Proposition 9

Let $\pi_1$ and $\pi_2$ be nonisomorphic irreducible unitary representations of G in Hilbert spaces $E_1$ and $E_2$. Suppose that $\pi_1$ and $\pi_2$ are square-integrable modulo the center and that the restrictions to Z of their central characters coincide. Then

for every $(x, x', y, y')\in E^2_1\times E^2_2$.

For $i= 1, 2$, let $w_i$ denote the morphism of Proposition 7 for the representation $\pi_i$. By Lemma 8, b) of V, p. 384 and assertion b) of Proposition 8 of V, p. 394, the image of $w_i$ is contained in the $\pi_i$-isotypical component of $\boldsymbol{\delta }_{G,\chi}$. By assertion a) of loc. cit., the image of $w_1$ is therefore orthogonal to the image of $w_2$. Consequently, one has $\langle w_1(x\otimes x')|w_2(y\otimes y')\rangle = 0$ for every $(x, x', y, y')\in E^2_1\times E^2_2$, which is the required formula.