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

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 $L(\pi ⊠{\sigma }^{\vee })$ 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.

arXiv comment29 pages