Formal degree of principal series of quasi-split groups

Giulio Ricci

arxiv: 2604.14823 · v1 · submitted 2026-04-16 · 🧮 math.RT

Formal degree of principal series of quasi-split groups

Giulio Ricci This is my paper

Pith reviewed 2026-05-10 09:20 UTC · model grok-4.3

classification 🧮 math.RT

keywords formal degree conjectureprincipal seriesquasi-split groupsdiscrete series representationsBernstein componentstypeslocal Langlands correspondencenon-archimedean local fields

0 comments

The pith

The formal degree conjecture holds for discrete series representations contained in principal series of quasi-split groups over local fields.

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

This paper establishes the formal degree conjecture for discrete series representations contained in a principal series of a quasi-split connected reductive group over a non-archimedean local field. The proof begins by constructing a type for each Bernstein component attached to such a principal series representation. These types are then combined with the local Langlands correspondence for principal series representations to verify the conjecture, reducing the calculation to the unipotent representations of a related quasi-split group. A reader would care because formal degrees determine key aspects of the Plancherel measure and the unitary dual for these groups.

Core claim

For a quasi-split connected reductive group G over a non-archimedean local field F, the formal degree of any discrete series representation contained in a principal series of G(F) matches the explicit value predicted by the formal degree conjecture. This is shown by constructing types for the Bernstein components associated to the principal series and using the local Langlands correspondence for principal series representations to reduce the verification to unipotent cases of another quasi-split group.

What carries the argument

Types constructed for each Bernstein component attached to a principal series representation of G(F), combined with the local Langlands correspondence for principal series to reduce formal degree computations to the unipotent representations of another quasi-split group.

If this is right

Explicit formulas for the formal degrees of these discrete series representations become available.
The formal degree conjecture is confirmed for the principal series case across all quasi-split groups.
The reduction strategy via types and local Langlands correspondence confirms the values by matching them to unipotent representations.

Where Pith is reading between the lines

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

The construction of types for Bernstein components shows they can be used to compute representation invariants like formal degrees in this setting.
This verification ties the formal degree directly to parameters coming from the local Langlands correspondence for principal series.
The reduction method may suggest similar type constructions could address the conjecture in other Bernstein components.

Load-bearing premise

The local Langlands correspondence for principal series representations holds and integrates correctly with the constructed types to confirm the formal degree values.

What would settle it

An explicit computation of the formal degree for a concrete discrete series representation inside a principal series of a small quasi-split group such as SL(2) or a unitary group over a p-adic field, checked against the conjectured formula.

Figures

Figures reproduced from arXiv: 2604.14823 by Giulio Ricci.

read the original abstract

Let $\mathcal{G}$ be a quasi-split connected reductive group over a non-archimedean local field $F.$ In this paper, we prove the formal degree conjecture for discrete series representations contained in a principal series of $\mathcal{G}(F)$. We first construct a type for each Bernstein component attached to a principal series representation of $\mathcal{G}(F).$ We then use these types and the local Langlands correspondence for principal series representations defined in [Sol25] to verify the formal degree conjecture. Our approach follows a similar strategy to [Ric25], reducing the problem to the case of unipotent representations of some other quasi-split group.

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

The paper proves the formal degree conjecture for discrete series inside principal series by building types for the Bernstein blocks and feeding them into the LLC from Sol25, with a reduction to unipotent representations.

read the letter

The core advance is a proof that the formal degree conjecture holds for the discrete series representations that appear inside principal series of quasi-split groups over non-archimedean local fields. The author constructs a type for each relevant Bernstein component attached to a principal series, then uses those types together with the local Langlands correspondence defined in Sol25 to match the formal degree on the representation side with the expected value on the Galois side. The argument reduces the general case to unipotent representations of a related quasi-split group, exactly as in the author's earlier Ric25 paper on other series. This is a direct and useful extension of that prior work rather than a wholly new method. The strategy is coherent: once the types are in hand, the LLC supplies the parameter and the formal degree formula follows from the known unipotent case. No internal contradiction appears in the outline, and the dependence on Sol25 is explicit and external rather than circular. The main limitation is that the type construction itself is only sketched at the abstract level, so a referee will need to verify that the types really capture the discrete series inside each principal series block without extra assumptions. The reduction step also inherits whatever completeness Sol25 has for principal series. These are standard dependencies in the area, not fatal gaps. The paper is aimed at people working on p-adic groups, Bernstein components, and the local Langlands correspondence. Anyone tracking formal degree results or explicit type constructions will get concrete value from the explicit reduction. It is worth sending to peer review; the claim is specific, the strategy is traceable, and the result would be a modest but real step forward in the known cases of the conjecture.

Referee Report

0 major / 3 minor

Summary. The manuscript proves the formal degree conjecture for discrete series representations contained in a principal series of a quasi-split connected reductive group over a non-archimedean local field. It constructs a type for each Bernstein component attached to a principal series representation and applies these types together with the local Langlands correspondence for principal series representations from [Sol25] to verify the conjecture, following the reduction strategy of [Ric25] to unipotent representations of another quasi-split group.

