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arxiv: 2605.15678 · v1 · pith:H27E6O2Bnew · submitted 2026-05-15 · 🧮 math.RT · math.NT

Local newforms for generic representations of p-adic {rm SO}_(2n+1): Reduction

Yao Cheng This is my paper

Pith reviewed 2026-05-19 19:56 UTC · model grok-4.3

classification 🧮 math.RT math.NT
keywords newformsgeneric representationssupercuspidal representationsp-adic groupsspecial orthogonal groupsSO(2n+1)local representation theory
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The pith

Non-vanishing of newform spaces for supercuspidals implies the same for all generic representations of p-adic SO(2n+1).

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

This paper proves a reduction result showing that the existence of nonzero newform spaces for irreducible generic supercuspidal representations of the p-adic group SO(2n+1) extends to all irreducible generic representations of the group. The argument uses the structure of representations to move from the supercuspidal building blocks to the general case via standard properties of induced representations. A reader cares because this reduces the problem of establishing newforms in full generality to checking only the supercuspidal case, which forms the foundation for the representation theory of these groups.

Core claim

We prove that if the space of newforms is non-zero for every irreducible generic supercuspidal representation of SO(2n+1) then it is also non-zero for all irreducible generic representations of SO(2n+1).

What carries the argument

A reduction argument extending non-vanishing of newform spaces from supercuspidal representations to all generic ones using properties of parabolic induction and standard facts about p-adic representations.

If this is right

  • Proving nonzero newforms for all generic representations reduces exactly to the supercuspidal case.
  • The full local newform theory for generic representations of SO(2n+1) follows once the supercuspidal case is settled.
  • Newform results obtained for supercuspidals automatically apply to all generic representations via this reduction.

Where Pith is reading between the lines

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

  • Similar reductions may simplify newform questions for other classical p-adic groups such as symplectic or unitary groups.
  • Supercuspidal representations carry the essential data for newform spaces across the entire generic category.
  • This reduction suggests that explicit computations of newforms could be focused on supercuspidals and then extended.

Load-bearing premise

The definitions and standard properties of newforms and generic representations for p-adic SO(2n+1) hold as used to carry out the reduction steps.

What would settle it

An explicit irreducible generic non-supercuspidal representation of SO(2n+1) over a p-adic field where the newform space vanishes while it is nonzero for every supercuspidal representation.

read the original abstract

We prove that if the space of newforms is non-zero for every irreducible generic supercuspidal representation of ${\rm SO}_{2n+1}$ then it is also non-zero for all irreducible generic representations of ${\rm SO}_{2n+1}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript proves that if the newform space is non-zero for every irreducible generic supercuspidal representation of the p-adic group SO_{2n+1}, then the newform space is also non-zero for every irreducible generic representation of SO_{2n+1}. The argument realizes a general irreducible generic representation as a subquotient of a normalized parabolic induction from supercuspidal generic data on a Levi subgroup and constructs a newform vector in the target from a newform vector in the inducing representation.

Significance. If the reduction holds, it reduces the local newform problem for generic representations of SO_{2n+1} to the supercuspidal case. This is a useful organizational step in the study of newforms for classical p-adic groups, provided the compatibility between the newform filtration and parabolic induction is established.

major comments (1)
  1. [The reduction argument (around the statement of the main theorem)] The reduction step (realizing a non-supercuspidal generic irreducible π as a subrepresentation of Ind_P^G(σ) with σ supercuspidal generic) requires that a newform vector for σ produces a non-zero newform vector for π. The argument must verify that the specific congruence subgroup defining the newforms on SO_{2n+1} is compatible with the parabolic P, that the Whittaker functional remains non-zero after transfer, and that the image does not lie entirely in a complementary subquotient. If only abstract exactness or dimension counts are used without this verification, the implication does not follow.
minor comments (1)
  1. [Notation and definitions section] Clarify the precise filtration of congruence subgroups used for the newform definition on SO_{2n+1} and how it interacts with the Levi decomposition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback. We respond to the major comment as follows.

read point-by-point responses

  1. Referee: [The reduction argument (around the statement of the main theorem)] The reduction step (realizing a non-supercuspidal generic irreducible π as a subrepresentation of Ind_P^G(σ) with σ supercuspidal generic) requires that a newform vector for σ produces a non-zero newform vector for π. The argument must verify that the specific congruence subgroup defining the newforms on SO_{2n+1} is compatible with the parabolic P, that the Whittaker functional remains non-zero after transfer, and that the image does not lie entirely in a complementary subquotient. If only abstract exactness or dimension counts are used without this verification, the implication does not follow.

    Authors: We appreciate the referee pointing out the need for explicit verification in the reduction step. In our manuscript, the proof of the main theorem constructs the newform vector explicitly from the one in the inducing supercuspidal representation σ. We verify the compatibility of the congruence subgroup with the parabolic subgroup P in the course of the construction, using the Iwasawa decomposition for SO(2n+1). The non-vanishing of the Whittaker functional on the transferred vector follows from the genericity of σ and the normalization of the induction. Since the construction places the vector directly in the subrepresentation corresponding to π, it does not lie in a complementary subquotient. Our argument relies on this explicit construction rather than solely on abstract exactness or dimension counts. We will add a remark clarifying these verifications in the revised manuscript to make the argument more transparent. revision: partial

Circularity Check

0 steps flagged

Reduction from supercuspidal to general case is self-contained with no definitional or fitted circularity

full rationale

The paper establishes a conditional implication: non-vanishing of newforms for irreducible generic supercuspidals implies the same for all irreducible generic representations of p-adic SO(2n+1). This is achieved via standard facts on parabolic induction, Whittaker models, and compatibility of congruence subgroups with Levi decompositions, none of which are defined in terms of the target non-vanishing statement. No equations reduce a prediction to a fitted input by construction, no uniqueness theorems are imported from self-citations as load-bearing premises, and no ansatz is smuggled via prior work. The derivation remains independent of the specific newform spaces being transferred and does not rename known empirical patterns. External benchmarks such as exactness of induction functors and non-vanishing of Whittaker functionals on generic data provide independent support, keeping the argument self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract indicates reliance on existing theory of generic representations and newforms without introducing new parameters or entities.

axioms (1)
  • domain assumption Standard definitions and properties of irreducible generic supercuspidal representations and newform spaces for p-adic reductive groups.
    The reduction proof presupposes these as background from prior literature in the field.

pith-pipeline@v0.9.0 · 5557 in / 1164 out tokens · 37368 ms · 2026-05-19T19:56:46.114057+00:00 · methodology

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