EQUIVARIANT DERIVED CATEGORIES, QUOTIENT STACKS, AND GEOMETRIC CATEGORIFICATION OF REPRESENTATIONS

We develop a derived-categorical framework for the categorification of representations arising from algebraic group actions. Our approach unifies localization, descent, and equivariant structures within a single triangulated setting. Central to the construction is the exact localization functor LG : D b G Coh(X) −→ D b Coh([X/G]), defined for any affine algebraic group G acting on a qcqs scheme X. Under standard coherent-descent finiteness hypotheses (summarized in §2.4), the functor LG is the atlasinduced descent equivalence D b G Coh(X) ≃ D b Coh([X/G]); no linear reductivity is required for this equivalence. Linear reductivity is used later only for the exactness of invariants (−) G when comparing equivariant/global sections. Building on this equivalence, we construct a categorified character theory for Ad(G)-equivariant complexes with Weil structure in the constructible ℓ-adic category D b c (GF q , Q ℓ) Ad(G), Weil. We define derived character sheaves and a geometric trace map recovering classical characters via Frobenius traces, providing a unified description of Deligne-Lusztig and Steinberg representations within the derived formalism. Throughout, all proofs are carried out in the triangulated setting; we explicitly separate geometric descent assumptions from those ensuring invariant exactness, and present counterexamples delineating the boundaries of the theory. The framework culminates in conjectural extensions to derived motivic and ∞-categorical contexts, offering a coherent route from equivariant localization to categorified representation theory.