An arXiv preprint dated 26 Aug 2026 gives a formal answer to a question in geometric Langlands: what the equivalence assigns to the constant sheaf on BunG. Its main theorem identifies that image with Poincspec applied to the trivial central spectral skyscraper. Put plainly, the paper identifies the spectral-side object corresponding to the constant sheaf. This is a categorical identity in a specified mathematical setting, rather than a numerical estimate.
The result also describes the image of arbitrary finite-rank local systems on BunG. The description takes the form of a commutative functor diagram linking the standard morphism denoted pi_BunG, the Langlands equivalences for G and its abelianized form, and Poincspec between the automorphic and spectral categories. For a general reader, commutative means that the permitted routes through the diagram agree.
A conjecture made precise
The paper frames the constant-sheaf result as confirmation of an unpublished conjecture of V. Lafforgue concerning that image. The conjecture is treated as a target, and the paper presents a formal proof in the stated mathematical setting.
The setting is exact: a smooth projective curve X and a reductive group G over an algebraically closed field of characteristic zero. That makes this a proof-based mathematical study, with no human or animal cohort and no empirical dataset. Readers should therefore read the result as a theorem within that setting, not as a measurement about the wider world.
Following the proof across two sides
To reach the theorem, the paper moves between automorphic and spectral descriptions. It constructs a canonical Langlands equivalence for possibly disconnected tori, identifying D-mod(BunT) with the corresponding spectral category of quasi-coherent sheaves. Under this torus equivalence, geometric Langlands intertwines the automorphic action associated with the abelianized group and the spectral convolution action on the corresponding local-system category.
A second step controls singular support, the part of the geometry that records the directions in which a sheaf can carry information. The inverse-Langlands image of Poincspec is shown to factor through the centered singular-support subcategory on BunG. The spectral Whittaker functor, Whitspec, vanishes on the irregular-nilpotent subcategory of the spectral ind-coherent category.
The proof then compares constant-term constructions on the two sides. On the automorphic side, the corrected negative constant-term functor is identified with pullback along the map from BunT to BunG, up to the stated relative-dimension shift. For the trivial central spectral skyscraper, the inverse-Langlands image lies in the zero-singular-support subcategory of D-mod(BunG).
On the spectral side, the constant term of Poincspec is the pullback along the map from the spectral side for G to that for T, followed by the stated grading shear. The argument uses a geometric Langlands compatibility that identifies the automorphic and spectral constant-term functors. Together, these formulas put the two calculations into the same comparison.
The final step uses the shifted pullback from BunG to BunT. On the zero-singular-support subcategory, that functor is conservative and t-exact. In ordinary language, conservative means it retains enough information to detect the relevant object, while t-exact means it respects the category's cohomological structure. This lets the proof use the matching constant terms to identify the constant-sheaf image.
A result with a defined boundary
The same framework gives the paper's broader claim: arbitrary finite-rank local-system images can be described through the commuting functor diagram, while the constant sheaf is identified with the specified Poincspec object. The contribution is therefore a structural description of an equivalence, with the general case organized around the same automorphic and spectral machinery.
The boundary is clear. The result is confined to smooth projective curves, reductive groups and algebraically closed fields of characteristic zero, and it does not establish that the categorical formulas continue to hold beyond those assumptions. No empirical outcomes or statistical uncertainty are reported, because the work analyzes mathematical objects rather than a human or animal sample.
The document is marked as an arXiv preprint, version one, dated 26 Aug 2026. Its acknowledgments thank Sam Raskin and Wyatt Reeves for intellectual assistance and do not name a grant or funding source.
Paper data and sources
Original title: The image of the constant sheaf under the geometric Langlands equivalence
Authors: Kenta Suzuki
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text