Functoriality of toric coherent-constructible correspondence

A morphism from a diagonalizable group $G$ to the torus of a toric variety $X$ induces an action of $G$ on $X$. We prove the category of ind-coherent sheaves on the quotient stack is equivalent to the category of sheaves on a cover of a real torus with singular supports contained in the FLTZ skeleton, extending Kuwagaki’s nonequivariant coherent-constructible correspondence arXiv:1610.03214. We also investigate the functoriality of such correspondence for toric morphisms and inclusions of orbit closures.