Derived Geometric Satake for the Basic Object

Definitions and mathematical statement

Let $G$ and $G^\vee$ denote the group and its Langlands dual intended in the supplied source. Let $X$ denote the geometric datum defining the basic object $\delta_X$, and let $M^\vee$ denote its proposed dual geometric object, equipped with the relevant $G^\vee$-action and grading. The source does not specify the admissible class of these data, the base field, or the sheaf coefficients.

Write $\mathcal C_X$ for the intended category containing $\delta_X$, and $\Phi$ for the applicable derived geometric Satake equivalence. Here $\mathcal C_X$ is a placeholder for a category whose precise identification is unresolved below. Write $(-)^{\mathrm{sh}}$ for the source's shearing operation, which reinterprets grading in cohomological terms; its degree conventions are not supplied. The notation $\mathcal O(M^\vee)$ denotes the structure sheaf or function algebra regarded as an object of the intended equivariant quasi-coherent category, whose precise realization also requires specification.

The intended conjectural assertion is that, for every datum in the source's unspecified range of applicability, \[ \Phi(\delta_X)\simeq \mathcal O(M^\vee)^{\mathrm{sh}}, \] and that the resulting identification of endomorphism algebras has the form \[ \operatorname{End}_{\mathcal C_X}(\delta_X) \simeq \left(\mathcal O(M^\vee)^{G^\vee}\right)^{\mathrm{sh}}. \] These formulas retain the source's notation: whether endomorphisms mean a derived endomorphism object, a graded algebra, or degree-zero endomorphisms, and whether invariants are ordinary or derived, is not determined by the record.

The extracted passage places $\delta_X$ in \[ \operatorname{Shv}_{G(D)}\left(G(D^\circ)/G(D)\right), \] where $D$ denotes a formal disk and $D^\circ$ its punctured disk, and places the proposed image in an equivariant quasi-coherent category over $\check{\mathfrak g}^{*}$, with $\check{\mathfrak g}=\operatorname{Lie}(G^\vee)$ and an additional $\mathbb G_m$-grading. Elsewhere the same extraction expresses the endomorphisms using $\operatorname{Shv}_{G_{\mathcal O}}(X_F)$ and $\operatorname{Shv}_{G_{\mathcal O}}(X_{\mathcal O})$, with objects $\delta_X$ and $\omega_X$, and describes the answer as equivariant Borel--Moore homology of $X_{\mathcal O}$, after shearing. The definitions of these spaces and objects, and the identifications connecting these categories, are absent. Consequently, the displayed conjecture is a conditional reconstruction of the intended target, not a fully specified universally quantified statement.

Short English statement

The proposed compatibility says that derived geometric Satake sends the basic object $\delta_X$ to the sheared structure sheaf of the proposed dual space $M^\vee$, with its endomorphisms corresponding to sheared invariant functions. The admissible geometric data, categories, shearing conventions, and meanings of endomorphisms and invariants require clarification. The supplied record labels the problem open; this draft does not independently verify that status.

Sources

• Local UnsolvedMath registry record supplied with this task: permanent ID $30006321$, title "Derived Geometric Satake for the Basic Object," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record assigns the year 2025 and names no individual proposer.
• Arbeitsgemeinschaft: Relative Langlands Duality (2025), Oberwolfach Reports, pp. 811--882, DOI 10.4171/owr/2025/18. Supplied source report and literature link: https://doi.org/10.4171/owr/2025/18. No narrower location for the conjecture is supplied.
• Relative Langlands Duality, supplied literature link: https://arxiv.org/abs/2409.04677. The supplied assessment refers to this as the BZSV paper; no author list or conjecture number is supplied.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open and states that the BZSV paper and the 2025 report present the basic-object assertion as a conjecture, provide examples, and report no general proof. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels its statement corrected_verified and reports that the official Oberwolfach report was checked and notation repaired. These are supplied provenance claims; no source visit or independent mathematical verification is asserted here.