Invariant Distribution Sheaves for Rank-One Lie Groups

Definitions and mathematical statement

Let $G$ be a connected simple real Lie group of real rank one, let $P=MAN$ be a minimal parabolic subgroup, and let $U\leq N$ be connected. The source considers a principal-series representation with generic induction parameter. Denote its smooth realization by $V_{\sigma,\lambda}^{\infty}$, where $\sigma$ records the inducing $M$-representation and $\lambda$ the $A$-parameter; normalized versus unnormalized induction and $\sigma$ are not specified. Distribution vectors $V_{\sigma,\lambda}^{-\infty}$ are continuous functionals on the smooth contragredient realization, with the induced $G$-action. Their $U$-invariant part is \[ (V_{\sigma,\lambda}^{-\infty})^U =\{v:\pi_{\sigma,\lambda}(u)v=v\text{ for every }u\in U\}. \]

Write $Z_U$ for the report's auxiliary manifold and $\mathscr F_{U,\sigma,\lambda}$ for the sheaf it constructs, with the stated identification \[ \Gamma(Z_U,\mathscr F_{U,\sigma,\lambda}) \cong (V_{\sigma,\lambda}^{-\infty})^U. \] Neither the construction of $Z_U$ nor the local sections and restriction maps of $\mathscr F_{U,\sigma,\lambda}$ are given in this record.

For a smooth complex vector bundle $E\to Z_U$, the standard sheaf of distributional sections assigns to an open set $O\subseteq Z_U$ \[ \mathscr D'(E)(O)= \left(C_c^\infty(O,E^*\otimes\operatorname{Dens}_{Z_U})\right)', \] with continuous dual and the usual restriction maps. The conjectural question is whether, for every $G,P,U$ as above and every parameter in the intended generic range, there is a suitable bundle $E=E_{U,\sigma,\lambda}$ and a canonical sheaf isomorphism \[ \mathscr F_{U,\sigma,\lambda}\ \cong\ \mathscr D'(E_{U,\sigma,\lambda}). \] This means compatible isomorphisms on every open set, commuting with restriction, with the intended topological or module structures if required by the report. An isomorphism of global-section vector spaces alone is insufficient, and the bundle is to be supplied by the intended construction rather than chosen without a specified relation to the representation.

The record does not define the excluded parameters, canonical choices, rank or construction of the bundle, or the sheaf category. Its supplied assessment dated 2026-08-21 explicitly says the surrounding definitions are insufficient to identify a settled theorem and reports no identified resolution of the isolated conjecture.

Short English statement

For generic rank-one principal series, the problem asks whether the sheaf describing invariant distribution vectors is canonically the distribution-section sheaf of a vector bundle. The supplied record names the objects but omits their local construction and the precise generic range.

Sources

• Local UnsolvedMath registry, permanent ID $30000862$, exact source code catalogue entry, proposed year 2007; no individual proposer is supplied.
• Harmonische Analysis und Darstellungstheorie Topologischer Gruppen (2008), Oberwolfach Reports, DOI 10.4171/owr/2007/49, pp. 2893--2942. https://doi.org/10.4171/owr/2007/49.
• The supplied corrected_verified note says the complete question was retained while parabolic, subgroup, and auxiliary-space notation was restored. The open label and assessment checked 2026-08-21 are dated supplied triage requiring independent verification; the assessment itself stresses missing surrounding definitions.