[Paper Review] The local Langlands conjecture for the $p$-adic inner form of Sp(4)

Kwangho Choiy|arXiv (Cornell University)|Oct 4, 2015

Advanced Algebra and Geometry26 references3 citations

TL;DR

This paper proves the local Langlands conjecture for the non quasi-split $p$-adic inner form $\mathrm{Sp}_{1,1}$ of $\mathrm{Sp}_4$ by restricting $L$-packets from the metaplectic group $\mathrm{GSp}_{1,1}$, leveraging the LLC for $\mathrm{GSp}_{1,1}$ and theta correspondence with dual groups. The key result establishes a canonical bijection between $L$-packets of $\mathrm{Sp}_{1,1}$ and irreducible representations of the $S$-group with specified central character, resolving the parameterization of $L$-packets for this non-split form.

ABSTRACT

This paper proves the local Langlands conjecture for the non quasi-split inner form Sp(1,1) of Sp(4) over a p-adic field of characteristic 0, by studying the restriction of representations from the non quasi-split inner form GSp(1,1) of GSp(4) to Sp(1,1). The L-packets for Sp(1,1) are constructed based on the earlier work on the local Langlands correspondence for GSp(1,1) by Gan and Tantono. To parameterize them in terms of so-called S-groups, we establish and utilize the local Langlands correspondence for reductive dual groups which participate in the theta correspondence with Sp(1,1) and GSp(1,1). An interesting phenomenon arises when two distinct members in an L-packet of GSp(1,1) are restricted to Sp(1,1).

Motivation & Objective

• To establish the local Langlands correspondence (LLC) for the non quasi-split inner form $\mathrm{Sp}_{1,1}$ of $\mathrm{Sp}_4$ over a $p$-adic field of characteristic 0.
• To construct $L$-packets for $\mathrm{Sp}_{1,1}$ using the restriction of $L$-packets from $\mathrm{GSp}_{1,1}$, building on prior work by Gan and Tantono.
• To parameterize $L$-packets of $\mathrm{Sp}_{1,1}$ via the $S$-group construction, incorporating the Kottwitz isomorphism and central character data.
• To resolve the structure of $L$-packets for $\mathrm{Sp}_{1,1}$ by analyzing the behavior of distinct $L$-packets of $\mathrm{GSp}_{1,1}$ under restriction to $\mathrm{Sp}_{1,1}$.

Proposed method

• Restriction of $L$-packets from $\mathrm{GSp}_{1,1}$ to $\mathrm{Sp}_{1,1}$, using the fact that distinct $L$-packets in $\mathrm{GSp}_{1,1}$ yield distinct images under restriction.
• Utilization of the local Langlands correspondence for dual groups $\mathrm{SO}_{2,2}$ and $\mathrm{SO}_{3,3}$, which participate in theta correspondence with $\mathrm{Sp}_{1,1}$ and $\mathrm{GSp}_{1,1}$.
• Establishment of a canonical isomorphism between the $S$-groups $\mathcal{S}_{\varphi,\mathrm{sc}}(\widehat{\mathrm{Sp}_{1,1}})$ and $\mathcal{S}_{\varphi,\mathrm{sc}}(\widehat{\mathrm{SO}^{*}_{1,1}})$, enabling transfer of representation data.
• Application of the Kottwitz isomorphism to relate central characters $\zeta_G$ to $F$-inner forms of dual groups, particularly $\mathrm{SO}_{2,2}$ and $\mathrm{SO}_{4,0}$.
• Use of the Jacquet-Langlands correspondence to relate representations of $\mathrm{SO}_{2,2}$ and $\mathrm{SO}_{4,0}$ to those of $\mathrm{Sp}_{1,1}$ via $\mathrm{GSp}_{1,1}$-data.
• Proof of the main theorem via case analysis on the structure of $S$-groups and their irreducible representations, showing $\Pi_\varphi(\mathrm{Sp}_{1,1}) \cong \mathrm{Irr}(\mathcal{S}_{\varphi,\mathrm{sc}}(\widehat{\mathrm{Sp}_{1,1}}), \zeta_G)$.

Experimental results

Research questions

• RQ1 How can the local Langlands correspondence be extended to non quasi-split inner forms of $\mathrm{Sp}_4$, such as $\mathrm{Sp}_{1,1}$?
• RQ2 What happens to $L$-packets of $\mathrm{GSp}_{1,1}$ when restricted to $\mathrm{Sp}_{1,1}$, and how does this affect the structure of $L$-packets for the latter?
• RQ3 How can the $S$-group parameterization of $L$-packets be adapted for non quasi-split groups, especially when the $S$-group is non-abelian?
• RQ4 What role does the theta correspondence with dual groups $\mathrm{SO}_{2,2}$ and $\mathrm{SO}_{3,3}$ play in constructing $L$-packets for $\mathrm{Sp}_{1,1}$?
• RQ5 How are the central characters $\zeta_G$ in the $S$-group construction related to the $F$-inner forms of the dual group?

Key findings

• The local Langlands correspondence is fully established for the non quasi-split inner form $\mathrm{Sp}_{1,1}$ of $\mathrm{Sp}_4$ over a $p$-adic field of characteristic 0.
• The $L$-packets of $\mathrm{Sp}_{1,1}$ are constructed via restriction of $L$-packets from $\mathrm{GSp}_{1,1}$, and distinct $L$-packets in $\mathrm{GSp}_{1,1}$ yield distinct images in $\mathrm{Sp}_{1,1}$.
• The $S$-group $\mathcal{S}_{\varphi,\mathrm{sc}}(\widehat{\mathrm{Sp}_{1,1}})$ is isomorphic to $\mathcal{S}_{\varphi,\mathrm{sc}}(\widehat{\mathrm{SO}^{*}_{1,1}})$, enabling transfer of representation data from dual groups.
• The $L$-packet $\Pi_\varphi(\mathrm{Sp}_{1,1})$ is in canonical bijection with $\mathrm{Irr}(\mathcal{S}_{\varphi,\mathrm{sc}}(\widehat{\mathrm{Sp}_{1,1}}), \zeta_G)$, where $\zeta_G$ is determined by the Kottwitz isomorphism and corresponds to $\mathrm{SO}_{2,2}$ or $\mathrm{SO}_{4,0}$.
• The proof relies on the theta correspondence and Jacquet-Langlands correspondence to relate representations of $\mathrm{SO}_{2,2}$ and $\mathrm{SO}_{4,0}$ to those of $\mathrm{Sp}_{1,1}$, with explicit bijections established via $\mathrm{GSp}_{1,1}$-data.
• The structure of $L$-packets for $\mathrm{Sp}_{1,1}$ is fully described, with the $S$-group being an elementary 2-group in all cases, and the parameterization is compatible with $\gamma$-factors and $L$-factors.