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Let $X$ be a complex algebraic variety with an action of a connected algebraic group $G$. The forgetful functor from the category of $G$-equivariant perverse sheaves on $X$ to the category of perverse sheaves on $X$ is fully faithful, and its image is contained in the subcategory of perverse sheaves which are constructible with respect to the orbit stratification (see this question).

Under what condition is it the case that every that constructible with respect to the orbit stratification is equivariant?

I know that this is the case, for example, for the action of $G(\mathbb{C}[[t]])$ on the affine Grassmannian of a reductive group $G$ (this is proposition 2.1 in [MV]).

[MV] Mirković, I.; Vilonen, K., Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. Math. (2) 166, No. 1, 95-143 (2007); erratum ibid. 188, No. 3, 1017-1018 (2018). ZBL1138.22013.

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There is a monodromy action of $\pi_1(G)$ on every $G$-constructible perverse sheaf. It must be trivial for equivariant sheaves.

Simplest example is $\mathbb{C}^*$ acting on $\mathbb{C}$. In the description of perverse sheaves as representations of a quiver with two vertices and two arrows $u$ and $v$ the monodromy acts by $(1-uv)$ and $(1-vu)$ on each vertex.

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  • $\begingroup$ Is this a sufficient condition for equivariance, or only necessary? $\endgroup$
    – Antoine Labelle
    Commented Jul 13, 2023 at 3:20
  • $\begingroup$ @AntoineLabelle This seems to be sufficient, see Proposition 6.2.17 of Pramod Achar's book. It basically says that for a connected algebraic group $G$ acting on $X$ any isomorphism between the two pullbacks of a perverse sheaf on $X$ to $G\times X$ gives an equivariant structure. $\endgroup$
    – Sergey Guminov
    Commented Oct 4 at 15:21

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