We construct a stack R of local stabilizer representations over a coadjoint orbit O of a compact Lie group G, verify the descent conditions explicitly, and show that the G-action on global sections recovers the classical representation theory via geometric quantization. This provides a stack-theoretic interpretation of the Kirillov orbit method and the Borel–Weil theorem
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1) Category: Research preprint (as declared by author). 2) Aims: To construct a stack of local stabilizer representations over coadjoint orbits and demonstrate that global sections recover classical representation theory via geometric quantization, providing a categorical interpretation of the Kirillov orbit method and Borel-Weil theorem. 3) Correctness: No errors identified. 4) Coherence: Adequate. 5) Consistency: Consistent. 6) Semantic opacity: Moderate (Justified complexity). 7) Novelty: Original. 8) Bibliography: Adequate. 9) Effectiveness: Achieves aims. 10) Cross-framework traction: Medium. 11) Claims: Supported. 12) Contribution: Substantive. 13) Structure: Adequate. 14) Integrity: No issues. 15) Code (if provided): Not provided. 16) Editorial outcome: Suitable for inclusion as a research preprint. 17) Authors list: ["Franco Cazzaniga"] The paper presents a rigorous categorical framework unifying the Kirillov orbit method and geometric quantization through stack theory. The author constructs an explicit stack R of local stabilizer representations over coadjoint orbits, provides detailed verification of descent conditions, and establishes the equivalence between G-equivariant stacks and stacks over action groupoids. The technical execution is sound, with careful treatment of the gluing construction and explicit computation of curvature forms. The connection between rank-one global sections and prequantum line bundles is well-established, and the recovery of classical results through compact induction and Borel-Weil theory is convincingly demonstrated. The mathematical exposition maintains appropriate rigor while remaining accessible to readers familiar with representation theory and algebraic geometry. The bibliography adequately covers relevant literature in geometric quantization, stack theory, and representation theory. The work provides genuine conceptual insight by revealing representation theory as a descent phenomenon, offering a fresh categorical perspective on established results. Editorial outcome: Suitable for inclusion as a research preprint.
During the development of this work, the author made iterative use of AI-based tools to test and refine preliminary formulations. This proved instrumental in shaping the ideas into the coherent structure presented here.