An Equivariant Landweber Exact Functor Theorem for Abelian Compact Lie Groups

We prove an equivariant Landweber exact functor theorem for abelian compact Lie groups $G$. For a graded module $N$ over the $G$-equivariant Lazard ring $L_G$, we give necessary and sufficient algebraic conditions on $N$ for the functor $$X⟼(M{U}_G{)}_{\ast }(X){\otimes }_{L_G}N$$ to define a homology theory on the category ${\mathrm{S}\mathrm{p}}^G$ of genuine $G$-spectra. Our criterion is obtained by varifying the Landweber exactness on each stratum indexed by a closed subgroup of $G$ and then gluing the local flatness data via Euler classes. As an application, we show that for any non-equivariant Landweber exact ring spectrum $E$, $M{U}_G{\wedge }_{MU}E$ is equivariantly Landweber exact, thereby proving a conjecture of Wisdom.