Epsilon Dichotomy for Generic General Linear Representations

Definitions and mathematical statement

Let $F$ denote the field appearing in the supplied statement, and write $\operatorname{GL}_6(F)$ for the group of invertible $6\times6$ matrices over $F$. The record discusses archimedean and $p$-adic cases but does not specify the precise allowed fields, their characteristics, or the category of representations.

Let $\pi$ be a representation described by the source as generic, with central character $\omega_\pi=\chi^2$. Under the usual interpretation of this hypothesis, $\chi$ is a character of $F^\times$, and \[ \omega_\pi(a)=\chi(a)^2\qquad(a\in F^\times). \] The source does not explicitly state irreducibility, admissibility, continuity or smoothness requirements, or a definition of genericity.

Retain the source notation \[ E(\pi,\chi):=\epsilon\!\left(\frac12,(\wedge^3\pi)\otimes\chi^{-1}\right). \] Here $\wedge^3\pi$ is the source's exterior-cube notation; its precise construction and the conventions for the epsilon factor are not supplied. Likewise, $m(\pi)$ is not defined in the record. The cited Ginzburg--Rallis reference suggests a model-multiplicity interpretation, but the relevant subgroup, character, and Hom space are not specified and are not reconstructed here.

Source-stated epsilon-dichotomy conjecture. For every generic $\pi$ and character $\chi$ satisfying the central-character hypothesis, within the intended but incompletely specified setting, do both equivalences hold? \[ \begin{aligned} m(\pi)=1&\quad\Longleftrightarrow\quad E(\pi,\chi)=1,\\ m(\pi)=0&\quad\Longleftrightarrow\quad E(\pi,\chi)=-1. \end{aligned} \] Both equivalences are retained as requested assertions. Calling the second equivalent to the first requires additional range information, such as $m(\pi)\in\{0,1\}$ and $E(\pi,\chi)\in\{1,-1\}$, which the supplied record does not establish. Until the field, representation category, multiplicity, and epsilon-factor conventions are specified, this is a qualified transcription of the proposed target rather than a fully determined mathematical conjecture.

Short English statement

The supplied conjecture predicts that a generic representation of $\operatorname{GL}_6(F)$ with central character $\chi^2$ has multiplicity one exactly when its twisted exterior-cube epsilon factor at $1/2$ is $+1$, and multiplicity zero exactly when that factor is $-1$. The intended multiplicity and several hypotheses and conventions remain unspecified. The record labels the problem partially solved; that assessment is supplied dated curation requiring independent verification.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30003515$, title "Epsilon Dichotomy for Generic General Linear Representations," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied proposed year is 2017; no individual proposer is identified.
• Supplied report citation: Harmonic Analysis and the Trace Formula (2018), Oberwolfach Reports, pp. 1551--1630, DOI 10.4171/owr/2017/25. https://doi.org/10.4171/owr/2017/25. The record describes these reports as documentation of mathematical workshops and their open-problem discussions. No specific page or named contribution within the report is supplied.
• Supplied literature source, labeled "Wan, local Ginzburg--Rallis model": https://sites.rutgers.edu/chen-wan/wp-content/uploads/sites/518/2020/08/GRgeneric.pdf.
• Supplied literature source, labeled "local trace formula for the generalized Shalika model": https://hal.science/hal-01941350v1/document.
• Supplied literature source, labeled "original OWR report": https://doi.org/10.4171/owr/2017/25.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports proofs in archimedean cases and broad $p$-adic tempered or non-discrete classes, with exceptional $p$-adic discrete-series cases and the full generic case remaining unresolved. These are supplied dated curation claims, not independently verified mathematical or literature conclusions. The record itself requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, with an editorial explanation that mathematical content was preserved while notation, typography, and sentence structure were repaired. This supplied label does not constitute independent mathematical review of this draft or verification of the cited sources.