Moy-Prasad Data and Langlands Parameters

Definitions and mathematical statement

The supplied record concerns the intended setting of representations of $p$-adic groups. Write $G(F)$ for the group of $F$-points under consideration, $\operatorname{Irr}_{\mathrm{sm}}(G(F))$ for its irreducible smooth representations, $W_F$ for the Weil group, and $G^\vee$ for the dual group. The precise hypotheses on $F$, $G$, and representation coefficients are not supplied.

Let $(X,r)$, with $r>0$, be a Moy--Prasad datum attached to $\pi\in\operatorname{Irr}_{\mathrm{sm}}(G(F))$ in the sense of the source. The excerpt attributes to Chen--DeBacker--Tsai a construction of a continuous homomorphism \[ \phi_X:W_F^{=r}\longrightarrow G^\vee, \] depending only on the "associativity class" of $X$ and a choice of $\psi_F$. Neither that class nor the normalization and nature of $\psi_F$ is defined in the supplied excerpt.

Conjectural compatibility. For every representation and attached positive-depth datum covered by the intended construction, and its corresponding Langlands parameter $\phi_\pi$, the proposed general compatibility is \[ \left.\phi_\pi\right|_{W_F^{>r}}=1, \qquad \left.\phi_\pi\right|_{W_F^{=r}}=\phi_X. \] Here the existence and normalization of the correspondence assigning $\phi_\pi$ are prerequisites whose general scope is not specified by the record.

The notation in the second equality needs clarification. Under the usual possible interpretation using upper-numbering ramification groups, \[ W_F^{>r}=W_F^{r+},\qquad W_F^{=r}=W_F^r/W_F^{r+}. \] With this interpretation, the second displayed equality means that, after the first equality makes descent possible, the map induced by $\left.\phi_\pi\right|_{W_F^r}$ on $W_F^r/W_F^{r+}$ equals $\phi_X$. This interpretation is conditional: the supplied excerpt does not define these symbols. It also does not specify the identification of the relevant parameter target with $G^\vee$, or whether equality means equality of chosen representatives or equality up to conjugacy.

The quoted source asserts this compatibility when $\phi_\pi$ is a supercuspidal Langlands parameter of Kaletha, referring to $[\mathrm{Kal19},\mathrm{Kal21}]$, and the datum is attached to $\pi$ "like before." The requested conjecture extends that compatibility beyond the stated case. The omitted preceding hypotheses and attachment procedure are essential unresolved qualifications; this draft does not assert a theorem for arbitrary fields, groups, or choices of datum.

Short English statement

The proposed conjecture says that a representation's Langlands parameter is trivial above the positive depth $r$, and that its depth-$r$ part agrees with the parameter constructed from its Moy--Prasad datum. The supplied excerpt states this compatibility for specified Kaletha supercuspidal parameters. A precise general statement still requires the omitted hypotheses, ramification conventions, and parameter normalizations.

Sources

• Local UnsolvedMath registry, permanent ID $30006116$, title Moy-Prasad Data and Langlands Parameters, exact source code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2024.
• Supplied report citation: Representations of $p$-adic Groups (2025), Oberwolfach Reports, pp. 3103--3178, DOI 10.4171/owr/2024/54. https://doi.org/10.4171/owr/2024/54. The report title's year is reproduced as supplied alongside the DOI's 2024 component.
• Supplied literature source, "OWR report": https://publications.mfo.de/bitstream/handle/mfo/4270/OWR_2024_54.pdf?sequence=1.
• Supplied literature source, "Chen--DeBacker--Tsai": https://arxiv.org/abs/2509.07780. The excerpt attributes the construction of $\phi_X$ and the stated compatibility case to Chen--DeBacker--Tsai.
• The excerpt cites Kaletha as $[\mathrm{Kal19},\mathrm{Kal21}]$; full bibliographic entries are not supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the 2025 Moy--Prasad-types work constructs restricted parameters and proves compatibility in specified cases, while general compatibility remains conjectural. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. These are supplied provenance claims, not independent verification performed for this draft.