On fixed point property for $L_p$-representations of Kazhdan groups

Let $G$ be a topological group with finite Kazhdan set, let $\mathrm{\Omega }$ be a standard Borel space and $\mu$ a finite measure on $\mathrm{\Omega }$. We prove that for any $p\in [1,\mathrm{\infty })$, any affine isometric action $G \curvearrowright L_p(\Omega, \mu)$ whose linear part arises from an ergodic measure-preserving action $G\curvearrowright (\mathrm{\Omega },\mu )$, has a fixed point.