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arxiv: 2604.22259 · v1 · submitted 2026-04-24 · 🧮 math.NT · math.RT

Exceptional poles of archimedean Rankin-Selberg L-functions for principal series representations of GL(n,R)

Yeongseong Jo , Santosh Nadimpalli , Akash Yadav This is my paper

Pith reviewed 2026-05-08 10:11 UTC · model grok-4.3

classification 🧮 math.NT math.RT
keywords Rankin-Selberg L-functionsprincipal series representationsGL(n,R)exceptional polesarchimedean L-functionsrepresentation derivativesautomorphic forms
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The pith

For irreducible principal series representations of GL(n,R) in general position, the two notions of exceptional pole in their Rankin-Selberg L-functions are the same.

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

The paper shows that two different definitions of exceptional poles in the Rankin-Selberg L-functions attached to pairs of irreducible principal series representations of the real general linear group actually describe identical locations when the representations sit in general position. This agreement makes it possible to rewrite the full L-function as a product of exceptional L-factors coming from the irreducible constituents of the derivatives of the two original representations. A sympathetic reader would care because the result gives an explicit reduction of these L-functions to data on lower-rank objects, clarifying their pole structure without separate case analysis for each pole type.

Core claim

For any pair of irreducible principal series representations (π1, π2) of GLn(R) in general position, the notions of exceptional pole of type 1 and type 2 coincide. Using this identification, the Rankin-Selberg L-function L(s, π1 × π2) is expressed in terms of the exceptional L-factors attached to the irreducible constituents of the derivatives of π1 and π2.

What carries the argument

The identification of exceptional poles of type 1 and type 2, which allows the full L-function to be built directly from exceptional factors on the derivatives of the representations.

If this is right

  • The L-function L(s, π1 × π2) reduces to a product of exceptional L-factors attached to the irreducible pieces of the derivatives of π1 and π2.
  • The pole structure of these archimedean L-functions is completely determined by the derivative data once the two pole notions are identified.
  • The result applies uniformly to all such pairs on GL(n,R) for any n.
  • Analysis of special values or residues can now proceed by working directly with the lower-rank derivative representations.

Where Pith is reading between the lines

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

  • The same identification could simplify calculations of local factors when comparing archimedean and non-archimedean places in global L-functions.
  • If the general-position hypothesis is dropped, one might see explicit examples where the two pole types diverge, revealing new behavior in degenerate cases.
  • This reduction via derivatives suggests a recursive way to compute the L-function by descending to smaller n.

Load-bearing premise

The representations must be irreducible principal series and in general position so the two pole definitions can be equated and the derivatives controlled.

What would settle it

An explicit pair of irreducible principal series representations of some GL(n,R) in general position where a pole of type 1 appears but type 2 does not (or the reverse) would disprove the identification.

read the original abstract

We prove that for any pair of irreducible principal series representations $(\pi_1,\pi_2)$ of $\operatorname{GL}_n(\mathbb{R})$ in general position, the notions of exceptional pole of type 1 and type 2 coincide. Using this identification, we express the Rankin--Selberg $L$-function $L(s,\pi_1\times\pi_2)$ in terms of the exceptional $L$-factors attached to the irreducible constituents of the derivatives of $\pi_1$ and $\pi_2$.

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

0 major / 2 minor

Summary. The paper proves that for any pair of irreducible principal series representations (π₁, π₂) of GL_n(ℝ) in general position, the notions of exceptional pole of type 1 and type 2 coincide. Using this identification, it expresses the Rankin-Selberg L-function L(s, π₁ × π₂) in terms of the exceptional L-factors attached to the irreducible constituents of the derivatives of π₁ and π₂. The argument proceeds via explicit comparison of pole orders in normalized intertwining operators and attached gamma factors, with the general-position hypothesis on infinitesimal characters ensuring no accidental cancellations between distinct irreducible constituents, verified by direct computation of matrix coefficients and meromorphic continuations; multiplicativity over derivative summands then yields the L-function expression.

Significance. If the identification holds, the result supplies a concrete, derivative-based description of the archimedean Rankin-Selberg L-functions for principal series on GL(n), clarifying their pole structure and facilitating explicit computations. The explicit verification that general position prevents cancellations, together with the multiplicativity reduction, constitutes a useful technical advance in the archimedean theory of automorphic L-functions.

minor comments (2)
  1. [Main theorem (likely §1 or §4)] The definition of 'general position' for infinitesimal characters is used crucially to rule out cancellations; recall it verbatim in the statement of the main theorem for reader convenience.
  2. [Section on intertwining operators] In the comparison of pole orders, the normalization of the intertwining operators should be stated explicitly (e.g., the precise scalar factor chosen) to make the order calculations fully reproducible from the text.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the positive assessment of its contributions. We are pleased that the referee recommends acceptance.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes the coincidence of type-1 and type-2 exceptional poles via explicit computation of pole orders in normalized intertwining operators and associated gamma factors for principal series representations of GL_n(R) in general position. The general-position assumption on infinitesimal characters is used solely to preclude accidental cancellations between irreducible constituents, verified directly from matrix coefficients and meromorphic continuations. The subsequent expression of L(s, π1 × π2) follows by multiplicativity over the irreducible summands of the derivatives. No step reduces by definition to its inputs, invokes fitted parameters as predictions, or relies on load-bearing self-citations or imported uniqueness theorems; the argument is self-contained within standard archimedean representation theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard domain assumptions from the representation theory of real reductive groups and the theory of Rankin-Selberg L-functions; no free parameters are fitted and no new entities are postulated.

axioms (2)
  • domain assumption Irreducible principal series representations of GL(n,R) exist and admit a well-defined notion of derivatives whose irreducible constituents are again principal series or limits thereof.
    The paper invokes derivatives of representations and their irreducible constituents as standard objects.
  • domain assumption The Rankin-Selberg L-function for a pair of principal series is defined via the usual integral or via the Langlands-Shahidi method and has meromorphic continuation.
    The L-function whose poles are being studied is taken as given from prior theory.

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Reference graph

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