Transfer using Fourier transform and minimal representation of $E_7$

TL;DR

This paper investigates the transfer map for a specific spherical variety using Fourier transform techniques, confirming conjectures and characterizing relatively cuspidal representations.

Contribution

It establishes the transfer map satisfying relative character identities for the rank-1 spherical variety and confirms its agreement with existing formulas, advancing understanding of the Sakellaridis-Venkatesh conjecture.

Findings

Transfer map satisfying relative character identities established

Confirmed agreement with Sakellaridis (2021) formula

Characterization of X-relatively cuspidal representations achieved

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.