Invariant four-variable automorphic kernel functions

TL;DR

This paper introduces a new invariant expansion for four-variable automorphic kernel functions using circle method techniques, enhancing understanding of their symmetry properties in automorphic representation theory.

Contribution

It presents an alternative, invariant expansion of four-variable automorphic kernel functions employing circle method ideas, improving symmetry analysis.

Findings

Provides an invariant expansion under GL_2(F)×GL_2(F)

Uses circle method to analyze automorphic kernel functions

Enhances understanding of automorphic representation symmetries

Abstract

Let $F$ be a number field, let $\mathbb{A}_F$ be its ring of adeles, and let $g_1,g_2,h_1,h_2 \in \mathrm{GL}_2(\mathbb{A}_F)$. Previously the author provided an absolutely convergent geometric expression for the four variable kernel function $$ \sum_{\pi} K_{\pi}(g_1,g_2)K_{\pi^{\vee}}(h_1,h_2)L(s,(\pi \times \pi^{\vee})^S), $$ where the sum is over isomorphism classes of cuspidal automorphic representations $\pi$ of $\mathrm{GL}_2(\mathbb{A}_F)$. Here $K_{\pi}$ is the typical kernel function representing the action of a test function on the space of the cuspidal automorphic representation $\pi$. In this paper we show how to use ideas from the circle method to provide an alternate expansion for the four variable kernel function that is visibly invariant under the natural action of $\mathrm{GL}_2(F) \times \mathrm{GL}_2(F)$.