The $L_p$-dual space of a semisimple Lie group

Let $G$ be a semisimple Lie group. We describe the irreducible representations of $G$ by linear isometries on $L_p$-spaces for $p\in (1,+\infty)$ with $p\neq 2.$ More precisely, we show that, for every such representation $\pi,$ there exists a parabolic subgroup $Q$ of $G$ such that $\pi$ is equivalent to the natural representation of $G$ on $L_p(G/Q)$ twisted by a unitary character of $Q.$ When $G$ is of real rank one, we give a complete classification of the possible irreducible representations of $G$ on an $L_p$-space for $p\neq 2,$ up to equivalence.