# Are equivariant perverse sheaves constructible with respect to the orbit stratification?

[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 $$X$$.

Question. Is it true that there exists a stratification $$\tau$$ of $$X$$ which is $$G$$-equivariant and such that $$F$$ is $$\tau$$-constructible?

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.

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.