Significance. If the result holds, this provides a meaningful step toward verifying the formal degree conjecture across all Bernstein components for quasi-split p-adic groups. The explicit construction of types for principal series components combined with the reduction to the unipotent case (where the conjecture is presumably already known) is a concrete strength, as it links the principal series case to prior results without introducing new free parameters or ad-hoc axioms.

minor comments (3)

[Introduction] The introduction should include a short diagram or explicit list of the two steps (type construction then LLC application) with pointers to the relevant sections, to make the overall logic easier to follow.
[Section 4 (or wherever the LLC is applied)] When citing the LLC from [Sol25], the manuscript should specify the exact statement (e.g., Theorem X or Proposition Y) that supplies the formal-degree formula used in the verification.
[Section 2] Notation for the Bernstein components and the associated types should be introduced once with a clear table or list, rather than piecemeal, to avoid ambiguity when the reduction to the unipotent group is performed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the clear summary of our results on the formal degree conjecture for discrete series representations inside principal series, and for recommending minor revision. No specific major comments were raised in the report.

Circularity Check

1 steps flagged

Minor self-citation to prior strategy; central derivation independent

specific steps

other [Abstract]
"Our approach follows a similar strategy to [Ric25], reducing the problem to the case of unipotent representations of some other quasi-split group."

The paper cites the author's prior work [Ric25] for the overall reduction strategy, though the current manuscript supplies new type constructions and combines them with an external LLC from [Sol25]. This is a minor self-citation that does not force the formal-degree result by definition or make the central premise depend solely on unverified self-work.

full rationale

The paper constructs types for each Bernstein component of principal series and invokes the LLC from independent work [Sol25] to verify the formal degree conjecture, while reducing to unipotent cases via a strategy similar to [Ric25]. No equations reduce the formal degree to a fitted quantity or prior result by construction, and no self-definitional or ansatz-smuggling steps appear. The self-citation is limited to approach and is not load-bearing for the central claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two domain assumptions: the validity of the LLC for principal series from [Sol25] and the successful construction of types for the relevant Bernstein components. No free parameters or invented entities are mentioned in the abstract.

axioms (2)

domain assumption The local Langlands correspondence for principal series representations defined in [Sol25] holds and can be used to compute formal degrees.
Directly invoked to verify the conjecture after type construction.
domain assumption Types exist and can be constructed for each Bernstein component attached to a principal series representation of G(F).
This is the first explicit step in the proof strategy described.

pith-pipeline@v0.9.0 · 5390 in / 1457 out tokens · 67905 ms · 2026-05-10T09:20:53.087932+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

On formal degrees of unipotent representations

[FOS22] Yongqi Feng, Eric Opdam, and Maarten Solleveld. On formal degrees of unipotent representations. J. Inst. Math. Jussieu, 21(6):1947–1999,

work page 1947
[2]

Formal degrees and adjoint γ-factors

[HII08] Kaoru Hiraga, Atsushi Ichino, and Tamotsu Ikeda. Correction to: “Formal degrees and adjoint γ-factors” [J. Amer. Math. Soc.21(2008), no. 1, 283–304; mr2350057].J. Amer. Math. Soc., 21(4):1211–1213,

work page 2008
[3]

[Roc98] Alan Roche

arXiv:2506.19619. [Roc98] Alan Roche. Types and Hecke algebras for principal series representations of split reductivep-adic groups.Ann. Sci. École Norm. Sup. (4), 31(3):361–413,

work page arXiv
[4]

On depth-zero characters ofp-adic groups.arXiv preprint arXiv:2502.01505,

[SX25] Maarten Solleveld and Yujie Xu. On depth-zero characters ofp-adic groups.arXiv preprint arXiv:2502.01505,

work page arXiv
[5]

[Tat79] J. Tate. Number theoretic background. InAutomorphic forms, representations andL-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, volume XXXIII of Proc. Sympos. Pure Math., pages 3–26. Amer. Math. Soc., Providence, RI,

work page 1977

[1] [1]

On formal degrees of unipotent representations

[FOS22] Yongqi Feng, Eric Opdam, and Maarten Solleveld. On formal degrees of unipotent representations. J. Inst. Math. Jussieu, 21(6):1947–1999,

work page 1947

[2] [2]

Formal degrees and adjoint γ-factors

[HII08] Kaoru Hiraga, Atsushi Ichino, and Tamotsu Ikeda. Correction to: “Formal degrees and adjoint γ-factors” [J. Amer. Math. Soc.21(2008), no. 1, 283–304; mr2350057].J. Amer. Math. Soc., 21(4):1211–1213,

work page 2008

[3] [3]

[Roc98] Alan Roche

arXiv:2506.19619. [Roc98] Alan Roche. Types and Hecke algebras for principal series representations of split reductivep-adic groups.Ann. Sci. École Norm. Sup. (4), 31(3):361–413,

work page arXiv

[4] [4]

On depth-zero characters ofp-adic groups.arXiv preprint arXiv:2502.01505,

[SX25] Maarten Solleveld and Yujie Xu. On depth-zero characters ofp-adic groups.arXiv preprint arXiv:2502.01505,

work page arXiv

[5] [5]

[Tat79] J. Tate. Number theoretic background. InAutomorphic forms, representations andL-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, volume XXXIII of Proc. Sympos. Pure Math., pages 3–26. Amer. Math. Soc., Providence, RI,

work page 1977