Langlands-Dual Parametrization of Shifted Quantum Affine Modules

Definitions and mathematical statement

For an algebra $A$, a simple $A$-module is a nonzero module whose only submodules are $0$ and itself. For a specified class $\mathcal C$ of $A$-modules, let $\operatorname{Irr}_{\mathcal C}(A)$ denote the set of isomorphism classes of simple modules belonging to $\mathcal C$. For a specified character expression $X$, let $\operatorname{Mon}(X)$ denote its monomial support, meaning the distinct monomials with nonzero coefficient. This support convention does not count repeated occurrences separately; whether multiplicities have any role in the intended parametrization is not specified by the record.

Conjectural target, with unresolved data. The supplied source proposes that the simple modules of a truncation of a shifted quantum affine algebra of non-simply-laced type admit an explicit parametrization by monomials in the Langlands-dual $q$-character of a finite-dimensional representation of the Langlands-dual quantum affine algebra.

To display the intended correspondence schematically, write $A$ for the truncated shifted quantum affine algebra, $\mathcal C$ for the intended class of its modules, $V$ for the finite-dimensional representation of the dual quantum affine algebra, and $\chi_q^{\mathrm L}(V)$ for the source's Langlands-dual $q$-character. The proposed parametrization would in particular give a bijection \[ \operatorname{Irr}_{\mathcal C}(A) \;\longleftrightarrow\; \operatorname{Mon}\left(\chi_q^{\mathrm L}(V)\right). \] Here $\chi_q^{\mathrm L}$ is a placeholder for the construction intended by the source, not a definition of that construction.

This display is a schematic formulation requiring further source specification. The record does not give the ground field, the allowed values of $q$, the precise non-simply-laced root datum or affine convention, the shift, the truncation parameters and relations, the module class $\mathcal C$, or the definition and normalization of the Langlands-dual character. It also does not specify how $V$ is determined by the truncation data, which truncations are admissible, or the explicit correspondence. Thus a fully quantified statement for all intended truncations cannot be recovered from this record alone. In particular, finite dimensionality is asserted for $V$, not for the simple $A$-modules being classified.

Short English statement

The conjecture predicts an explicit classification, up to isomorphism, of simple modules for suitable truncated shifted quantum affine algebras of non-simply-laced type using monomials of an associated Langlands-dual $q$-character. The supplied record omits the data needed to specify the algebras, module class, associated representation, and correspondence precisely. Its partially solved label is supplied dated curation, not an independently verified conclusion.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30004660$, exact source problem code catalogue entry, title Langlands-Dual Parametrization of Shifted Quantum Affine Modules, in the Oberwolfach Reports Open Problems collection. The proposed year is recorded as 2021; no individual proposer is supplied.
• Supplied report citation: Algebraic Groups (2022), Oberwolfach Reports, pp. 1087--1148, DOI 10.4171/owr/2021/20. https://doi.org/10.4171/owr/2021/20. The background also calls this Oberwolfach Report 20/2021 and describes the collection as drawn from workshop reports and their open-problem discussions. The differing year labels are retained without reconciliation.
• Hernandez, representations of shifted quantum affine algebras, as labeled in the supplied literature sources. https://arxiv.org/abs/2010.06996.
• Hernandez--Zhang, Jordan--Hölder property, as labeled in the supplied literature sources. https://arxiv.org/abs/2501.16859.
• Supplied literature assessment, checked on 2026-08-22: the record labels the problem partially solved and states that Baxter polynomiality proves the predicted descent and finiteness consequences and provides substantial evidence, while the proposed Langlands-dual parametrization of all non-simply-laced simple truncation modules remains conjectural. This is supplied dated curation derived from the OWR material, not independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified. Its accompanying verification note reports manual semantic, typographic, and LaTeX review and claims that the clean statement is self-contained and preserves the mathematical target without unsupported additions. This is a supplied review assertion, not verification performed for this draft.