Semisimple types for quaternionic forms of p-adic classical groups and compatible beta-extensions

Daniel Skodlerack; Shuyang Ye

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Quaternionic forms of p-adic classical groups admit Bushnell-Kutzko-Stevens types for every Bernstein block along with compatible beta-extensions.

2026-07-02 02:32 UTC pith:BYXFCSHN

load-bearing objection This paper extends Bushnell-Kutzko-Stevens types to quaternionic forms of p-adic classical groups by reduction via transfer, covering every Bernstein block plus compatible beta-extensions.

arxiv 2607.01074 v1 pith:BYXFCSHN submitted 2026-07-01 math.RT

Semisimple types for quaternionic forms of p-adic classical groups and compatible beta-extensions

Daniel Skodlerack , Shuyang Ye This is my paper

classification math.RT

keywords Bushnell-Kutzko-Stevens typesBernstein blocksquaternionic formsp-adic classical groupsbeta-extensionsBruhat-Tits buildingsmooth representationstransfer maps

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a Bushnell-Kutzko-Stevens type for every Bernstein block in the smooth complex representations of a quaternionic form G of a p-adic classical group with p odd. It also produces a system of compatible beta-extensions parametrized by the points of a chamber in the Bruhat-Tits building of the centralizer G_beta, with these extensions related to each other by transfer. The work extends the classical-group case by using transfer maps to reduce the quaternionic situation to the known machinery. A reader would care because such types decompose the representation category into blocks that can be studied via Hecke algebras and endomorphism rings.

Core claim

Let G be a quaternionic form of a p-adic classical group with p odd. We construct a Bushnell-Kutzko-Stevens type for every Bernstein block of the category of smooth complex representations of G. Further we construct a system of compatible beta-extensions, i.e. a family of beta-extensions parametrised by the points of a chamber of the Bruhat-Tits building of the centralizer G_beta which are related via transfer.

What carries the argument

Bushnell-Kutzko-Stevens type together with a system of compatible beta-extensions parametrized by a chamber of the Bruhat-Tits building of G_beta and related by transfer maps.

Load-bearing premise

The quaternionic form G reduces to the classical-group case via the given transfer maps, and the prior Bushnell-Kutzko-Stevens machinery extends without additional obstructions when p is odd.

What would settle it

Existence of a Bernstein block for such a G in which no Bushnell-Kutzko-Stevens type exists or in which beta-extensions cannot be chosen compatibly via transfer.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Every Bernstein block of smooth representations of G decomposes according to the constructed type.
The beta-extensions supply a consistent choice of extensions across the chamber in the Bruhat-Tits building.
Transfer maps carry the types and beta-extensions from the classical case to the quaternionic case.
The construction applies uniformly to all Bernstein blocks when p is odd.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar transfer arguments might apply to other non-split forms once the classical case is settled.
The resulting types could be used to describe the Hecke algebras attached to these blocks explicitly.
Compatibility via transfer may give a way to compare representations across different inner forms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

This paper extends Bushnell-Kutzko-Stevens types to quaternionic forms of p-adic classical groups by reduction via transfer, covering every Bernstein block plus compatible beta-extensions.

read the letter

The core result is a construction of BKS types for all Bernstein blocks of smooth representations of these quaternionic forms, together with a family of beta-extensions parametrized by points in a chamber of the centralizer's Bruhat-Tits building and linked by transfer.

The new part is the application to quaternionic forms. Prior work handled the classical groups, and this paper imports that machinery through transfer maps. The claim that this works for every block and produces the compatible system is the concrete advance. It sits inside the larger program of classifying representations of p-adic classical groups and their inner forms.

The approach looks solid on its own terms. The reduction to the classical case is the load-bearing step, and the abstract states that the constructions go through for odd p. No internal contradiction appears in the stated claim, and the beta-extension compatibility is defined explicitly via the building. If the transfer preserves the necessary properties without extra obstructions, the argument holds.

A minor point is how much of the proof is new technical work versus direct citation of earlier results on classical groups. The paper does not appear to introduce fundamentally different ingredients beyond the transfer, but that is not a flaw if the reduction is clean.

This is for people already working on types and the local Langlands correspondence for p-adic groups. It fills a specific remaining case. I would send it to peer review; the result is targeted enough and grounded enough in existing machinery to merit referee attention.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim is a construction of Bushnell-Kutzko-Stevens types and compatible β-extensions for quaternionic forms via reduction to the classical-group case using transfer maps. The abstract and reader's summary provide no equations, fitted parameters, or self-citations that reduce any result to its own inputs by construction. The derivation relies on prior established machinery for classical groups (with p odd), which is independent external support rather than a self-referential loop. No load-bearing step exhibits self-definition, renaming of known results, or uniqueness imported solely from the authors' prior unverified work.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No details on free parameters, axioms, or invented entities are supplied by the abstract; the constructions presumably rely on standard background results in the theory of types for p-adic groups.

pith-pipeline@v0.9.1-grok · 5609 in / 1231 out tokens · 48722 ms · 2026-07-02T02:32:20.886149+00:00 · methodology

0 comments

read the original abstract

Let $G$ be a quaternionic form of a $p$-adic classical group ($p$ odd). We construct a Bushnell-Kutzko-Stevens type for every Bernstein block of the category of smooth complex representations of $G$. Further we construct a system of compatible $\beta$-extensions, i.e. a family of $\beta$-extensions parametrised by the points of a chamber of the Bruhat-Tits building of the centralizer $G_\beta$ which are related via transfer.

