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Let $X$ be a CW complex with a torus action $T$. Is there an established definition in equivariant stable homotopy theory of $T$-equivariant Morava K-theory, $K_p(n)^*_T(X)$? Any explicit references or discussion of this would be appreciated.

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    $\begingroup$ For finite $G$ there are several good reasons to define $K(p,n)_G^*(X)=K(p,n)^*(\phi^G(X))$ for $G$-spectra $X$. Here $\phi^G$ is the geometric fixed point functor, which satisfies $\phi^G(\Sigma^\infty Y)=\Sigma^\infty Y^G$ for $G$-spaces $Y$. I think that this is also the appropriate definition for compact Lie groups $G$, but I am less able to produce a detailed justification for that. $\endgroup$
    – Neil Strickland
    Commented Mar 31 at 21:42

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