[Paper Review] Some local-global non-vanishing results of theta lifts for symplectic-orthogonal dual pairs

Shuichiro Takeda|arXiv (Cornell University)|Oct 16, 2008

Advanced Algebra and Geometry21 references3 citations

TL;DR

This paper establishes local-global non-vanishing results for theta lifts in symplectic-orthogonal dual pairs by replacing the temperedness condition in Roberts' work with a weaker boundedness condition on local representations. The key contribution is a characterization of global non-vanishing theta lifts in terms of local non-vanishing and non-vanishing of incomplete standard L-functions at s=1, with applications to functoriality and genericity of GSp(4) automorphic representations without requiring the base field to be totally real.

ABSTRACT

Following the approach of B. Roberts, we characterize the non-vanishing of global theta lifts for symplectic-orthogonal dual pairs in terms of its local counterpart. In particular, we replace the temperedness assumption present in Robert's work by a certain weaker assumption, and apply our results to small rank similitude groups. Among our applications is a certain instance of Langlands functorial transfer of a (non-generic) cuspidal automorphic representation of $GSp(4)$ to GL(4).

Motivation & Objective

• To generalize Roberts' local-global non-vanishing results for theta lifts in symplectic-orthogonal dual pairs by weakening the temperedness assumption.
• To characterize global non-vanishing of theta lifts in terms of local non-vanishing and non-vanishing of incomplete standard L-functions at s=1.
• To apply the results to small-rank similitude groups, particularly to establish Langlands functorial transfer from GSp(4) to GL(4) for non-generic cuspidal representations.
• To provide an alternative proof of the multiplicity one theorem for generic GSp(4) representations, removing the totally real base field assumption present in prior work.

Proposed method

• Uses the notion of boundedness of local representations, defined via the exponents in the Langlands classification of standard modules.
• Applies Roberts' global theta lifting framework but replaces his temperedness assumption with the boundedness condition, showing the arguments extend under this weaker hypothesis.
• Employs the theory of standard modules and Langlands quotients to analyze the local exponents of representations and relate them to L-function behavior.
• Analyzes theta lifts for symplectic-to-orthogonal and orthogonal-to-symplectic dual pairs using local non-vanishing and L-function non-vanishing at s=1.
• Applies the results to similitude groups, particularly GO(V) with dim V = 4, to study theta lifts from GO(V) to GSp(4).
• Uses the tower property of theta lifting and the main theorem of Gan-Gross-Prasad to deduce global genericity and functoriality results.

Experimental results

Research questions

• RQ1 Can the temperedness assumption in Roberts' local-global non-vanishing theorem for theta lifts be weakened while preserving the global non-vanishing conclusion?
• RQ2 Under what conditions does the non-vanishing of the incomplete standard L-function L^S(s,σ) at s=1 imply the non-vanishing of the global theta lift Θ_n(V_σ) to Sp(2n,A)?
• RQ3 Does the boundedness condition on local representations suffice to ensure the global non-vanishing of theta lifts in symplectic-orthogonal dual pairs?
• RQ4 Can the global non-vanishing of theta lifts from GSp(4) to GL(4) be established without assuming the base field is totally real?
• RQ5 Is the multiplicity one theorem for generic cuspidal representations of GSp(4) valid without the totally real base field assumption?

Key findings

• The global theta lift Θ_n(V_σ) to Sp(2n,A) does not vanish if σ_v has a non-zero theta lift to Sp(2n,F_v) at all places v and L^S(s,σ) does not vanish at s=1.
• If L^S(s,σ) has a pole at s=1, then the global theta lift Θ_{n-1}(V_σ) to Sp(2n-2,A) does not vanish, provided n ≥ 1.
• The global theta lift Θ_{V_r}(V_π) to O(V_r,A) does not vanish if π_v has a non-zero theta lift to O(V_r,F_v) at all places v and L^S(s,π) does not vanish at s=1.
• If L^S(s,π,χ) has a pole at s=1, then the global theta lift Θ_{V_{r-1}}(V_π) to O(V_{r-1},A) does not vanish, provided r ≥ 1.
• The proof of Theorem 1.5 (genericity of non-CAP GSp(4) representations) is completed without assuming the base field is totally real, removing a key restriction from prior work.
• The multiplicity one theorem for generic cuspidal representations of GSp(4) is now valid without the totally real base field assumption, as the prior dependency on this assumption is removed.