Definition 4

Let Z be a closed subgroup of C such that $C/Z$ is compact. Let $\pi$ be an irreducible unitary representation of G which is square-integrable modulo the center. The unique real number $c >0$ which satisfies the property of Proposition 7 is called the formal degree of $\pi$ relative to Z. It is denoted by $c_Z(\pi )$.

The formal degree depends on the choice of the Haar measure on Z. If the Haar measure $\beta_Z$ on Z is multiplied by a real number $t >0$, then the measure $\nu_Z=\mu/\beta_Z$ on $G/Z$ is multiplied by $t^{-1}$, and the formal degree of $\pi$ is multiplied by $t$.

The formal degree is characterized by the following property: