The Twisted Fourier–Jacobi Gan–Gross–Prasad Conjecture

Definitions and mathematical statement

Let $F$ be a local or global field, and let $E$ and $K$ be quadratic étale $F$-algebras. Write $a\mapsto\bar a$ for the canonical involution of $E/F$. Under the usual interpretation of a skew-Hermitian space, $V$ is a finite-dimensional $E$-module with a nondegenerate sesquilinear form $h$ satisfying \[ h(av,bw)=a\,h(v,w)\,\bar b, \qquad h(w,v)=-\overline{h(v,w)}. \] Its unitary group consists of the $E$-linear automorphisms preserving $h$. Interpret \[ V_K=V\otimes_F K \] as a skew-Hermitian space over $E\otimes_F K$, with the scalar-extended form and involution $\bar{\phantom{a}}\otimes\operatorname{id}_K$. Scalar extension gives the natural inclusion of $U(V)$ into the group denoted $U(V_K)$ in the source. These are conventional interpretations of notation that the supplied record does not explicitly define.

For local $F$, let $\Pi$ be an irreducible representation of $U(V_K)$, and let $\omega_{V,\psi,\mu}$ denote the Weil representation of $U(V)$ specified in the source. Once an appropriate representation category and the auxiliary data $\psi,\mu$ have been fixed, define the branching space and its multiplicity by \[ \mathcal H_V(\Pi;\psi,\mu) =\operatorname{Hom}_{U(V)} \left(\Pi|_{U(V)},\omega_{V,\psi,\mu}\right), \qquad m_V(\Pi;\psi,\mu) =\dim_{{\mathbb{C}}}\mathcal H_V(\Pi;\psi,\mu). \] Here the Hom space consists of intertwining maps in the chosen category; the record does not specify that category or establish finiteness of its dimension.

Requested local and global assertions. For the data to which the twisted Fourier--Jacobi Gan--Gross--Prasad conjectures apply, determine $m_V(\Pi;\psi,\mu)$ locally and determine the corresponding global period when $F$ is global, in accordance with the conjectural twisted GGP formulas. The supplied record does not reproduce those formulas or define the global period. It also omits the restrictions on the fields, representations, and auxiliary characters needed for their precise formulation. Consequently, this is a statement of the supplied research target, not a complete conjectural identity: neither a specific multiplicity formula nor a specific global period identity can be recovered from this record alone.

Short English statement

The problem asks for the twisted Fourier--Jacobi Gan--Gross--Prasad formulas describing the restriction of $\Pi$ from $U(V_K)$ to $U(V)$, measured against a Weil representation, and the corresponding global period. The supplied record omits the formulas and several necessary hypotheses, so the exact conjectures require human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004780$, title: The Twisted Fourier--Jacobi Gan--Gross--Prasad Conjecture; exact source problem code: catalogue entry. The recorded proposed year is 2021; no individual proposer is supplied.
• Supplied report citation: Automorphic Forms, Geometry and Arithmetic (2022), Oberwolfach Reports, pp. 2089--2156, DOI 10.4171/owr/2021/39. https://doi.org/10.4171/owr/2021/39. The background describes Oberwolfach Reports as documenting workshops and their open-problem discussions.
• Supplied literature reference: Twisted GGP Problems and Conjectures. https://arxiv.org/abs/2204.10108.
• Supplied literature reference: Local twisted GGP for U(V_K)/U(V). https://arxiv.org/abs/2511.01301.
• Supplied literature reference: Twisted GGP for unramified quadratic extensions. https://arxiv.org/abs/2307.15234.
• Supplied literature assessment, checked 2026-08-22: the record labels the problem partially solved and reports precise formulations and verified low-rank and principal-series evidence, while describing the general local and global conjectures as open. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its standalone formulation and hypotheses were checked against the official Oberwolfach report. This verification claim is supplied metadata and has not been independently checked here.