[Paper Review] Galois self-dual cuspidal types and Asai local factors

U. K. Anandavardhanan, Robert Kurinczuk+3 more|arXiv (Cornell University)|Jul 20, 2018

Advanced Algebra and Geometry46 references10 citations

TL;DR

This paper establishes an explicit integral representation for the Asai L-function of a $ ceil$-self-dual cuspidal representation $π$ of $ ceil_n( ceil)$, proving that it arises as a single Asai integral with an explicit test vector. It further computes the Asai root number as 1 when $π$ is $ ceil_n( ceil_o)$-distinguished, and proves that the local Asai epsilon factor matches the Langlands–Shahidi epsilon factor up to explicit factors involving the central character and local constants.

ABSTRACT

Let $F/F_{\mathsf{o}}$ be a quadratic extension of non-archimedean locally compact fields of odd residual characteristic and $σ$ be its non-trivial automorphism. We show that any $σ$-self-dual cuspidal representation of ${ m GL}_n(F)$ contains a $σ$-self-dual Bushnell--Kutzko type. Using such a type, we construct an explicit test vector for Flicker's local Asai $L$-function of a ${ m GL}_n(F_{\mathsf{o}})$-distinguished cuspidal representation and compute the associated Asai root number. Finally, by using global methods, we compare this root number to Langlands--Shahidi's local Asai root number, and more generally we compare the corresponding epsilon factors for any cuspidal representation.

Motivation & Objective

• To construct an explicit test vector for Flicker's local Asai L-function of a $ ceil$-self-dual cuspidal representation.
• To compute the Asai root number for such representations, particularly when they are $ ceil_n( ceil_o)$-distinguished.
• To compare Flicker's Asai epsilon factor with the Langlands–Shahidi epsilon factor for cuspidal representations.
• To establish that the Asai L-function of a distinguished cuspidal representation is realized via a single Asai integral with a canonical test vector.

Proposed method

• Uses Bushnell–Kutzko type theory to construct a $ ceil$-self-dual type for $ ceil$-self-dual cuspidal representations of $ ceil_n( ceil)$.
• Constructs an explicit Whittaker function ${ ceil}_0$ in the Whittaker model of $π$ such that the Asai integral with the characteristic function of $Ο_o^n$ gives the Asai L-function.
• Applies global methods to compare the Flicker Asai epsilon factor with the Langlands–Shahidi epsilon factor.
• Employs a duality argument via the theory of induced representations and intertwining operators to relate the Asai factor to the Langlands–Shahidi factor.
• Uses the theory of types and the structure of parabolic inductions to analyze the space of ${ ceil}^\sigma$-invariant linear forms.
• Verifies the compatibility of the local Asai factor with the functional equation of the Asai L-function.

Experimental results

Research questions

• RQ1 Does every $ ceil$-self-dual cuspidal representation of $ ceil_n( ceil)$ admit a $ ceil$-self-dual Bushnell–Kutzko type?
• RQ2 Can the Asai L-function of a $ ceil$-self-dual cuspidal representation be realized as a single Asai integral with an explicit test vector?
• RQ3 What is the value of the Asai root number when the representation is $ ceil_n( ceil_o)$-distinguished?
• RQ4 How do Flicker's Asai epsilon factor and the Langlands–Shahidi epsilon factor relate for cuspidal representations?
• RQ5 Is the Asai L-function of a distinguished cuspidal representation independent of the choice of non-degenerate character?

Key findings

• The Asai L-function of a $ ceil$-self-dual cuspidal representation $π$ is realized as a single Asai integral: $\operatorname{I}_{\rm As}(s,\Phi_0,{\rceil}_0)={\rceil}_{{\rm As}}(s,\pi)$, where $\Phi_0$ is the characteristic function of $\u039f_o^n$.
• For a $ ceil_n(\rceil_o)$-distinguished cuspidal representation, the Asai root number satisfies $\epsilon_{{\rm As}}(1/2,\pi,\psi_o,\delta)=1$, confirming a conjecture.
• The Flicker Asai epsilon factor is related to the Langlands–Shahidi factor by $\epsilon_{{\rm As}}(s,\pi,\psi_o,\delta)=\omega_\pi(\delta)^{n-1}\cdot|\delta|^{\frac{n(n-1)}{2}(s-1/2)}\cdot\lambda({\rceil}/{\rceil_o},\psi_o)^{-\frac{n(n-1)}{2}}\cdot\epsilon_{{\rm As}}^{{\rm LS}}(s,\pi,\psi_o)$.
• The space of ${\rceil}^\sigma$-invariant linear forms on a cuspidal representation has dimension 1, and coincides with the space of ${\rceil}_n(\rceil_o)$-invariant forms if the representation is distinguished.
• The Asai L-function is non-trivial if and only if the representation has a distinguished unramified twist.
• The construction of the test vector ${\rceil}_0$ is canonical and independent of the choice of non-degenerate character $\psi$.