Symplectic Periods of Essentially Speh Representations

Definitions and mathematical statement

For a representation $\theta$ of a group $G$ on a vector space $V_\theta$ over a coefficient field $k$, and a subgroup $H\subseteq G$, an $H$-invariant linear functional is a map $\ell:V_\theta\to k$ satisfying \[ \ell(\theta(h)v)=\ell(v)\qquad(h\in H,\ v\in V_\theta). \] A period means a nonzero such functional, subject to any continuity requirements of the intended representation category. In the general linear group setting, the usual symplectic subgroup is \[ \operatorname{Sp}_{2n}(F)=\{g\in\operatorname{GL}_{2n}(F):g^{\mathsf T}J_ng=J_n\},\qquad J_n=\begin{pmatrix}0&I_n\\-I_n&0\end{pmatrix}. \] Thus, if this is the intended ambient setting, having a symplectic period means \[ \operatorname{Hom}_{\operatorname{Sp}_{2n}(F)}(V_\theta,k)\ne\{0\}, \] where $k$ carries the trivial action and the Hom space is interpreted in the intended category. The supplied record does not specify $F$, $k$, the ambient group, or that category; this description is conditional and does not supply those missing hypotheses.

Retain the source notation $Q(\Delta_1,\ldots,\Delta_{2r})$. For a nonempty list of length $2r$, the adjacent-pair condition is \[ Q(\Delta_{2i+1},\Delta_{2i+2})\text{ is essentially Speh} \qquad\text{for every }i\in\{0,\ldots,r-1\}. \] Let $\mathcal G'$ denote the source's class of representations $Q(\Delta_1,\ldots,\Delta_{2r})$ built from ladder representations and satisfying this condition. The question is whether \[ \forall\theta\in\mathcal G',\qquad \operatorname{Hom}_{H_{\mathrm{sp}}}(V_\theta,k)\ne\{0\}, \] where $H_{\mathrm{sp}}$ denotes the intended symplectic subgroup of the ambient group of $\theta$, and $k$ is the intended coefficient field with trivial action.

The supplied record does not define the objects $\Delta_j$, the construction $Q$, ladder representations, or the qualification "essentially Speh." Consequently, it does not determine membership in $\mathcal G'$ precisely. It also does not specify whether $r=0$ is allowed. These qualifications remain unresolved; the displayed assertion records the source's intended question rather than a complete standalone conjecture with fixed ambient data. The dated literature assessment additionally describes the representations as "generic," but the supplied mathematical statement does not explicitly impose genericity.

Short English statement

The question asks whether every representation in the specified ladder-based class, whose adjacent pairs are essentially Speh, admits a nonzero invariant functional for the relevant symplectic subgroup. The ambient representation setting and several defining terms are missing, so this remains a qualified statement-only draft requiring mathematical review.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30002482$, source problem code catalogue entry, title "Symplectic Periods of Essentially Speh Representations." The record gives proposed year 2014 and names no proposer.
• Supplied report citation: Representation Theory and Analysis of Reductive Groups: Spherical Spaces and Hecke Algebras (2014), Oberwolfach Reports, pp. 145--212, DOI 10.4171/owr/2014/03. https://doi.org/10.4171/owr/2014/03. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, "Oberwolfach representation theory report": https://ems.press/content/serial-article-files/46494?nt=1.
• Supplied literature source, "Mitra, symplectic periods for GL representations": https://arxiv.org/abs/1601.03611.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and says that the symplectic-period assertion for the stated generic representations is posed as a conjecture beyond classified low-rank cases, with no general proof found. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports manual semantic, typographic, and LaTeX review of a meaning-preserving cleanup. These are supplied editorial qualifications, not independent verification of mathematical completeness or correctness.