Remark 2

Let $\kappa ': G\rightarrow \mathbf{R}_+^*$ be such that

for every $(x, h)\in G\times H$. Let $\nu '$ be the quasi-invariant measure $(\kappa '\cdot \mu)/\beta$ on $G/H$. The function $\kappa '\kappa^{-1}$ defines, by passing to quotients, a continuous function $\xi : G/H\rightarrow \mathbf{R}_+^*$ such that $\nu '=\xi \cdot \nu$. The endomorphism $\alpha$ of $\mathscr{K}_{\pi}(G)$ defined by $f\mapsto (\kappa '\kappa^{-1})^{1/p}f$ satisfies

and makes it possible to identify the spaces $\mathscr{L}_{\pi}^p(G, \nu ')$ and $\mathscr{L}_{\pi}^p(G, \nu )$, as well as the spaces $L^p_{\pi}(G, \nu ')$ and $L^p_{\pi}(G, \nu )$. Moreover, $\alpha$ induces an isometric isomorphism of the representations Ind$^G_H(\pi , \kappa ')$ and Ind$^G_H(\pi , \kappa )$.