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

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