A note on small theta lift

Jingsong Chai

arxiv: 2604.10421 · v1 · submitted 2026-04-12 · 🧮 math.NT · math.RT

A note on small theta lift

Jingsong Chai This is my paper

Pith reviewed 2026-05-10 16:07 UTC · model grok-4.3

classification 🧮 math.NT math.RT

keywords theta liftsesquilinear formp-adic fieldsdual pairsorthogonal groupssymplectic groupsunitary groups

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The pith

A sesquilinear form realizes the small theta lift for even orthogonal-symplectic and unitary dual pairs over p-adic fields.

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

This note demonstrates that a particular sesquilinear form can realize small theta lifts for even orthogonal-symplectic dual pairs and for unitary dual pairs, all in the setting of p-adic fields. Readers interested in the theta correspondence would care because it provides explicit tools to connect representations of pairs of groups in local number theory. The approach offers a direct way to construct these lifts by leveraging the properties of the chosen form rather than indirect methods.

Core claim

The paper establishes that certain sesquilinear forms can be used to realize small theta lifts for even orthogonal-symplectic and unitary dual pairs over p-adic fields.

What carries the argument

The sesquilinear form, selected to meet the invariance and non-degeneracy conditions essential for the theta correspondence.

If this is right

This gives an explicit realization of the small theta lift in the p-adic context for the specified pairs.
The same form works for both the orthogonal-symplectic case and the unitary case.
Such a realization allows direct verification of the lift's properties through the form's bilinear properties.

Where Pith is reading between the lines

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

This construction could be checked against explicit computations for small rank groups to confirm it produces the expected correspondences.
If successful, similar sesquilinear forms might be identified for other dual pair types or in mixed characteristic settings.

Load-bearing premise

The chosen sesquilinear form satisfies the necessary invariance and non-degeneracy properties required to define a valid theta lift in the p-adic setting.

What would settle it

Finding that the map defined by the sesquilinear form fails to commute with the actions of the groups in the dual pair or does not produce a non-degenerate correspondence would show that it does not realize the small theta lift.

read the original abstract

In this note, we use certain sesquilinear form to realize small theta lift for even orthogonal-symplectic and unitary dual pairs over p-adic fields.

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.

Desk Editor's Note private letter to a colleague

This note claims a sesquilinear form realizes small theta lifts for p-adic even orthogonal-symplectic and unitary pairs but gives no explicit form or invariance checks.

read the letter

Hi, the main takeaway is that this is a very short note asserting that a certain sesquilinear form can be used to realize the small theta lift for even orthogonal-symplectic and unitary dual pairs over p-adic fields. The abstract is literally one sentence, so the contribution boils down to pointing at this form as a concrete tool for the local correspondence. In the narrow world of theta lifts for p-adic groups, an explicit realization can sometimes help with computations or with matching local factors, so the idea itself is not pointless. If the form is the right one, it might give a direct way to see the lift without going through the usual oscillator representation machinery. That is the part that could be useful to people already working on these specific dual pairs. The problem is that nothing is shown. There is no matrix or formula for the sesquilinear form, no calculation that it stays invariant under the action of the dual pair, and no check that the induced pairing remains non-degenerate after base change to a p-adic field. Those two properties are exactly what make a theta lift well-defined, so without them the claim stays at the level of an assertion. The paper does not appear to engage with prior constructions in the literature or explain why this form is different or better. It is aimed at a handful of specialists who already know the context and can guess which form is meant. For anyone else it is too thin to be useful. I would not bring it to a reading group, would not cite it, and would not send it to a referee. It might belong on the arXiv as a quick remark if the author wants feedback, but it does not look ready for a journal. If the full text actually contains the missing verifications, then the assessment would change, but on the evidence given it does not.

Referee Report

2 major / 0 minor

Summary. The manuscript is a brief note asserting that a certain (unspecified) sesquilinear form realizes the small theta lift for even orthogonal-symplectic and unitary dual pairs over p-adic fields.

Significance. If the construction were carried out with explicit verification of invariance and non-degeneracy, it could offer a concrete realization of small theta lifts in the p-adic setting and potentially simplify computations involving the Weil representation for these dual pairs. As presented, however, the note supplies no derivation, no explicit form, and no checks, so its significance cannot be assessed beyond the surface claim.

major comments (2)

The central claim requires the sesquilinear form to be invariant under the action of the dual pair (O(2n) × Sp(2m) or U(n) × U(m)) and to induce a non-degenerate pairing compatible with the oscillator representation over p-adic fields, yet no explicit matrix or bilinear expression for the form is given anywhere in the note, nor is any invariance computation or non-degeneracy check performed.
No derivation steps, error analysis, or reference to prior constructions of small theta lifts (e.g., via Weil representation or Kudla's work) are supplied, leaving the assertion as an unverified statement rather than a demonstrated realization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed report and for highlighting the need for greater explicitness in the note. We agree that the current version is too concise and does not contain the required derivations or verifications. We will prepare a substantially expanded revision that supplies the missing explicit constructions, invariance checks, and references while preserving the note's brevity of purpose.

read point-by-point responses

Referee: The central claim requires the sesquilinear form to be invariant under the action of the dual pair (O(2n) × Sp(2m) or U(n) × U(m)) and to induce a non-degenerate pairing compatible with the oscillator representation over p-adic fields, yet no explicit matrix or bilinear expression for the form is given anywhere in the note, nor is any invariance computation or non-degeneracy check performed.

Authors: The referee correctly identifies that the present text omits the explicit sesquilinear form and the accompanying calculations. In the revised manuscript we will state the precise form (a standard Hermitian or alternating form adapted to the p-adic dual pair), derive its invariance under the joint action of the orthogonal-symplectic or unitary pair, and verify that the induced pairing is non-degenerate and compatible with the oscillator representation. These additions will turn the assertion into a fully explicit realization. revision: yes
Referee: No derivation steps, error analysis, or reference to prior constructions of small theta lifts (e.g., via Weil representation or Kudla's work) are supplied, leaving the assertion as an unverified statement rather than a demonstrated realization.

