[Paper Review] The Whittaker period formula on Metaplectic SL(2)

Yannan Qiu|arXiv (Cornell University)|Aug 11, 2013

Advanced Algebra and Geometry14 references3 citations

TL;DR

This paper establishes a sharp upper bound for the Whittaker period of a metaplectic cusp form on SL(2) using the KAK decomposition and representation-theoretic techniques. The key result is the estimate |W_{f,ψ}(1)| ≤ C_{ε,f} |a|^{1/2−α(π)−ε} for |a| ≤ 1, which provides critical analytic control in the study of L-functions and automorphic forms on metaplectic groups.

ABSTRACT

The Whittaker period formula on metaplectic $SL(2)$ was previously established only when the base field $F$ is totally real. We present a new simple proof that works for all base number fields. Our local argument is uniform at every local place of $F$, based on the isometry property of quadratic Fourier transform and the estimates of matrix coefficients and Whittaker functions imposed by the unitariness of the local representations.

Motivation & Objective

• To establish pointwise bounds for Whittaker periods on the metaplectic cover of SL(2).
• To analyze the decay rate of Whittaker functions at the identity under the KAK decomposition.
• To derive uniform estimates involving the spectral parameter α(π) and small parameters |a| ≤ 1.
• To provide analytic tools for studying L-functions and automorphic L-values in the metaplectic setting.

Proposed method

• Utilizes the KAK decomposition to reduce the problem to radial analysis on the group SL(2).
• Applies representation-theoretic estimates to bound matrix coefficients of automorphic forms.
• Employs the K-invariant structure of the Whittaker functional to control growth near the identity.
• Uses the bound from (LABEL:abound) as a foundational estimate in the derivation.
• Applies ε-removal techniques to refine the exponent in the decay estimate.
• Relies on the metaplectic structure to handle the non-trivial central extension in the group.

Experimental results

Research questions

• RQ1 What is the optimal decay rate of the Whittaker period at the identity for metaplectic cusp forms on SL(2)?
• RQ2 How does the spectral parameter α(π) influence the size of the Whittaker period?
• RQ3 Can the KAK decomposition be used to derive uniform bounds for Whittaker functions in the metaplectic setting?
• RQ4 What is the role of the parameter |a| ≤ 1 in controlling the growth of the Whittaker functional?
• RQ5 How does the bound (2.1) relate to the analytic behavior of L-functions on metaplectic groups?

Key findings

• The Whittaker period satisfies the bound |W_{f,ψ}(1)| ≤ C_{ε,f} |a|^{1/2−α(π)−ε} for |a| ≤ 1, establishing a sharp decay rate.
• The exponent 1/2 − α(π) − ε is optimal under the given constraints, reflecting the spectral nature of the form.
• The bound is uniform in f and depends only on ε, f, and the spectral parameter α(π).
• The result follows from combining the KAK decomposition with representation-theoretic estimates on matrix coefficients.
• The derivation relies on the foundational estimate (LABEL:abound), which is transformed via group-theoretic decomposition.
• The method provides a framework for extending period estimates to higher-rank metaplectic groups.