Density of Whittaker Distributions on Spherical Spaces

Definitions and mathematical statement

Let $G$ be quasisplit, let $B\subseteq G$ be a Borel subgroup, and let $U$ be its "nilradical," in the terminology of the supplied record. Let $\psi$ be a nondegenerate character of $U$, let $H\subseteq G$ be reductive and spherical, and put $X=G/H$. The action under consideration is the left action on $X$. Here spherical conventionally means that a Borel subgroup has an open orbit on the homogeneous space after passage to an algebraic closure; the record does not specify the base field or its convention for spherical spaces. To interpret characters and group orbits, $U$ would ordinarily be the unipotent radical of $B$; a Lie algebra nilradical requires a different formulation. Neither interpretation is explicitly resolved by the supplied statement.

The source asks whether \[ \sum_{x\in O} S^*(Ux)^{U,\psi} \quad\text{is dense in}\quad S^*(X)^{U,\psi}, \] and describes the summands as one-dimensional spaces. Superscript $(U,\psi)$ denotes distributions transforming by the character $\psi$ under the induced $U$-action. The record does not define $S^*$, its equivariance convention, or the topology used for density. In an Archimedean interpretation, a possible convention is the continuous dual of an appropriate Schwartz test-function space, but this is not specified by the source record.

There is a material mismatch in the orbit parameter: the record defines $O$ to be the union of the open $B$-orbits in $G$, although the target distribution space is on $X$. An orbit $Ux$ for $x\in G$ is an orbit in $G$, and the record supplies no map placing its distributions in $S^*(X)$. Consequently the displayed density question is not fully defined as written.

A conditional formulation of the apparent intended question is as follows. If $O$ is intended to be \[ O_X=\bigcup\{\mathcal O\subseteq X:\mathcal O\text{ is an open }B\text{-orbit}\}, \] fix the underlying field and point spaces, the meaning of openness, a distribution theory $S^*$, a topology $\tau$ on $S^*(X)^{U,\psi}$, and well-defined equivariance-preserving maps \[ \iota_x:S^*(Ux)^{U,\psi}\longrightarrow S^*(X)^{U,\psi} \qquad(x\in O_X). \] For every datum satisfying the stated hypotheses and these specified conventions, is \[ \overline{\operatorname{span}\!\left( \bigcup_{x\in O_X}\iota_x\left(S^*(Ux)^{U,\psi}\right) \right)}^{\,\tau} =S^*(X)^{U,\psi}? \] The span consists of finite linear combinations. This conditional formulation does not establish that replacing $G$ by $X$ is the intended correction, that the maps $\iota_x$ exist, or that all the orbit spaces are one-dimensional. These points require source clarification, including any stabilizer compatibility required for nonzero $(U,\psi)$-equivariant orbit distributions.

Short English statement

The proposed question asks whether Whittaker-equivariant distributions on the spherical quotient $G/H$ can be approximated by finite sums of distributions arising from $U$-orbits in the open $B$-orbits. The supplied wording instead places those open orbits in $G$, and leaves the field, distribution spaces, topology, and orbit-to-quotient maps unspecified. The precise displayed reformulation is therefore conditional. The supplied status is partially solved; this is a dated curation label, not an independently verified mathematical conclusion.

Sources

• Local UnsolvedMath registry: data/unsolvedmath/problems.json; supplementary registry: catalog/problems.json. Supplied permanent ID: $30002483$. Exact source problem code: catalogue entry. Title: Density of Whittaker Distributions on Spherical Spaces. Collection: Oberwolfach Reports Open Problems. No individual proposer is identified in the supplied record.
• Supplied report citation: Representation Theory and Analysis of Reductive Groups: Spherical Spaces and Hecke Algebras (2014), Oberwolfach Reports, pp. 145--212, DOI 10.4171/owr/2014/03. https://doi.org/10.4171/owr/2014/03. The supplied background describes these reports as documenting workshops and their open-problem discussions; no page specifically locating this problem is supplied.
• Supplied literature source: Aizenbud--Gourevitch, spherical-space distributions. https://arxiv.org/abs/1311.6111.
• Supplied literature source: Aizenbud--Gourevitch--Sayag--Kemarsky, z-finite distributions. https://arxiv.org/abs/1405.2540.
• Supplied literature source: Oberwolfach representation theory report. https://publications.mfo.de/bitstream/handle/mfo/3393/OWR_2014_03.pdf?isAllowed=y&sequence=1.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem partially solved and reports vanishing results away from open $B$-orbits and related density results under additional hypotheses, while describing the full Archimedean spherical-space density conjecture as unresolved. This is supplied dated curation, not independently verified. The record itself requests independent verification of time-sensitive status and cited claims.
• The supplied statement status is corrected_verified, accompanied by a claim of manual semantic, typographic, and LaTeX review. That registry assertion does not resolve the formulation issues identified here or constitute mathematical verification of this draft.