
## 9. Highest Weight Structure on Categories of Perverse Sheaves

1. #### Problem 9.1.

[Mautner] Is there a geometric proof (i.e. avoiding Geometric Satake) that $\operatorname{Perv}_{\mathrm{sph}}(Gr, k)$ is highest-weight?
• #### Problem 9.2.

More generally, when is a category of perverse sheaves constructible with respect to a fixed stratification highest-weight?
• #### Problem 9.3.

[Braden–Mautner] Given a symplectic resolution $\pi\colon Y \to X$, with $G = \operatorname{Aut}(X, \omega)$, consider the subcategory of $\operatorname{Perv}_G(X, k)$ generated by subquotients of $\pi_*\underline{k}_Y$. When is this highest-weight?

Cite this as: AimPL: Sheaves and modular representations of reductive groups, available at http://aimpl.org/sheavemodular.