The standard construction for cocycle twisted and braided tensor product W$^*$-algebras

Given a locally compact quantum group $\mathbb{G}$ and a (generalized) dual unitary $2$-cocycle $\hat{\Omega}$, any W$^*$-algebra $A$ with a $\mathbb{G}$-action can be twisted into a new W$^*$-algebra $A_{\hat{\Omega}}$ with an action by the cocycle twist $\mathbb{G}_{\hat{\Omega}}$ of $\mathbb{G}$. We show how, in general, the standard space $L^2(A_{\hat{\Omega}})$, with its standard $\mathbb{G}_{\hat{\Omega}}$-representation, can be seen as a twist of $L^2(A)$ with its standard $\mathbb{G}$-representation. We then apply this general result in the special case of (generalized) Drinfeld doubles.