[Moved here from MSE]


Consider a variety $X$ over a field $k$ (complex numbers is fine) with the action of a group scheme $G$, and a $G$-equivariant perverse sheaf $F$ over $X$.



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> **Question.** Is it true that there exists a stratification $\tau$ of $X$ which is $G$-equivariant and such that $F$ is $\tau$-constructible?
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For example, one could inspect the orbit stratification.


I am trying to use the characterization of invariant perverse sheaves as those perverse sheaves such that $act^\* F\simeq pr\_2^\*F$ where $act \colon G\times\_k X\to X$ is the action and $pr\_2:G\times X\to X$ is the second projection. But I cannot find the solution.



Are equivariant perverse sheaves constructible with respect to the orbit stratification?
Every complex of sheaves has a unique maximal open subset on which it is locally constant, because if it is locally constant on two open sets, it is locally constant on their union.


Let $U$ be then maximal open subset of $X$ where $F$ is locally constant. Then $U$ is $G$-invariant, because $gU$ is also the maximally open set on which $F$ is locally constant and so $U= gU$.


The complement $X - U$ of $X$ is also $G$-invariant, and $F$ remains $G$-equivariant on restriction to $X - U$. Now we can induct - take the maximal open subset of $X - U$ on which $G$ is locally constant, check it is $G$-invariant, and restrict $F$ to its complement, and so on.


