[Paper Review] On local zeta-integrals for GSp(4) and GSp(4) x GL(2)

David Loeffler|arXiv (Cornell University)|Nov 30, 2020

Advanced Algebra and Geometry11 references4 citations

TL;DR

This paper establishes the compatibility of Novodvorsky's local zeta-integral definition with the standard $L$-factor for generic representations of $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ over non-archimedean local fields when the $\mathrm{GL}(2)$ representation is non-supercuspidal. It further identifies the Langlands parameter conditions underlying exceptional and subregular poles of $L$-factors, and derives consequences for branching laws in Gan–Gross–Prasad type problems, particularly for reducible or non-generic representations.

ABSTRACT

We prove that Novodvorsky's definition of local L-factors for generic representations of GSp(4) x GL(2) is compatible with the local Langlands correspondence when the GL(2) representation is non-supercuspidal. We also give an interpretation in terms of Langlands parameters of the "exceptional" poles of the GSp(4) x GL(2) L-factor, and of the "subregular" poles of the GSp(4) L-factor studied in recent work of Roesner and Weissauer.

Motivation & Objective

• To prove that Novodvorsky’s zeta-integral definition of $L$-factors for $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ agrees with the standard $L$-factor when the $\mathrm{GL}(2)$ representation is non-supercuspidal.
• To interpret the exceptional poles of the $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ $L$-factor in terms of Langlands parameters, particularly via unramified summands in the tensor product of $\mathrm{GSp}(4)$ and $\mathrm{GL}(2)$ parameters.
• To provide a conceptual interpretation of subregular poles of $\mathrm{GSp}(4)$ $L$-factors using Langlands parameters, distinguishing between 1-dimensional and 2-dimensional self-dual unramified summands.
• To deduce consequences for Gan–Gross–Prasad-type branching laws, especially for reducible or non-generic representations of $\mathrm{GSp}(4)$.

Proposed method

• Uses the relation between Novodvorsky’s zeta-integral for $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ and Piatetski-Shapiro’s zeta-integral for $\mathrm{GSp}(4)$, depending on a choice of split Bessel model.
• Applies the local Langlands correspondence and Shahidi’s method to compare $L$-factors, leveraging known agreement between these constructions.
• Analyzes poles of $L$-factors via the condition $\chi_\pi \chi_\sigma |\cdot|^{2s_0} = 1$, linking exceptional poles to unramified summands in the Langlands parameter of $\phi_\pi \otimes \phi_\sigma$.
• Uses the ratio $\frac{L(\pi \times \sigma, s) L(\pi \times \sigma, s+1)}{L(\pi \times \sigma \times \mathrm{St}, s+\frac{1}{2})}$ as a criterion to detect exceptional poles, based on Lemma 7.1.
• Applies results from Rösner and Weissauer on $\mathrm{GSp}(4)$ $L$-factors and Moeglin–Waldspurger’s multiplicity results to analyze branching laws for $\mathrm{SO}(4,F)$-invariant homomorphisms.
• Employs filtration techniques on the representation $\Xi = \Sigma \boxtimes \Sigma$ to study the dimension of $\mathrm{Hom}_H(\pi \otimes \Xi, \mathbb{C})$, particularly focusing on the behavior at $s = -\frac{1}{2}$.

Experimental results

Research questions

• RQ1 Does Novodvorsky’s zeta-integral definition of the $L$-factor for $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ agree with the standard $L$-factor when the $\mathrm{GL}(2)$ representation is non-supercuspidal?
• RQ2 What is the Langlands parameter-theoretic interpretation of the exceptional poles of the $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ $L$-factor?
• RQ3 Under what conditions on the Langlands parameter of $\pi$ does $s_0$ become a subregular pole of the $\mathrm{GSp}(4)$ $L$-factor?
• RQ4 What are the implications of these $L$-factor pole structures for the multiplicity of $\mathrm{SO}(4,F)$-invariant homomorphisms in tensor products involving $\mathrm{GSp}(4)$ representations?

Key findings

• Theorem A confirms that Novodvorsky’s $L^\mathrm{Nov}$-factor agrees with the standard $L$-factor for $\mathrm{GSp}(4) \times \mathrm{GL}(2)$ when the $\mathrm{GL}(2)$ representation is non-supercuspidal, including principal series and Steinberg representations.
• Theorem B establishes that exceptional poles of $L^\mathrm{Nov}(\pi \times \sigma, s)$ correspond precisely to poles of the ratio $\frac{L(\pi \times \sigma, s) L(\pi \times \sigma, s+1)}{L(\pi \times \sigma \times \mathrm{St}, s+\frac{1}{2})}$, under the same non-supercuspidal assumption.
• Theorem C shows that a pole $s_0$ of the $\mathrm{GSp}(4)$ $L$-factor is subregular if and only if the Langlands parameter of $\pi$ contains a 1-dimensional unramified summand or a 2-dimensional self-dual unramified twist of the Steinberg parameter.
• For $\pi$ generic and $\sigma = \Sigma \boxtimes \sigma$ with $\Sigma$ the Steinberg representation of $\mathrm{SO}(3,F)$, the space $\mathrm{Hom}_{\mathrm{SO}(4,F)}(\pi \otimes \sigma, \mathbb{C})$ has dimension at most 1, and is exactly 1 unless $s = -\frac{1}{2}$ is a subregular pole of $L(\pi, s)$.
• The canonical non-zero homomorphism $\mathfrak{z}$ in $\mathrm{Hom}_H(\pi \otimes \Xi, \mathbb{C})$ restricts non-trivially to $\Xi_{00} = \mathrm{St} \boxtimes \mathrm{St}$ if and only if $s = -\frac{1}{2}$ is not a subregular pole of $L(\pi, s)$.
• When $s = -\frac{1}{2}$ is not a subregular pole, $\mathrm{Hom}_H(\pi \otimes \Xi, \mathbb{C})$ is 1-dimensional and spanned by $\mathfrak{z}$, and every non-generic subquotient of $\Xi$ has trivial $H$-invariant homomorphism space with $\pi$.