Categorical Functors from Soergel Bimodules to Flag Hilbert Schemes

Definitions and mathematical statement

Write \[ \mathcal A_n=K_b(\operatorname{SBim}_n),\qquad \mathcal B_n=D_b\left(\operatorname{Coh}^T(\operatorname{FHilb}_n({\mathbb{C}}))\right). \] Here $K_b$ denotes the bounded homotopy category, and $D_b$ denotes the bounded derived category. The notation $\operatorname{SBim}_n$ refers to the intended category of Soergel bimodules, while $\operatorname{Coh}^T(X)$ denotes the category of $T$-equivariant coherent sheaves on $X$. The source does not specify the realization, coefficient ring, or grading conventions for the Soergel bimodules; the scheme denoted by $\operatorname{FHilb}_n({\mathbb{C}})$; or the group $T$ and its action. The displayed scheme notation is retained exactly without substituting a different ambient space.

The supplied question asks whether there exist adjoint functors between $\mathcal A_n$ and $\mathcal B_n$ such that the functor designated as forward is monoidal and satisfies \[ F\left(L_1\otimes\cdots\otimes L_{k-1}\right) \cong C(FT_k). \] The symbols $L_i$, $FT_k$, and $C$ are retained from the source: their definitions, category membership, and conventions are not supplied. In particular, interpreting the $L_i$ as objects of $\mathcal B_n$ and $C(FT_k)$ as an object of $\mathcal A_n$ would require a functor \[ F:\mathcal B_n\longrightarrow\mathcal A_n, \qquad G:\mathcal A_n\longrightarrow\mathcal B_n. \] This is a conditional typing of the displayed requirement, not a verified identification of the intended direction. The record does not determine whether the intended adjunction is $F\dashv G$ or $G\dashv F$. Its title suggests functors from Soergel bimodules to flag Hilbert schemes, but does not resolve which functor is designated as forward in the displayed condition.

The monoidal structures, the meaning of monoidal functor, and the relevant equivariant and derived tensor conventions must also be specified. No ranges or quantifiers for $n$ and $k$, relation between these indices, or boundary convention for the tensor product are supplied. Thus the displayed existence question records the mathematical target, but these unresolved qualifications prevent it from being a fully specified conjecture.

Short English statement

The problem asks for adjoint functors relating Soergel bimodules and equivariant coherent sheaves on a flag Hilbert scheme, with one functor preserving the intended monoidal structure and carrying a specified tensor product to $C(FT_k)$. The supplied record leaves essential objects, functor directions, and index ranges undefined, so this is a statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30003223$, source problem code catalogue entry, titled "Categorical Functors from Soergel Bimodules to Flag Hilbert Schemes." The record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2016.
• Supplied source citation: Singularities (2017), Oberwolfach Reports, DOI 10.4171/owr/2016/46, pp. 2625--2691. https://doi.org/10.4171/owr/2016/46. This is also the literature source labeled "Oberwolfach Report 46." The background describes the reports as records of mathematical workshops and their open-problem discussions. The report itself has not been independently consulted for this draft.
• Supplied literature source: Gorsky--Negut--Rasmussen, Flag Hilbert schemes. https://doi.org/10.1016/j.aim.2020.107542. No conjecture number, page, or passage identifying the exact formulation is supplied.
• Literature assessment date: 2026-08-21. This is supplied dated curation, not independently verified. The supplied status is "open." The supplied assessment says that the proposed adjoint functors remain conjectural and that later work develops the functor and consequences without establishing the full stated conjecture. The background explicitly describes this as dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review claiming self-containment and preservation of the target. These are supplied review labels, not independent confirmation that the unresolved definitions and directions above are determined.