Relative Character Asymptotics Beyond Stability for $\mathrm{PGL}_2 \times \mathrm{GL}_1$

TL;DR

This paper extends the understanding of relative character asymptotics for the pair (PGL_2, GL_1) over non-archimedean fields, overcoming previous stability limitations and handling cases with conductor dropping.

Contribution

It introduces a new method to analyze relative character asymptotics beyond the stability regime, accommodating significant conductor dropping in the non-archimedean setting.

Findings

Derived asymptotics for relative characters with conductor dropping

Extended analysis to non-archimedean local fields

Overcame stability restrictions in previous methods

Abstract

The asymptotics of relative characters for real Lie groups were studied for representations $(\pi, \sigma)$ arising from Gan-Gross-Prasad pairs $(G,H)$ by Nelson and Venkatesh. They successfully compute the asymptotics of relative characters whenever the conductor of the associated Rankin-Selberg $L$-function $L(\pi \boxtimes \sigma^\vee)$ lies in a stable locus, i.e. away from conductor dropping. In this paper, we express asymptotics for relative characters in the non-archimedean setting for $(\mathrm{PGL}_2, \mathrm{GL}_1)$. The key new innovation is that our method overcomes the stability hypothesis and allows for significant conductor dropping.