Reproducing Subgroups of Metaplectic–Heisenberg Representations

Definitions and mathematical statement

Fix $d\geq1$ and $\mathcal H=L^2({\mathbb{R}}^d,{\mathbb{C}})$, with inner product linear in the first variable. Let $U=\widetilde\mu_e$ denote the metaplectic--Heisenberg representation of the ambient group $G$ meant in the report. The supplied record does not specify $G$, its central extension or covering convention, or the formula for $U$; these are part of the data still requiring source recovery.

For a locally compact subgroup $H\leq G$, equipped with the intended Haar measure ${\,\mathrm{d}} h$, a reproducing or admissible window is a vector $0\ne\varphi\in\mathcal H$ such that the same window works for all $f\in\mathcal H$: \[ f=\int_H\langle f,U(h)\varphi\rangle U(h)\varphi\,{\,\mathrm{d}} h. \] A precise weak interpretation is \[ \forall f,g\in\mathcal H,\qquad \langle f,g\rangle =\int_H\langle f,U(h)\varphi\rangle \langle U(h)\varphi,g\rangle\,{\,\mathrm{d}} h. \] Equivalently, under this interpretation the coefficient map \[ C_\varphi:\mathcal H\longrightarrow L^2(H,{\,\mathrm{d}} h),\qquad (C_\varphi f)(h)=\langle f,U(h)\varphi\rangle \] is an isometry, or \[ \int_H|\langle f,U(h)\varphi\rangle|^2\,{\,\mathrm{d}} h=\|f\|_2^2 \quad\text{for every }f\in\mathcal H. \] The cross integral then exists by the Cauchy--Schwarz inequality and is recovered from the norm identity by polarization. A strong or Bochner interpretation of the vector-valued integral would require additional justification.

The problem is to classify the subgroups $H$ for which such a window exists, for the intended metaplectic--Heisenberg ambient representation in general dimension. This asks for a single-window reproducing identity with constant one on all of $L^2({\mathbb{R}}^d)$, not merely a frame inequality, reconstruction on an invariant proper subspace, or a formula using a separate dual window. The source leaves the subgroup class and equivalence relation for classification unspecified.

The supplied assessment checked 2026-08-21 reports classifications in dimension one and important two-dimensional classes in the metaplectic setting, but no complete general-dimensional classification. Its scope must not be expanded to all subgroups of an unspecified metaplectic--Heisenberg extension.

Short English statement

Which subgroups make the orbit of one window into a continuous Parseval family for the entire Hilbert space? The exact ambient extension, subgroup class, Haar convention, and interpretation of the reproducing integral must be fixed to turn the requested classification into a fully determined problem.

Sources

• Local UnsolvedMath registry, permanent ID $30000804$, exact source code catalogue entry, proposed year 2007; no individual proposer is supplied.
• Wavelet and Multiscale Methods (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/36, pp. 2079--2172. https://doi.org/10.4171/owr/2007/36.
• Supplied references: Alberti--Balletti--De Mari--De Vito, "algebraic classification," https://arxiv.org/abs/1109.6789; "admissible vectors in dimension two," https://arxiv.org/abs/1212.2743.
• The supplied corrected_verified note says the full reproducing question and notation were preserved. The open label and assessment checked 2026-08-21 are dated supplied triage requiring independent verification of status and exact classification scope.