Equivariant D-module/perverse sheaf equivalence in characteristic zero

Establish an equivalence of categories between the equivariant bounded derived category of G-equivariant holonomic algebraic D-modules on a G-variety X over a characteristic-zero field and the equivariant bounded derived category of perverse sheaves on X. Concretely, for a complex algebraic group G acting on a variety X (over a characteristic-zero field, e.g., K = C), construct a canonical equivalence D_G^b(Hol(D_X)) ≃ D_G^b(Per(X(K), K–Vect)), where D_G^b(Hol(D_X)) denotes the equivariant bounded derived category of holonomic D-modules on X and D_G^b(Per(X(K), K–Vect)) denotes the equivariant bounded derived category of perverse sheaves of K-vector spaces on the analytic space X(K).

Background

Beilinson’s Theorem identifies, in the non-equivariant setting, the bounded derived category of perverse sheaves with the constructible derived category, and in characteristic zero relates holonomic D-modules to perverse sheaves via Riemann–Hilbert-type results. This paper establishes several equivariant analogues for perverse sheaves (and related settings), proving equivalences between equivariant derived categories and equivariant derived categories of perverse sheaves.

However, the D-module counterpart in the equivariant setting is not addressed here. The author notes the absence of an equivariant analogue of Beilinson’s D-module/perverse equivalence and conjectures that such an equivalence should hold. A key technical obstacle mentioned is the lack of a constructed pseudofunctor of derived categories of complexes of D-modules with bounded holonomic cohomology suited to the paper’s pseudofunctorial framework.

Resolving this problem would complete the equivariant extension of Beilinson’s framework by providing an equivariant Riemann–Hilbert-type equivalence between holonomic D-modules and perverse sheaves under group actions in characteristic zero.

References

The reader familiar with will note that we are missing the equivariant analogue of the last statement of Theorem 1.3: the (suggested) equivalence of the equivariant derived category of equivariant holonomic $D$-modules and the equivariant derived category of perverse sheaves for a characteristic zero field. While I conjecture such an equivalence holds and the proof of Theorem \ref{Thm: Equivariant Beilinson} may be adapted to even prove ssaid euivalence, it is nothing more than my ignorance of the $D$-module formalism and the nature or existence of a candidate pseudofunctor of derived categories of complexes of $D$-modules with bounded holonomic cohomology that led to the absence of such a result here.

— On the Equivariant Derived Category of Perverse Sheaves (2401.10174 - Vooys, 18 Jan 2024) in Remark, Section “Equivariant Beilinson’s Theorem and Ext Functors”