DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Global sections and localization at a fixed infinitesimal character

Definition

Fix $κ \in h^{*}$ and $A_{κ}$ , $D_{κ}$ as in Flag-variety twisted differential operators and their modules. By Global sections of twisted differential operators on the flag variety, the sheaf $D_{κ}$ is a right $A_{κ}$ -module by multiplication by global sections, and $Γ (X, -)$ takes values in left $A_{κ}$ -modules through the infinitesimal action. Define ${Loc}_{κ} (V) : = D_{κ} \otimes_{A_{κ}} V, V \in A_{κ} - M o d,$ where the tensor product is the sheaf tensor product. Define $Γ_{κ} (M) : = Γ (X, M)$ for a quasi-coherent left $D_{κ}$ -module $M$ . Localization is right exact as a tensor functor. The next item proves the adjunction and verifies that these constructions land in the stated categories.

Depends on

Used by

Dependency tree · two levels

21 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Masaki Kashiwara, Representation theory and D-modules on flag varieties, Astérisque 173–174 (1989) (standard reference, not scraped)
Yi Sun, D-modules and representation theory (standard reference, not scraped)