Transfer using Fourier transform and minimal representation of $E_7$

Abstract

In this paper, we study the Sakellaridis-Venkatesh conjecture for the rank-1 spherical variety $X={\text{Spin}}_9\backslash F_4$ using an exceptional theta correspondence. We establish the correct transfer map satisfying relative character identities in this case and show that our transfer map agrees with the formula in (Sakellaridis, 2021). We also formulate the local relative characters for the degenerate Whittaker period of (Mao-Wan-Zhang, 2026a) associated with $X$. Moreover, we show how our techniques lead to a characterization of $X$-relatively cuspidal representations.