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

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