Theta Correspondence of Automorphic Characters

Abstract

This paper describes the lifting of automorphic characters of $\GO(3)(\A)$ to $\SLT(\A)$. It does so by matching the image of this lift with the lift of automorphic characters from $\GO(1)(\A)$ to $\SLT(\A)$. Our matching actually gives a matching of individual automorphic forms, and not just of representation spaces. Let $\V$ be a $3-$ dimensional quadratic vector space and $\U$ a certain $1-$ dimensional quadratic space. To an automorphic form $I_{\V}(\chi,\phi)$ determined by the Schwartz function $\phi\in \Sc(\V(\A))$ in the lift of the character $\chi$ we match an automorphic form $I_{\U}(\mu,\phi_{0})$ determined by the Schwartz function $\phi_{0}\in \Sc(\U(\A))$ in the lift of the character $\mu$. Our work shows that, the space $\U$ is explicitly determined by the character $\chi$. The character $\mu$ is explicitly determined by the space $\V$ and the function $\phi_{0}$ is given by an orbital integral involving $\phi$.