Indefinite Theta Series and Real Jacobi Representations

Definitions and mathematical statement

A lattice $L$ is a discrete subgroup spanning a finite-dimensional real vector space $V$. A quadratic form $Q\colon V\to{\mathbb{R}}$ is indefinite if it takes both positive and negative values. These definitions explain the terminology; the supplied record does not specify a particular lattice, quadratic form, or signature.

Write $\Theta$ symbolically for an indefinite theta series formed from a lattice sum over cones, and $\widehat\Theta$ for a modular completion associated with it. Let $\mathcal C_J$ denote the real-Jacobi Casimir operator intended by the source. The stated condition on the completion is \[ \mathcal C_J\widehat\Theta=0. \] These symbols do not specify a summation formula, a completion procedure, or a normalization of the operator.

Question. Can indefinite theta series of this cone-sum type, together with their modular completions satisfying the displayed equation, be realized through principal-series representations of the real Jacobi group?

Unresolved formulation. The original wording asks whether series "of the described type" can be "connected to" principal-series representations; the supplied rewritten statement uses "realized through." The record does not supply the description needed to identify that class or define either relationship. It specifies neither the dimension and convention for the real Jacobi group nor the inducing data for its principal series. It also omits the cones, summation kernels, variables, convergence or regularization conditions, modular transformation laws, completion data, and Casimir formula. The quantifier is unresolved: the question could concern every series in an intended class, a subclass characterized by additional hypotheses, or particular examples. Thus the displayed condition and the question preserve the supplied target, but the record does not determine a fully quantified representation-theoretic assertion.

Short English statement

The question asks whether cone-sum indefinite theta series and their modular completions annihilated by a real-Jacobi Casimir operator admit a principal-series representation interpretation. The intended class, quantifiers, and meaning of realization remain unspecified.

Sources

• Local registry: supplied complete UnsolvedMath record, permanent ID $30003033$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The project registry locations are data/unsolvedmath/problems.json and catalog/problems.json; their contents have not been independently inspected for this draft.
• Source report: Lattices and Applications in Number Theory (2016), Oberwolfach Reports, DOI 10.4171/owr/2016/3, pp. 87--154. https://doi.org/10.4171/owr/2016/3. This is also the supplied literature reference labeled "Oberwolfach Report 3/2016." The record gives proposed year 2016 and names no individual proposer.
• Supplied literature reference: Indefinite zeta functions and indefinite theta methods. https://doi.org/10.1007/s40687-021-00252-9.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and reports connections of indefinite theta series with modular, mock, and Jacobi-form structures in important cases, while stating that the precise principal-series connection asked here is not established generally. This is supplied dated curation derived from the OWR material, 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 the official passage was checked and rewritten to preserve its mathematical content. That verification claim is supplied metadata; the report passage has not been independently visited or checked for this draft.