Localization of Types in Bruhat–Tits Buildings

Definitions and mathematical statement

Let $G$ be a connected reductive $p$-adic group, let $\mathcal B(G)$ denote its Bruhat--Tits building, and let $x\in\mathcal B(G)$. The source also refers to a finite-length representation $\tau(y,g)$ arising in the Mackey decomposition associated with a type, but does not specify the type, the inducing data, or the parameters $y,g$.

The proposed geometric statement uses additional source-dependent objects $H,G^0,\operatorname{proj}_0,\Omega_{t+},\Xi_t$, and $s_0$, whose definitions are absent from the supplied record. If $H$ acts on the building, its fixed-point set is \[ \mathcal B(G)^H=\{u\in\mathcal B(G):hu=u\text{ for every }h\in H\}. \] For the intended auxiliary data, write \[ A=\operatorname{proj}_0^{-1}\left(\mathcal B(G^0)\setminus\{x\}\right),\qquad C_t=\Omega_{t+}\setminus\left(\Omega_{t+}\cap\Xi_t\right) \quad(0<t<s_0). \] These expressions retain the source notation; they do not supply the missing definitions or identifications of buildings.

The proposed geometric assertion is that there exists a bounded convex neighborhood $\Gamma\subseteq\mathcal B(G)$ of $x$ such that \[ \mathcal B(G)^H\subseteq\Gamma \] and, for every $z\in\mathcal B(G)\setminus\Gamma$ and every geodesic segment $\gamma$ from $x$ to $z$, \[ \gamma\cap A\ne\varnothing \quad\text{or}\quad \exists\,t\in\mathbb R\;\left(0<t<s_0\ \text{ and }\ \gamma\cap C_t\ne\varnothing\right). \] Here the geodesic segment is the source's $[x,z]$; the existential parameter $t$ may depend on the segment. The intended building, metric, topology, and admissible auxiliary data must be specified before this becomes a fully determined conjecture. In particular, containment in a bounded $\Gamma$ requires $\mathcal B(G)^H$ to be bounded, a property not justified by the supplied hypotheses.

The source further proposes that this geometric condition excludes additional types in $\tau(y,g)$ far from $x$. The relation between $z$, the Mackey parameters $(y,g)$, and occurrence of a type is not supplied, and neither is a definition of the claimed small neighborhood. Consequently this representation-theoretic consequence cannot be formulated precisely from the record alone.

Short English statement

The proposed conjecture asks for a bounded convex neighborhood of $x$, containing the $H$-fixed points, such that every geodesic from $x$ to a point outside it intersects one of the specified obstruction sets. This is intended to localize the occurrence of types in Mackey components. Essential auxiliary definitions and the precise representation-theoretic consequence are missing, so this is a statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath problem registry, permanent ID $30004252$, exact source problem code catalogue entry, title Localization of Types in Bruhat--Tits Buildings; collection: Oberwolfach Reports Open Problems. The record supplies proposed year 2019 and names no proposer.
• Supplied report citation: New Developments in Representation Theory of p-adic Groups (2020), Oberwolfach Reports, pp. 2739--2819, DOI 10.4171/owr/2019/44. https://doi.org/10.4171/owr/2019/44. The supplied curation identifies the assertion as Conjecture 2; the report text and its surrounding definitions were not supplied or independently consulted.
• Supplied literature reference: Unicity of types for toral supercuspidals. https://arxiv.org/abs/1801.06721. The record does not supply authors or a precise theorem supporting the assessment.
• Literature assessment checked on 2026-08-22: supplied dated curation, not independently verified. The record labels the problem open and states that the general anisotropic-torus case is poorly understood, with later results covering special unicity cases without establishing the full conjecture. Its background explicitly requests independent verification of time-sensitive status and cited claims. The record also labels the statement corrected_verified, explaining that notation, typography, and sentence structure were repaired while preserving mathematical content; this label is not an independent mathematical verification of this draft.