Plancherel Volumes of Rankin–Selberg Metric Balls

Definitions and mathematical statement

The supplied record concerns irreducible supercuspidal representations of $G=\operatorname{GL}_n(F)$. It does not specify the local field $F$, the admissible values of $n$, the representation category, or whether representations are required to be unitary. Write $X_n(F)$ for the intended set of isomorphism classes, once these choices are fixed.

A pseudometric on a set $X$ is a function $d:X\times X\to[0,\infty)$ such that, for all $x,y,z\in X$, \[ d(x,x)=0,\qquad d(x,y)=d(y,x),\qquad d(x,z)\le d(x,y)+d(y,z). \] Distinct elements may have distance zero. Let $d_{\mathrm{RS}}$ denote the local Rankin--Selberg pseudometric intended by the source. Its defining formula and normalization are absent from the supplied record; the axioms above do not identify that pseudometric.

For a center $\pi\in X_n(F)$ and a real radius $r\ge0$, the two possible ball conventions are \[ B_r^{<}(\pi)=\{\sigma\in X_n(F):d_{\mathrm{RS}}(\pi,\sigma)<r\}, \qquad B_r^{\le}(\pi)=\{\sigma\in X_n(F):d_{\mathrm{RS}}(\pi,\sigma)\le r\}. \] The record does not choose between them or specify the intended range of radii. In particular, the distinction matters at $r=0$.

Question, conditional on the missing specifications. For the intended local field, rank, representation space, Rankin--Selberg pseudometric, and ball convention, determine \[ \mu_{\mathrm{Pl}}\left(B_r(\pi)\right) \] as a function of the center $\pi$ and the allowed radius $r$, where $\mu_{\mathrm{Pl}}$ is the intended Plancherel measure. For the usual group Plancherel interpretation, one must specify the relevant unitary representation space and the normalization of Haar measure on $G$. The record does not explain how its supercuspidal representation space is related to that measure, nor does it supply a measurability assertion for these balls. Consequently, this is a formulation schema preserving the supplied question, rather than a fully specified measure-computation problem.

Short English statement

The question asks for the Plancherel measure of a ball centered at a supercuspidal representation of $\operatorname{GL}_n(F)$, using the local Rankin--Selberg pseudometric. The supplied record leaves the pseudometric formula, representation domain, measure normalization, and ball boundary convention unspecified. These choices require clarification before a precise numerical or functional answer is determined.

Sources

• Local UnsolvedMath registry: supplied complete source record, permanent ID $30004503$, source problem code catalogue entry, title "Plancherel Volumes of Rankin--Selberg Metric Balls." Project registry references are data/unsolvedmath/problems.json and catalog/problems.json; the supplied material does not identify which file contains this entry.
• Supplied report citation: Automorphic Forms and Arithmetic (2021), Oberwolfach Reports, pp. 1317--1383, DOI 10.4171/owr/2020/26. https://doi.org/10.4171/owr/2020/26. The background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions. No individual proposer is supplied; the proposed year is recorded as 2020.
• Supplied literature source: "Oberwolfach Report 26/2020," https://doi.org/10.4171/owr/2020/26. The literature assessment checked on 2026-08-22 labels the problem open and describes the Plancherel-measure question as a natural question with no cited computation resolving it. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its time-sensitive status and cited claims require independent verification.
• The registry assigns the statement status corrected_verified and says that mathematical content was preserved while notation, typography, omitted definitions, or duplicate clauses were repaired. This is a supplied editorial claim, not mathematical verification of this draft.