discussion (0)

Reference graph

Works this paper leans on

6 extracted references

[1]

Le “centre

[Ber84] J. Bernstein. “Le “centre” de Bernstein”. Représentations des groupes réductifs sur un corps local . Ed. by P. Deligne. Travaux en Cours. Paris: Hermann, 1984, pp. 1–32. [BK93] C. Bushnell and P. Kutzko. The admissible dual of GL(N ) via compact open subgroups. Vol

1984
[2]

Bushnell and Philip C

[BK94] Colin J. Bushnell and Philip C. Kutzko. The admissible d ual of SL(N ). II. Proceedings of the London Mathematical Society . 3rd ser. 68.2 (1994), pp. 317–

1994
[3]

Bushnell and P

[BK98] C. Bushnell and P. Kutzko. Smooth representations of re ductivep-adic groups: structure theory via types. Proc. Lond. Math. Soc. 77 (1998), pp. 582–634. [CGP15] B. Conrad, O. Gabber, and G. Prasad. Pseudo-reductive Groups. 2nd ed. Vol

1998
[4]

Goldberg and A

[GR02] D. Goldberg and A. Roche. Types in SLn. Proceedings of the London Mathe- matical Society. 3rd ser. 85.1 (2002), pp. 119–138. [GR05] D. Goldberg and A. Roche. Hecke algebras and SLn-types. Proceedings of the London Mathematical Society . 3rd ser. 90.1 (2005), pp. 87–131. [KS20] R. Kurinczuk and S. Stevens. Cuspidal ℓ-modular representations of p-adi...

2002
[5]

Kurinczuk, D

[KSS21] R. Kurinczuk, D. Skodlerack, and S. Stevens. Endo pa rameters for p-adic clas- sical groups. Invent. Math. 223.2 (2021), pp. 597–723. [Mor99] L. Morris. Level Zero G-Types. Comp. Math. 118 (1999), pp. 135–157. [MS12] M. Miyauchi and S. Stevens. Semisimple types for p-adic classical groups. Math. Ann. 358.1-2 (2012), pp. 257–288. [MS14] A. Mìngues ...

2021
[6]

Berlin: Springer-Verlag, 1985

Grundlehren der Math- ematischen Wissenschaften. Berlin: Springer-Verlag, 1985 . [Séc05] V. Sécherre. Représentations lisses de GL(m,D ) II : β -extensions. Compositio Mathematica 141.6 (2005), pp. 1531–1550. [Sko13] D. Skodlerack. The centralizer of a classical group and Bruhat-Tits buildings. Ann. de l’Institut Fourier 63.2 (2013), 525–546. [Sko20] D. S...

1985

[1] [1]

Le “centre

[Ber84] J. Bernstein. “Le “centre” de Bernstein”. Représentations des groupes réductifs sur un corps local . Ed. by P. Deligne. Travaux en Cours. Paris: Hermann, 1984, pp. 1–32. [BK93] C. Bushnell and P. Kutzko. The admissible dual of GL(N ) via compact open subgroups. Vol

1984

[2] [2]

Bushnell and Philip C

[BK94] Colin J. Bushnell and Philip C. Kutzko. The admissible d ual of SL(N ). II. Proceedings of the London Mathematical Society . 3rd ser. 68.2 (1994), pp. 317–

1994

[3] [3]

Bushnell and P

[BK98] C. Bushnell and P. Kutzko. Smooth representations of re ductivep-adic groups: structure theory via types. Proc. Lond. Math. Soc. 77 (1998), pp. 582–634. [CGP15] B. Conrad, O. Gabber, and G. Prasad. Pseudo-reductive Groups. 2nd ed. Vol

1998

[4] [4]

Goldberg and A

[GR02] D. Goldberg and A. Roche. Types in SLn. Proceedings of the London Mathe- matical Society. 3rd ser. 85.1 (2002), pp. 119–138. [GR05] D. Goldberg and A. Roche. Hecke algebras and SLn-types. Proceedings of the London Mathematical Society . 3rd ser. 90.1 (2005), pp. 87–131. [KS20] R. Kurinczuk and S. Stevens. Cuspidal ℓ-modular representations of p-adi...

2002

[5] [5]

Kurinczuk, D

[KSS21] R. Kurinczuk, D. Skodlerack, and S. Stevens. Endo pa rameters for p-adic clas- sical groups. Invent. Math. 223.2 (2021), pp. 597–723. [Mor99] L. Morris. Level Zero G-Types. Comp. Math. 118 (1999), pp. 135–157. [MS12] M. Miyauchi and S. Stevens. Semisimple types for p-adic classical groups. Math. Ann. 358.1-2 (2012), pp. 257–288. [MS14] A. Mìngues ...

2021

[6] [6]

Berlin: Springer-Verlag, 1985

Grundlehren der Math- ematischen Wissenschaften. Berlin: Springer-Verlag, 1985 . [Séc05] V. Sécherre. Représentations lisses de GL(m,D ) II : β -extensions. Compositio Mathematica 141.6 (2005), pp. 1531–1550. [Sko13] D. Skodlerack. The centralizer of a classical group and Bruhat-Tits buildings. Ann. de l’Institut Fourier 63.2 (2013), 525–546. [Sko20] D. S...

1985