Spectral Expansion of Twisted Orbital Integrals

Definitions and mathematical statement

Retain the source notation \[ H\in\{\operatorname{Sp}(2n),\operatorname{SO}(2n)\},\qquad H\text{ split}. \] Here $\operatorname{GL}(V)$ denotes the group of invertible linear transformations of the vector space $V$. The source does not specify the ground field, the admissible values of $n$, the dimension of $V$, or the relationship between $V$ and $H$.

Recorded assertion, incomplete. For such $H$, the twisted orbital integral associated with the relevant bilinear-form element in $\operatorname{GL}(V)$ admits the spectral expansion referred to as Theorem 1.4. Its terms are described as twisted characters of functorial lifts, weighted by formal-degree gamma factors and the elementary tempered measure.

Unresolved qualifications. Neither Theorem 1.4 nor the definitions needed to express its identity are supplied. In particular, the bilinear form and associated element, twisting operation, test-function space, orbital-integral normalization, spectral indexing, functorial lifts, gamma factors, and tempered measure are unspecified. Consequently, no precise integral identity, quantifiers over test functions or representations, or convergence statement can be recovered from this record.

The supplied registry labels the entry open, but also marks its statement unrecoverable. Its supplied verification note describes the extraction as a theorem-level spectral-expansion result used in the proof of Theorem 1.3, rather than an open problem. These are source annotations, not independently verified mathematical conclusions; the record does not identify a complete open target.

Short English statement

The record describes a spectral expansion of twisted orbital integrals for split symplectic and even special orthogonal groups. The referenced theorem and essential definitions are missing, and the supplied annotations conflict about whether this is an open problem or an established result. A precise standalone statement requires recovery of the original theorem.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004778$, source code catalogue entry, title Spectral Expansion of Twisted Orbital Integrals, in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the recorded proposed year is 2021.
• Supplied report citation: Automorphic Forms, Geometry and Arithmetic (2022), Oberwolfach Reports, pp. 2089--2156, DOI 10.4171/owr/2021/39. The literature-source label is Oberwolfach Report 39/2021. https://doi.org/10.4171/owr/2021/39. The supplied verification note identifies the extraction as a theorem-level result used in the proof of Theorem 1.3; the extracted statement refers to Theorem 1.4. The complete theorem text is not supplied.
• Supplied literature source: Hiraga--Ichino--Ikeda conjecture for classical groups. https://arxiv.org/abs/2508.08470. No theorem text or precise connection establishing the recorded spectral expansion is supplied.
• Literature assessment dated exactly 2026-08-22: supplied dated curation, not independently verified. It labels the entry open and describes the extracted passage as an incomplete continuation of a proof setup, with the associated general formal-degree and twisted-orbital-integral claim unresolved as extracted. The background explicitly requests independent verification of time-sensitive status and cited claims. No source visit or independent literature assessment is asserted here.