# Relative character identities and theta correspondence

## TL;DR

This paper explores the factorization of global periods for a specific spherical variety using theta correspondence, connecting it to the Sakellaridis-Venkatesh conjecture and providing explicit formulas and local analysis.

## Contribution

It establishes the factorization of global periods for the spherical variety via theta correspondence and details the local Plancherel formula and transfer formulas.

## Key findings

- Determined the factorization of global periods for the spherical variety.
- Established the Plancherel formula and relative character identities for $L^2(X)$.
- Provided explicit integral formulas for transfer between $X$-side and Whittaker-side.

## Abstract

In this paper, we will determine the factorization of global period attached to the spherical variety $X:=\mathrm{SO}(n-1)\backslash \mathrm{SO}(n)$, which is a special case of the Sakellaridis-Venkatesh conjecture. The main idea is to build a connection between the periods of $X$ and the periods of the Whittaker case $N,\psi\backslash \mathrm{SL}_2$ (n even) or $N,\psi\backslash \mathrm{Mp}_2$ (n odd) using the tool of theta correspondence. In the local setting, we determine the Plancherel formula of $L^2(X)$, the relative character identities and give an explicite integral formula of transfer between $X$-side and Whittaker-side which coincides with the theory of transfer developed by Sakellaridis.