Theta Correspondence of Automorphic Characters

This paper describes the lifting of automorphic characters of $\text{GO}(3)(\text{A})$ to $\text{SLT}(\text{A})$. It does so by matching the image of this lift with the lift of automorphic characters from $\text{GO}(1)(\text{A})$ to $\text{SLT}(\text{A})$. Our matching actually gives a matching of individual automorphic forms, and not just of representation spaces. Let $\text{V}$ be a $3-$ dimensional quadratic vector space and $\text{U}$ a certain $1-$ dimensional quadratic space. To an automorphic form $I_{\text{V}}(\chi ,\varphi )$ determined by the Schwartz function $\varphi \in \text{Sc}(\text{V}(\text{A}))$ in the lift of the character $\chi$ we match an automorphic form $I_{\text{U}}(\mu ,{\varphi }_0)$ determined by the Schwartz function $\phi_{0}\in \Sc(\U(\A))$intheliftofthecharacter$\mu$. Our work shows that, the space $\text{U}$ is explicitly determined by the character $\chi$. The character $\mu$ is explicitly determined by the space $\text{V}$ and the function ${\varphi }_0$ is given by an orbital integral involving $\varphi$.