Proposition 2

Let $\pi$ be an irreducible unitary representation of G in a Hilbert space E. The dimension of E is finite and equal to the formal degree of $\pi$.

Since the representation $\pi$ is square-integrable (lemma 1), its formal degree $c_\mu(\pi )$ relative to $\mu$ is defined (def. 4 of V, p. 423); it is a strictly positive real number. Let $(e_i)_{i\in I}$ be a finite orthonormal family in E. Let $x$ be an element of E of norm 1 (there exists one since E is nonzero). For $i\in I$, one has

(prop. 8 of V, p. 424). Let us sum this formula over $i\in I$. We obtain

Card(I) $=c_\mu(\pi )\int_G\sum_{i\in I}|\langle e_i|\pi (g)x\rangle |^2d\mu(g)$

by Bessel's inequality (EVT, V, p. 21, prop. 4). Hence the cardinal of I is bounded above by $c_\mu(\pi )$. This implies that the dimension of E is finite.

One may then apply the foregoing to an orthonormal basis $(e_i)_{i\in I}$ of E. One obtains, by EVT, V, p. 22, prop. 5,

dim(E) $=c_\mu(\pi )\int_G\sum_{i\in I}|\langle e_i|\pi (g)x\rangle |^2d\mu(g)$

which completes the proof.

In particular, it is therefore permissible to speak of the character $\chi_{\pi}$ of an irreducible unitary representation $\pi$ of G. It is a continuous function on G.