Definition 3

Let $\pi$ be an irreducible unitary representation of G in a Hilbert space E. One says that $\pi$ is square-integrable modulo the center if the function on $G/C$ deduced from the function $|f_{x,y}|$ by passing to the quotient belong to $\mathscr{L}^2(G/C)$ for every $(x, y)\in E\times E$.

This condition does not depend on the choice of a Haar measure on $G/C$.

If the matrix coefficients of $\pi$ belong to $\mathscr{L}^2$(G), one says that $\pi$ is square-integrable; the existence of an irreducible unitary square-integrable representation of G implies that the center of G is compact (exercise 5 of V, p. 487).

There exist groups G which admit no square-integrable representation, even modulo the center (cf. exercise 32 of V, p. 516).