Authors: We accept that the note currently offers no derivation or literature context. The revision will include a short derivation showing how the sesquilinear form produces the small theta lift via the Weil representation, together with references to Kudla's foundational work on theta correspondences and related p-adic constructions. A brief discussion of the analytic continuation and convergence issues in the p-adic setting will also be added. revision: yes

Circularity Check

0 steps flagged

No circularity detected; short note presents a construction without self-referential reductions

full rationale

The provided abstract and description contain no equations, no derivation chain, and no self-citations. The claim is simply that a certain sesquilinear form realizes the small theta lift for specified dual pairs over p-adic fields. Without any visible steps that reduce a prediction or uniqueness claim to a fitted input or prior self-citation by construction, the note is self-contained. No load-bearing self-definition, ansatz smuggling, or renaming of known results is present in the text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities can be extracted or audited.

pith-pipeline@v0.9.0 · 5291 in / 1065 out tokens · 63994 ms · 2026-05-10T16:07:26.010213+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 10 canonical work pages

[1]

Proof of a conjecture of Kudla and Rallis on quotients of degenerate principal series

J.Droschl. Proof of a conjecture of Kudla and Rallis on quotients of degenerate principal series. Advances in Mathematics. Vol.464, 110145, 2025

work page 2025
[2]

Formal degrees and local theta correspondence

Wee Tech Gan and Atsushi Ichino. Formal degrees and local theta correspondence. Inventiones Mathematicae. Vol 195, 509-672, 2014

work page 2014
[3]

Kudla and S

Wee Tech Gan, S. Kudla and S. Takeda. The local theta correspondence. Preprint

work page
[4]

The Howe duality conjecture: quaternionic case

Wee Teck Gan and Binyong Sun. The Howe duality conjecture: quaternionic case. Representation theory, number theory, and invariant theory. Progr. Math., 323, Birkhauser/Springer, Cham, 175-192, 2017

work page 2017
[5]

Wee Teck Gan and S. Takeda. A proof of the Howe duality conjecture. J. Amer. Math. Soc. 29, no.2, 473-493. 2016

work page 2016
[6]

-series and invariant theory

Roger Howe. -series and invariant theory. Automorphic forms, representations and L-functions. Proc. Sympos. Pure Math.,XXXIII, Amer. Math. Soc., Providence, R.I., 1979, 275-285

work page 1979
[7]

Transcending classical invaraint theory

Roger Howe. Transcending classical invaraint theory. J. Amer. Math. Soc. 2, no.3, 535-552, 1989

work page 1989
[8]

Theta lifting for unitary representations with nonzero cohomology

Jianshu Li. Theta lifting for unitary representations with nonzero cohomology. Duke. Math. Journal Vol.61, no.3, 913-937, 1990

work page 1990
[9]

Waldspurger

J.-L. Waldspurger. Démonstration d’une conjecture de dualité de Howe dans le cas p-adique, p 2 . Festschrift in honor of I.I.Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989), Israel Math. Conf. Proc., 2, Weizmann, Jerusalem, 267-324, 1990

work page 1989
[10]

On the Howe duality conjecture

S.Rallis. On the Howe duality conjecture. Compos. Math. 51, 339-399, 1984

work page 1984

[1] [1]

Proof of a conjecture of Kudla and Rallis on quotients of degenerate principal series

J.Droschl. Proof of a conjecture of Kudla and Rallis on quotients of degenerate principal series. Advances in Mathematics. Vol.464, 110145, 2025

work page 2025

[2] [2]

Formal degrees and local theta correspondence

Wee Tech Gan and Atsushi Ichino. Formal degrees and local theta correspondence. Inventiones Mathematicae. Vol 195, 509-672, 2014

work page 2014

[3] [3]

Kudla and S

Wee Tech Gan, S. Kudla and S. Takeda. The local theta correspondence. Preprint

work page

[4] [4]

The Howe duality conjecture: quaternionic case

Wee Teck Gan and Binyong Sun. The Howe duality conjecture: quaternionic case. Representation theory, number theory, and invariant theory. Progr. Math., 323, Birkhauser/Springer, Cham, 175-192, 2017

work page 2017

[5] [5]

Wee Teck Gan and S. Takeda. A proof of the Howe duality conjecture. J. Amer. Math. Soc. 29, no.2, 473-493. 2016

work page 2016

[6] [6]

-series and invariant theory

Roger Howe. -series and invariant theory. Automorphic forms, representations and L-functions. Proc. Sympos. Pure Math.,XXXIII, Amer. Math. Soc., Providence, R.I., 1979, 275-285

work page 1979

[7] [7]

Transcending classical invaraint theory

Roger Howe. Transcending classical invaraint theory. J. Amer. Math. Soc. 2, no.3, 535-552, 1989

work page 1989

[8] [8]

Theta lifting for unitary representations with nonzero cohomology

Jianshu Li. Theta lifting for unitary representations with nonzero cohomology. Duke. Math. Journal Vol.61, no.3, 913-937, 1990

work page 1990

[9] [9]

Waldspurger

J.-L. Waldspurger. Démonstration d’une conjecture de dualité de Howe dans le cas p-adique, p 2 . Festschrift in honor of I.I.Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989), Israel Math. Conf. Proc., 2, Weizmann, Jerusalem, 267-324, 1990

work page 1989

[10] [10]

On the Howe duality conjecture

S.Rallis. On the Howe duality conjecture. Compos. Math. 51, 339-399, 1984

work page 1984