$. So every irreducible unitary representation of $G$ is in the discrete series.

Non-example: the irreducible unitary representations of $G = \R$ are the characters $x \mapsto e^{ixt}$ with $t \in \R$. Each matrix coefficient has constant absolute value, so it is not square-integrable on $\R$. So $\R$ has no discrete series, and $L^2(\R)$ decomposes as a continuous direct integral by Fourier analysis.

For groups with non-compact center, such as $\mathrm{GL}_n(\R)$, one asks instead that the matrix coefficients be square-integrable modulo the center. For connected semisimple $G$ with finite center and maximal compact subgroup $K$, discrete series exist if and only if $\rank G = \rank K$, by a theorem of Harish-Chandra. For example, $\mathrm{SL}_2(\R)$ has discrete series, while $\mathrm{SL}_2(\C)$ has none.