Plancherel Support of Spherical Homogeneous Spaces

Definitions and mathematical statement

The supplied record concerns a spherical homogeneous space $X=H\backslash G$, but does not specify the ground field, the class of groups, the meaning of sphericity, or the measure defining $L^2(X)$. Consequently, its intended generality cannot be fixed from this record alone. In the usual algebraic terminology, sphericity means that a Borel subgroup of a reductive algebraic group has an open orbit on the homogeneous space, with the field and geometric interpretation still requiring specification.

The following gives a conditional analytic formulation, rather than asserting hypotheses absent from the source. Suppose that $G$ is a second-countable locally compact group of type I, that $H\subseteq G$ is closed, and that $X=H\backslash G$ has a specified nonzero $G$-invariant Radon measure $\mu_X$. Let $\widehat G$ be the set of unitary equivalence classes of irreducible unitary representations of $G$, equipped with the Fell topology. The unitary right regular representation on $L^2(X,\mu_X)$ is \[ (R_X(g)f)(Hx)=f(Hxg). \] Its Plancherel decomposition has the form \[ L^2(X,\mu_X)\simeq \int_{\widehat G}^{\oplus} \mathcal H_\pi\otimes\mathcal M_X(\pi)\,\mathrm d\nu_X(\pi), \] where $\mathcal H_\pi$ carries $\pi$, $\mathcal M_X(\pi)$ is a multiplicity space, and the spectral measure class is taken on the part where the multiplicity space is nonzero. Define the Fell support by \[ \operatorname{Supp}_{\mathrm{Fell}}(R_X) =\{[\pi]\in\widehat G:\pi\prec R_X\}, \] where $\pi\prec R_X$ denotes weak containment. Explicitly, after fixing a left Haar measure on $G$, weak containment means \[ \|\pi(a)\|\leq\|R_X(a)\| \quad\text{for every }a\in C_c(G), \qquad \pi(a)=\int_G a(g)\pi(g)\,\mathrm dg, \] with the analogous definition for $R_X(a)$.

Question. For the spherical homogeneous spaces intended by the source, characterize precisely \[ \operatorname{Supp}_{\mathrm{Fell}}(R_X). \] The source also asks which irreducible representations occur in the Plancherel decomposition. This wording needs qualification: a measurable spectral decomposition is specified only up to null sets, whereas Fell support is a closed set and can contain representations without individual positive spectral mass. Thus a classification of almost-everywhere spectral constituents and a characterization of Fell support must be distinguished. If the intended target includes the former, its measure-theoretic meaning must also be specified. The source's claimed equivalence does not resolve this distinction.

Short English statement

For a spherical quotient $H\backslash G$, determine exactly which irreducible unitary representations belong to the Fell support of the representation on $L^2(H\backslash G)$. The source leaves the field, group hypotheses, measure, and precise meaning of Plancherel occurrence unspecified; the displayed analytic framework is conditional and requires human review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30001740$, title Plancherel Support of Spherical Homogeneous Spaces, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record assigns the year $2011$ and supplies no individual proposer.
• Supplied report citation: Automorphic Forms: New Directions (2011), Oberwolfach Reports, pp. 719--779, DOI 10.4171/owr/2011/14. Supplied source URL and literature-source link, labeled "original OWR report": https://doi.org/10.4171/owr/2011/14. No problem-specific page or named contributor is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and describes it as a related unresolved question concerning the precise Plancherel support of $X$-distinguished representations. This is supplied dated curation, not independently verified; its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports that its mathematical content was checked against the official report, with notation, context, and question form normalized. This is a supplied verification claim, not an independent verification performed for this draft.