Solved-Conjecture Tail for Spherical Varieties

Definitions and mathematical statement

The supplied record does not contain a standalone conjecture: its extraction begins with "This conjecture" and omits the antecedent. Consequently, a precise general mathematical statement cannot be recovered from the supplied evidence.

The sole displayed example in the extraction has the reported data \[ G=\mathbb{G}_m,\qquad X=\mathbb{A}^{1},\qquad Z=\mathbb{G}_m,\qquad \check{G}_X=\mathbb{G}_m,\qquad V_X=T^{*}\mathbb{A}^{1}. \] Here $\mathbb{G}_m$ denotes the multiplicative algebraic group, $\mathbb{A}^{1}$ the affine line, and $T^{*}\mathbb{A}^{1}$ its cotangent bundle; the base field or base scheme is unspecified. The roles and definitions of $Z$, $\check{G}_X$, and $V_X$, the action of $G$ on $X$, and the nature, domain, and codomain of $\mu$ are not supplied. In particular, the phrase that $\mu$ is "as conjectured" states no recoverable condition on $\mu$.

The excerpt reports that the omitted conjecture is known in many cases due to references [3, 4], with a further reference to [1, § 9], and that the talk verified three examples. Only the first example is present in this record. These are supplied source claims, not a formulation or independent verification of the omitted conjecture. The missing hypotheses, quantifiers, and asserted relationship must be recovered before a mathematical conjecture can be stated.

Short English statement

This record preserves the end of an omitted conjecture and one reported example involving the multiplicative group and the affine line. It supplies no standalone mathematical assertion to formalize. Its partially_solved label and reports of verified cases are supplied curation; the intended conjecture remains unrecoverable from this record.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json, permanent ID $30006315$, source problem code catalogue entry, title "Solved-Conjecture Tail for Spherical Varieties." The supplied record gives proposed year 2025 and names no proposer.
• Arbeitsgemeinschaft: Relative Langlands Duality (2025), Oberwolfach Reports, pp. 811--882, DOI 10.4171/owr/2025/18: https://doi.org/10.4171/owr/2025/18. The extraction includes a page marker for p. 841. The supplied statement-verification note says that the official report confirms this is the concluding sentence of an earlier conjecture, followed by examples, and contains no standalone formulation. This verification account is supplied metadata and has not been independently checked here.
• Ben-Zvi, Sakellaridis, and Venkatesh, Relative Langlands Duality: https://www.math.ias.edu/~akshay/research/BZSVpaperV1.pdf. This is a supplied literature reference; the record does not identify it with any particular numbered reference in the excerpt.
• The excerpt attributes known cases to [3, 4] and also cites [1, § 9]. Their bibliographic identities are not supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved, says that the conjecture is known in many cases and that the report verifies three examples, and says that the general relative-Langlands statement is not proved in the cited material. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.