Sur les représentations mixtes des groupes de Lie résolubles exponentiels

Let $G$ be an exponential solvable Lie group, $H$ and $A$ two closed connected subgroups of $G$ and $\sigma$ a unitary and irreducible representation of $H$. We prove the orbital spectrum formula of the Up-Down representation $\rho(G,H,A,\sigma)=\operatorname{Ind}_H^G\sigma_{\vert A}$. When $G$ is nilpotent, the multiplicities of such representation turns out to be uniformly infinite or finite and bounded. A necessary and sufficient condition for the finiteness of the multiplicities is given. The same results are obtained when $G$ is exponential solvable Lie group, $H$ and $A$ are invariant.