Stratified Equivalence for Bernstein Components

Definitions and mathematical statement

Conventional framework and unresolved scope. The record concerns representations of a reductive group over a nonarchimedean local field. To explain its notation, use the conventional framework in which $F$ is such a field, $\mathbf G$ is a connected reductive $F$-group, $G=\mathbf G(F)$, and representations are smooth complex representations. These conventions are an interpretation: the record does not specify the field, its characteristic, the precise group hypotheses, or the coefficient field, so the intended scope requires confirmation.

An inertial class $s=[M,\sigma]_G$ is the equivalence class of a pair consisting of a Levi subgroup $M$ and an irreducible supercuspidal representation $\sigma$ of $M$, under $G$-conjugacy and twisting by unramified characters. Write $\operatorname{Irr}(G)_s$ for the isomorphism classes of irreducible smooth representations of $G$ with inertial supercuspidal support $s$. In the conventional Bernstein description, let \[ T_s=\{[\sigma\otimes\chi]:\chi\in X_{\mathrm{nr}}(M)\} \cong X_{\mathrm{nr}}(M)/X_{\mathrm{nr}}(M,\sigma), \] where $X_{\mathrm{nr}}(M)$ is the group of unramified complex characters and $X_{\mathrm{nr}}(M,\sigma)$ is the subgroup fixing the isomorphism class of $\sigma$. Let $W_s$ be the finite subgroup of $N_G(M)/M$ preserving this unramified-twist family, acting on $T_s$. The algebra $\mathcal O(T_s)$ is its complex coordinate algebra, and $\mathcal O(T_s)\rtimes W_s$ denotes the ordinary crossed product for that action.

For $t\in T_s$, write $W_{s,t}$ for its stabilizer. Given compatible stabilizer cocycles $\alpha_{s,t}$ and transport maps, the twisted spectral extended quotient has the form \[ (T_s/\!/W_s)_\alpha = \left\{(t,\rho):t\in T_s,\ \rho\in\operatorname{Irr}({\mathbb{C}}[W_{s,t},\alpha_{s,t}])\right\}/W_s, \] where ${\mathbb{C}}[W_{s,t},\alpha_{s,t}]$ is the twisted group algebra and the quotient uses the specified transport maps. The source does not supply these cocycles or maps. Write $\mathcal H(G,s)$ for the source's Bernstein algebra associated with the block $s$; its precise algebra model is also unspecified.

Source question, subject to these qualifications. For every inertial class $s$, does the Aubert--Baum--Plymen--Solleveld (ABPS) bijection $\mu_s$ between \[ \operatorname{Irr}(G)_s \quad\text{and}\quad (T_s/\!/W_s)_\alpha \] arise from a stratified equivalence between \[ \mathcal O(T_s)\rtimes W_s \quad\text{and}\quad \mathcal H(G,s), \] in the sense that the induced correspondence on irreducible representations agrees with the specified $\mu_s$?

The record does not define stratified equivalence, specify the particular ABPS bijection, or explain the identification of the twisted quotient with the relevant representation spectrum. In particular, the displayed crossed product is untwisted, whereas the quotient is described as twisted. An ordinary crossed product supplies untwisted stabilizer representation data; a nontrivial twist requires additional justification or a modified algebra. Thus this is a qualified statement of the supplied question, and the record does not determine a fully specified general conjecture.

Short English statement

For each Bernstein component, the question asks whether its proposed ABPS parametrization comes from a stratified equivalence of the two displayed algebras. The field and group hypotheses, cocycle data, algebra model, and meaning of stratified equivalence require clarification, especially because the source pairs a twisted quotient with an untwisted crossed product.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30004256$, title Stratified Equivalence for Bernstein Components, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record supplies proposed year $2019$ and names no individual proposer.
• Supplied source 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 record attributes the problem to workshop open-problem discussions documented in this report; it gives no precise problem page or passage.
• Supplied literature source, "OWR 2019 44": https://ems.press/content/serial-article-files/46824.
• Supplied literature source, "Baum--Aubert--Plymen--Solleveld": https://celebratio.org/Baum_P/article/714/.
• Literature assessment checked on 2026-08-22: the supplied curation labels the problem partially solved and reports results for inner forms of general linear groups and some principal series, while reporting no general result for arbitrary reductive $p$-adic groups. This is supplied dated curation derived from the OWR material, not an independently verified assessment. Its time-sensitive status and cited claims require independent verification.
• The supplied statement status is corrected_verified; its accompanying explanation says that mathematical content was preserved while notation, typography, and sentence structure were repaired. This is registry metadata, not independent mathematical verification of this draft.