Spherical Plancherel Decomposition of $p$-Adic Symmetric Varieties

Definitions and mathematical statement

Let $X_v=H_v\backslash G_v$ be the supplied $p$-adic symmetric variety, let $dx$ be its fixed invariant measure, and let $K_v\subseteq G_v$ be its specified maximal compact subgroup. Define \[ \mathcal H=L^2(X_v,dx),\qquad (R(g)f)(x)=f(xg), \] and \[ \mathcal H^{K_v}=\{f\in\mathcal H:R(k)f=f\text{ for every }k\in K_v\}. \] Here invariance and equality are understood in the Hilbert-space sense.

Formulation qualification. In general, $\mathcal H^{K_v}$ is not preserved by the full right action of $G_v$. Consequently, the supplied request to decompose this space into irreducible representations of $G_v$ does not literally specify a representation to decompose. A standard ambient-representation interpretation of the requested problem is the following; confirmation of this interpretation is required.

Determine a Plancherel decomposition of the right unitary representation on $\mathcal H$, together with its restriction to $K_v$-fixed vectors, in the form \[ \mathcal H\simeq \int_{\widehat{G_v}}^{\oplus} \mathcal M_{X_v}(\pi)\otimes\mathcal V_\pi\, d\mu_{X_v}(\pi), \qquad R(g)\simeq\int_{\widehat{G_v}}^{\oplus} I_{\mathcal M_{X_v}(\pi)}\otimes\pi(g)\, d\mu_{X_v}(\pi), \] so that \[ \mathcal H^{K_v}\simeq \int_{\widehat{G_v}}^{\oplus} \mathcal M_{X_v}(\pi)\otimes\mathcal V_\pi^{K_v}\, d\mu_{X_v}(\pi). \] Here $\widehat{G_v}$ denotes the unitary dual, $(\pi,\mathcal V_\pi)$ is an irreducible unitary representation, $\mathcal M_{X_v}(\pi)$ is its multiplicity Hilbert space, and \[ \mathcal V_\pi^{K_v}=\{u\in\mathcal V_\pi:\pi(k)u=u\text{ for every }k\in K_v\}. \] The requested determination concerns the representations contributing nonzero fibers, their multiplicities, and their spectral measure, including continuous and discrete contributions when present. No multiplicity-one assertion is assumed.

A numerical Plancherel density requires an explicitly specified reference measure $\nu$ and compatible spectral normalizations. On any part where $\mu_{X_v}\ll\nu$, the density means \[ \rho_{X_v}=\frac{d\mu_{X_v}}{d\nu}. \] The record supplies no such reference measure or spectral normalization; the fixed measure $dx$ alone does not specify them.

The record does not specify the underlying local field, the precise algebraic hypotheses on $G_v$ and $H_v$, or the involution defining the symmetric structure. The displayed formulation is therefore an interpretation of the intended spectral task, not a fully specified uniform theorem for an explicitly delimited class of symmetric varieties.

Short English statement

For the supplied $p$-adic symmetric quotient with fixed invariant measure and maximal compact subgroup, determine which irreducible unitary representations contribute to its spherical $L^2$ spectrum, with their multiplicities and Plancherel measure. The $K_v$-fixed space generally has no full $G_v$-action, so the intended decomposition must be clarified through an ambient representation or another specified action. A density also requires reference-measure and normalization choices.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001106$, source problem code catalogue entry, title "Spherical Plancherel Decomposition of $p$-Adic Symmetric Varieties," collection "Oberwolfach Reports Open Problems." No individual proposer is supplied; the proposed year is recorded as 2008.
• Supplied report citation: Mini-Workshop: Symmetric Varieties and Involutions of Algebraic Groups (2009), Oberwolfach Reports, pp. 2985--3028, DOI 10.4171/owr/2008/53. https://doi.org/10.4171/owr/2008/53. The supplied citation uses 2009, while the proposed-year metadata and DOI contain 2008; this discrepancy is unresolved.
• Supplied literature reference, labeled "Sakellaridis--Venkatesh": https://arxiv.org/abs/1203.0039.
• Supplied literature reference, labeled "low-rank verification": https://arxiv.org/abs/1111.6694.
• Supplied literature reference, labeled "Hermitian-matrix Plancherel formula": https://doi.org/10.1016/j.jnt.2021.06.001.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem "partially solved" and reports that Sakellaridis--Venkatesh developed a general Plancherel framework and proved formulas up to discrete boundary-degeneration spectra, that many low-rank and symmetric cases are resolved, and that the general direct-integral description remains conjectural. This is supplied dated curation derived from the OWR record, not an independently verified assessment. The record explicitly calls for independent verification of time-sensitive status and cited claims.
• The supplied statement carries the registry label corrected_verified, described there as meaning-preserving semantic, typographic, and LaTeX cleanup with no unsupported source-dependent additions. This records the supplied editorial assessment and does not establish mathematical verification of this draft.