[Paper Review] A Cuspidality Criterion for the Exterior Square Transfer of Cusp Forms on GL(4)

Mahdi Asgari, A. Raghuram|ArXiv.org|Dec 28, 2007

Advanced Algebra and GeometryMathematics22 references19 citations

TL;DR

This paper establishes a complete cuspidality criterion for the exterior square transfer of cuspidal automorphic representations on GL(4), proving that the transfer to GL(6) is non-cuspidal if and only if the original representation arises from a smaller group—such as GL(2) × GL(2), GSp(4), or via automorphic induction or Asai transfer from a quadratic extension. The key result identifies precisely when the exterior square L-function fails to be cuspidal, using representation-theoretic and Galois-theoretic conditions including self-duality and self-twists.

ABSTRACT

For a cuspidal automorphic representation Πof GL(4,A), H. Kim proved that the exterior square transfer \wedge^2Πis an isobaric automorphic representation of GL(6,A). In this paper we characterize those representations Πfor which \wedge^2Πis cuspidal.

Motivation & Objective

• To determine when the exterior square transfer of a cuspidal automorphic representation on GL(4) remains cuspidal or becomes isobaric non-cuspidal.
• To characterize the automorphic representations on GL(4) whose exterior square transfer to GL(6) is not cuspidal.
• To clarify the exceptional case in the self-dual condition where the transfer remains non-cuspidal despite satisfying the basic self-duality criterion.
• To provide a Galois-theoretic explanation for the exceptional case in the cuspidality criterion, particularly in relation to dihedral representations.

Proposed method

• Use of the strong multiplicity one theorem to uniquely identify the exterior square transfer as an isobaric automorphic representation on GL(6).
• Explicit computation of the isobaric decomposition of ∧²Π for each candidate case (tensor products, Asai transfers, GSp(4) lifts, automorphic induction).
• Application of the Langlands–Shahidi method to analyze L-functions and derive conditions on the cuspidality of the exterior square transfer.
• Use of Galois representations and Clifford theory to analyze self-dual and self-twist properties of the underlying 4-dimensional representation.
• Leverage the structure of symmetric and alternating square representations on 4-dimensional representations to detect invariance under subgroups and determine reducibility.
• Employ the theory of similitudes and nondegenerate forms on ∧²V to characterize G-invariant subspaces and relate them to self-duality types.

Experimental results

Research questions

• RQ1 When is the exterior square transfer of a cuspidal automorphic representation on GL(4) non-cuspidal?
• RQ2 What are the precise automorphic or Galois-theoretic conditions that ensure the exterior square transfer fails to be cuspidal?
• RQ3 Why does the standard self-duality condition fail to fully characterize non-cuspidality, and what is the nature of the exceptional case?
• RQ4 How do self-twists and self-duality types (symplectic vs. orthogonal) relate to the reducibility of the exterior square representation?
• RQ5 Can the exceptional case in the self-duality criterion be explained via Galois representations and their symmetries?

Key findings

• The exterior square transfer ∧²Π is non-cuspidal if and only if Π arises as a functorial transfer from GL(2)×GL(2), GSp(4), or via Asai or automorphic induction from a quadratic extension.
• Equivalently, ∧²Π is non-cuspidal if and only if Π is essentially self-dual (Π ≅ Π∨⊗χ) or has a nontrivial self-twist (Π ≅ Π⊗χ), except for a specific class of Asai transfers of nondihedral representations.
• The exceptional case in the self-duality criterion corresponds to Asai transfers of dihedral representations, which are essentially self-dual of improper orthogonal type and have a nontrivial quadratic self-twist.
• The reducibility of ∧²Π is detected by the existence of G-invariant 3-dimensional subspaces in the exterior square, which occurs precisely when the underlying 4-dimensional representation is essentially self-dual of proper orthogonal type.
• The presence of a nontrivial quadratic self-twist in Π is equivalent to the existence of two G-invariant isotropic 3-spaces in ∧²V, which are permuted by a subgroup of index two.
• The Galois-theoretic interpretation reveals that the exceptional case arises when the associated 2-dimensional representation is dihedral, which ensures the exterior square is reducible despite satisfying the self-duality condition.