Affine Springer Cohomology and Quantum-Group Centers

Definitions and mathematical statement

Let $\mathcal F\ell^J_\gamma$ and $u^J_\zeta$ denote, respectively, the affine Springer fiber and small quantum group object appearing in the supplied conjecture. The record does not specify their complete defining data or admissibility hypotheses; the statement below therefore concerns the intended source setting, rather than arbitrary affine Springer fibers and arbitrary small quantum groups. In particular, the underlying group, coefficient field, element $\gamma$, index $J$, root of unity $\zeta$, and meaning of the superscript $J$ on the quantum group require clarification.

Write $\check\Lambda$ for the coweight lattice designated in the clean statement, $W_{\mathrm{af}}$ for the affine Weyl group, and $\check G$ and $\check T$ for the group and torus acting on the quantum group center in the source. Their precise group forms and root datum are not supplied. For an associative algebra $A$, its center is \[ Z(A)=\{a\in A:ab=ba\text{ for every }b\in A\}. \] For a group $K$ acting on a vector space or algebra $V$, write \[ V^K=\{v\in V:kv=v\text{ for every }k\in K\}. \] The notation $H^\bullet(X)$ denotes the cohomology intended in the source; its coefficient field and precise cohomology theory are unspecified here.

The excerpt describes a left lattice action on $\mathcal F\ell^J_\gamma$, obtained by identifying the coweight lattice with a subgroup of the centralizer of $\gamma$. It says that the induced left action on cohomology extends to $W_{\mathrm{af}}$, with the finite Weyl group acting by monodromy through a map $\pi_1(\mathfrak g^{\mathrm{rs}})\to W$. The excerpt alternates between $\Lambda$ and $\check\Lambda$; their identification must be checked. The actions on the quantum group center are named but not constructed in the supplied record.

Conjectural target. With the intended source data and actions fixed, are there canonical isomorphisms \[ H^\bullet(\mathcal F\ell^J_\gamma)^{\check\Lambda} \cong Z(u^J_\zeta)^{\check T}, \qquad H^\bullet(\mathcal F\ell^J_\gamma)^{W_{\mathrm{af}}} \cong Z(u^J_\zeta)^{\check G}? \] The record does not specify the category of these isomorphisms, the relevant grading on the centers, or the compatibility properties meant by "canonical." These qualifications remain unresolved.

Related source qualifications. The extracted passage also contains a corollary involving loop rotation by ${\mathbb{C}}^\times$ on an affine Grassmannian and a different, corrupted affine Grassmannian expression. Its relationship to the displayed conjecture cannot be recovered reliably from this record. It must not be read as a proof of the conjecture.

For the unindexed fiber $\mathcal F\ell_\gamma$, the excerpt announces that the authors will check, using results of the second author with Losev, \[ \dim\left(H^\bullet(\mathcal F\ell_\gamma)^{W_{\mathrm{af}}}\right)=(h+1)^r, \] where $h$ is the Coxeter number and $r$ is the rank, and, for $G$ of type $A$, \[ H^\bullet(\mathcal F\ell_\gamma)^{W_{\mathrm{af}}} =H^\bullet(\mathcal F\ell_\gamma)^{\check\Lambda}. \] These are supplied source announcements with incompletely specified hypotheses. The passage states that the conjecture would imply triviality of the $\check G$-action on the center when $\check G$ has type $A$, together with the conjectures labeled [2, Conjecture 4.9] and [3, Conjecture 3.14]; those bibliography entries are absent.

The subsequent deformed discussion explicitly restricts to $J=\varnothing$, so that $\mathcal F\ell^J=\mathcal F\ell$, and concerns torus-equivariant cohomology. It sketches a flat lattice-graded algebra whose special fiber is graded Morita equivalent to $u^0_\zeta$, using a regular-weight summand of a completed quantum group deformation, and begins an alternative construction from a noncommutative Springer resolution. That discussion is truncated and does not supply definitions sufficient to complete the present conjecture.

Short English statement

In the intended source setting, the conjecture identifies two invariant parts of affine Springer fiber cohomology with the torus-invariant and group-invariant parts of a small quantum group center. Essential hypotheses, coefficient choices, actions, and the meaning of canonical isomorphism are missing. The supplied dated curation labels the problem partially solved; this draft does not independently verify that assessment.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30004887$, title "Affine Springer Cohomology and Quantum-Group Centers," exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2021 and names no individual proposer.
• Supplied report citation: Enveloping Algebras and Geometric Representation Theory (2022), Oberwolfach Reports, pp. 2827--2891, DOI 10.4171/owr/2021/52. https://doi.org/10.4171/owr/2021/52. The collection describes these reports as documenting mathematical workshops and their open-problem discussions. No precise page or named contributor for this conjecture is supplied.
• Supplied literature reference labeled Bezrukavnikov--Boixeda Alvarez--Shan--Vasserot: https://arxiv.org/abs/2205.05951.
• Supplied literature reference labeled Hemelsoet--Kivinen--Lachowska: https://arxiv.org/abs/2205.09700.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified here. It labels the problem partially solved and reports an established equivariant cohomology/center isomorphism, an injective nonequivariant invariant map, and partial confirmation of the conjectured invariant isomorphism, including type-$A$ progress. The supplied background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that the official report was checked directly. This is supplied provenance, not a source visit or verification performed for this draft. The extracted passage additionally refers to results of the "second author with Losev" and to [2, Conjecture 4.9] and [3, Conjecture 3.14], without supplying the corresponding identities or bibliography entries.