Verma-to-Wakimoto Isomorphisms for Jet-Level Characters

Definitions and mathematical statement

The supplied record uses $\mathfrak h_n$ for a jet-level Cartan object and $\mathfrak h_n^*$ for its dual, whose elements $\theta$ are called jet-level characters. It denotes the associated Verma module by $V(\theta)$, an associated parameter by $\chi_\theta$, and the desired Wakimoto module by $W(\chi_\theta)$. These are source notations: the record does not define the underlying Lie algebra, the jet construction, the allowed indices $n$, the representation level, the module category, or the assignment $\theta\mapsto\chi_\theta$. In particular, no specific truncated-current-algebra convention for $\mathfrak h_n$ is assumed here.

Statement requested by the source, conditional on specifying this framework. For every allowed jet index $n$ and every character $\theta\in\mathfrak h_n^*$, construct a Wakimoto module $W(\chi_\theta)$ together with a natural nonzero module homomorphism \[ \Phi_\theta:V(\theta)\longrightarrow W(\chi_\theta), \qquad \Phi_\theta\ne 0. \] Identify explicit additional hypotheses on $\theta$ under which $\Phi_\theta$ is an isomorphism. The original statement asks for suitable sufficient conditions; the cleaned statement asks to characterize the conditions for isomorphism, which may require a stronger, necessary-and-sufficient result. This difference is unresolved. Neither the additional hypotheses nor the meaning of naturality is supplied, so the displayed request is a schematic target rather than a fully specified conjecture.

The supplied dated literature assessment labels the problem partially solved and reports that the construction and a natural nonzero morphism are established for general jets, that the asserted isomorphism is known when $n=0$, and that the higher-jet isomorphism remains conditional. The hypotheses attached to those claims are not given. These are supplied curation claims, not independently verified mathematical conclusions.

Short English statement

For each jet-level character $\theta$, the source asks for a Wakimoto module and a natural nonzero map from the corresponding Verma module, together with explicit hypotheses making that map an isomorphism. The underlying representation-theoretic framework and the required hypotheses are missing. Supplied dated curation reports partial progress, including an $n=0$ isomorphism claim whose hypotheses require review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30002007$, title: Verma-to-Wakimoto Isomorphisms for Jet-Level Characters. Exact source problem code: catalogue entry. Collection: Oberwolfach Reports Open Problems. The record gives the proposed year as 2012 and names no proposer.
• Supplied report citation: Enveloping Algebras and Geometric Representation Theory (2012), Oberwolfach Reports, pp. 733--809, DOI 10.4171/owr/2012/13. https://doi.org/10.4171/owr/2012/13. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions; no problem-specific page or speaker attribution is supplied.
• Supplied literature reference: Compatibility of the Feigin-Frenkel and Harish-Chandra Isomorphisms. https://arxiv.org/abs/1308.2995.
• Supplied literature reference, labeled "Published Compatibility Paper": https://doi.org/10.1090/S0002-9947-2014-06419-2.
• Literature assessment date: 2026-08-21. This is supplied dated curation derived from the supplied OWR curation, not an independently verified assessment. It labels the problem partially solved and reports a general-jet construction and natural nonzero Verma-to-Wakimoto morphism, an isomorphism in the $n=0$ case, and a conditional higher-jet isomorphism. The supplied background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and supplies a manual semantic, typographic, and LaTeX review assertion describing the cleanup as meaning-preserving, self-contained, and unsupported-content-free. Those are supplied editorial assertions; they do not resolve the missing mathematical definitions or establish mathematical correctness.