Derived Equivalence for Spherical Hamiltonian Duality

Definitions and mathematical statement

Let $k$ be either a finite field or ${\mathbb{C}}$, and put $\mathfrak o=k[[t]]$, the ring of formal power series in $t$ over $k$. The source uses a field $F$ without defining it. If the intended convention is $F=\operatorname{Frac}(\mathfrak o)$, then $F=k((t))$, the field of formal Laurent series; this interpretation requires confirmation.

The source refers to objects $G$, $X$, $\check G$, $\check G_X$, and $\check V_X$, and to a Hamiltonian space described by \[ \check M\simeq\check V_X\times_{\check G_X}\check G. \] Their definitions, base fields, actions, and hypotheses are not supplied. In particular, the title suggests spherical data, but the source does not specify the class of spherical varieties or other data to which the question applies. The displayed product is retained exactly as supplied: whether it denotes a contracted product or a fiber product, and the actions or maps needed to define it, remain unresolved.

For purposes of recording the question, write $\mathcal D_{G(\mathfrak o),\mathrm{constr}}(X(F))$ for the source's intended derived category of $G(\mathfrak o)$-equivariant constructible sheaves on $X(F)$, and write $\mathcal C^{\mathrm{dg}}_{\check G,\mathrm{coh}}(\check M)$ for its intended differential graded category of $\check G$-equivariant coherent sheaves on $\check M$. These are descriptive placeholders, not fully specified categorical constructions.

Question as supplied. For the intended data and for each allowed choice of $k$, does there exist an equivalence \[ \mathcal D_{G(\mathfrak o),\mathrm{constr}}(X(F)) \simeq \mathcal C^{\mathrm{dg}}_{\check G,\mathrm{coh}}(\check M), \qquad \check M\simeq\check V_X\times_{\check G_X}\check G? \] A precise conjecture requires specification of the admissible data, the sheaf theory and coefficients, the topology or geometric model underlying $X(F)$, the boundedness and finiteness conditions on both categories, their enhancements, and the meaning of equivalence. The supplied formulation compares a derived category with a differential graded category without specifying a common categorical setting. No particular resolution of these issues is asserted here.

Short English statement

The source asks whether equivariant constructible sheaves on $X(F)$, for $\mathfrak o=k[[t]]$ with $k$ finite or complex, admit a derived equivalence with equivariant coherent sheaves on the displayed dual Hamiltonian space. The geometric data and categorical conventions are missing, so this records the intended question with explicit qualifications rather than a fully determined conjecture.

Sources

• Local UnsolvedMath registry, permanent ID $30004254$, source problem code catalogue entry, title "Derived Equivalence for Spherical Hamiltonian Duality," in the collection "Oberwolfach Reports Open Problems." The supplied proposed year is 2019; no individual proposer is identified.
• Supplied report citation: New Developments in Representation Theory of p-adic Groups (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/44, pp. 2739--2819. https://doi.org/10.4171/owr/2019/44. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. No exact page or passage for this question is supplied.
• Supplied literature source, labeled "OWR 2019 44": https://ems.press/content/serial-article-files/46824.
• Supplied literature source, labeled "Feng--Wang 2025": https://link.springer.com/article/10.1007/s00039-025-00702-4. No article title or specific supporting result is supplied.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem "partially solved" and states that the Hamiltonian-duality framework and several semiclassical and low-rank cases are established, while the stated derived equivalence for general spherical data remains conjectural. This is supplied dated curation derived from the OWR record, not an independently verified assessment; its time-sensitive status and cited claims require independent verification.