[Paper Review] Symmetry of Arthur parameters under Aubert involution

Ban, Dubravka|ArXiv.org|Jan 20, 2004

Advanced Algebra and Geometry30 references8 citations

TL;DR

This paper investigates the symmetry of Arthur parameters under the Aubert involution for nontempered representations of $SO(2n+1,F)$ over a $p$-adic field. Using explicit computation of Langlands data and $A$-parameters via recent results of Jiang-Soudry, Harris-Taylor, and Henniart, it shows that symmetry—defined by swapping the two $SL(2,\mathbb{C})$ factors in the parameter—holds if and only if the parameter's rank $k$ is odd and the induced representation $i_{G,M}(\nu^\alpha\rho\otimes\sigma)$ is irreducible at $\alpha = \frac{1}{2}$ or $\alpha = 0$, but fails when $k$ is even or at $\alpha = 1$, demonstrating that the Aubert involution does not generally preserve $A$-parameter symmetry.

ABSTRACT

We consider a nontempered $A$-parameter $ψ$ of $SO(2n+1, F)$ of a certain type and the base point representation $π$ in the $A$-packet of $ψ$. Let $\hatπ$ be the Aubert involution of $π$. We compute explicitly the Langlands data of $\hatπ$ and the $A$-parameter $\hatψ$ of $\hatπ$. We investigate whether $ψ$ and $\hatψ$ are symmetric. Although symmetry holds for large classes of parameters, it does not hold in general.

Motivation & Objective

• To investigate whether the Aubert involution preserves symmetry of Arthur parameters for nontempered $A$-packets of $SO(2n+1,F)$.
• To compute explicitly the Langlands data and $A$-parameters of the Aubert dual $\hat{\pi}$ of a base point representation $\pi$ associated to a specific $A$-parameter $\psi$.
• To determine under what conditions $\psi$ and $\hat{\psi}$ are symmetric, i.e., $\hat{\psi}(w,x,y) = \psi(w,y,x)$, generalizing the Zelevinsky involution behavior.
• To clarify the failure of symmetry in certain cases, particularly when $k$ is even or $\alpha = 1$, and to identify when the Aubert involution maps base points to non-base points.

Proposed method

• The study uses the Langlands-Arthur formalism to compute the Langlands data of the Aubert dual $\hat{\pi}$ of a base point representation $\pi$ associated to an $A$-parameter $\psi$.
• It applies recent results from Jiang-Soudry on $L$-parameters for generic representations of $SO(2\ell+1,F)$ and the structure of induced representations.
• The method relies on analyzing Jacquet modules $r_{M,G}(\pi)$ and $r_{M,G}(\hat{\pi})$ to determine the structure of induced representations and their components.
• The paper uses induction on $m$ to prove recursive structure of $\hat{\pi}$ in terms of standard modules $L_s(\delta[\nu^{-m}\rho,\nu^{-m+1}\rho], \dots, \tau)$.
• It compares the $L$-parameters of $\pi$ and $\hat{\pi}$ to determine whether $\hat{\psi}$ matches the symmetric form $\phi \otimes S_2 \otimes S_k \oplus \bigoplus \phi_i \otimes S_1 \otimes S_1$.
• The analysis distinguishes cases based on the reducibility point $\alpha$ of the induced representation $i_{G,M}(\nu^\alpha\rho \otimes \sigma)$, particularly $\alpha = \frac{1}{2}, 0, 1$.

Experimental results

Research questions

• RQ1 Does the Aubert involution preserve symmetry of $A$-parameters for nontempered representations of $SO(2n+1,F)$, i.e., does $\hat{\psi}(w,x,y) = \psi(w,y,x)$ hold?
• RQ2 Under what conditions on the parameter $\psi = \phi \otimes S_k \otimes S_2 \oplus \bigoplus \phi_i \otimes S_1 \otimes S_1$ does symmetry of $\psi$ and $\hat{\psi}$ fail?
• RQ3 How does the reducibility point $\alpha$ of the induced representation $i_{G,M}(\nu^\alpha\rho \otimes \sigma)$ affect the symmetry of $A$-parameters under Aubert involution?
• RQ4 Is the Aubert involution of a base point representation always a base point? If not, when does it fail?
• RQ5 What is the precise $L$-parameter of the Aubert dual $\hat{\pi}$, and how does it differ from the symmetric form expected from the Zelevinsky involution?

Key findings

• For $\alpha = \frac{1}{2}$, symmetry holds if $k$ is odd, yielding $\hat{\psi} = \phi \otimes S_2 \otimes S_k \oplus \bigoplus \phi_i \otimes S_1 \otimes S_1$, but fails if $k$ is even, giving $\hat{\psi} = \phi \otimes S_1 \otimes S_{k+1} \oplus \phi \otimes S_1 \otimes S_{k-1} \oplus \bigoplus \phi_i \otimes S_1 \otimes S_1$.
• For $\alpha = 0$, symmetry holds when $k$ is odd, but fails when $k$ is even, with the same parameter structure as in the $\alpha = \frac{1}{2}$ case.
• For $\alpha = 1$, the symmetry fails regardless of $k$'s parity, and the $L$-parameter of $\hat{\pi}$ is not of the symmetric form, indicating a breakdown of the expected involution behavior.
• When $k = 2$, the Aubert involution maps the base point $\pi$ to a tempered, non-generic representation $\tau$, which is not a base point, showing that the involution does not preserve the base point property.
• The $L$-parameter of $\hat{\pi}$ in the $\alpha = 1$ case is $\hat{\varphi} = \bigoplus_{j=\frac{3}{2}}^{m+\frac{1}{2}} (|\cdot|^j\phi \otimes S_2 \oplus |\cdot|^{-j}\phi \otimes S_2) \oplus \phi \otimes S_3 \oplus \phi \otimes S_1 \oplus \bigoplus \phi_i \otimes S_1$, missing the $j = \frac{1}{2}$ term, so it is not symmetric.
• The $A$-parameter $\psi'$ of the Aubert dual $\hat{\pi}$ at $k=2$ is $\phi \otimes S_3 \otimes S_1 \oplus \phi \otimes S_1 \otimes S_1 \oplus \bigoplus \phi_i \otimes S_1 \otimes S_1$, and its dual $\hat{\psi}'$ is $\phi \otimes S_1 \otimes S_3 \oplus \phi \otimes S_1 \otimes S_1 \oplus \bigoplus \phi_i \otimes S_1 \otimes S_1$, showing that $\psi'$ and $\hat{\psi}'$ are symmetric, even though $\psi$ and $\hat{\psi}$ are not.