Invariant quantum measure on $q$-deformed twisted adjoint orbits

Let $U$ be a compact semisimple Lie group with complexification $G$ and associated Cartan involution $\mathrm{\Theta }$. Let $\nu$ be an involutive complex Lie group automorphism of $G$ commuting with $\mathrm{\Theta }$, and consider the associated semisimple real Lie group $G_{\nu }=\{g\in G\mid \nu (g)=\mathrm{\Theta }(g)\}$. We consider $q$-deformed analogues of the $U$-orbits of the quotient space $G_{\nu }\mathrm{\backslash }G$, and determine for these the associated von Neumann algebra and invariant state.