Theta Correspondence of Automorphic Characters

This paper describes the lifting of automorphic characters of O(3)(A) to S̃L2(A). It does so by matching the image of this lift with the lift of automorphic characters from O(1)(A) to S̃L2(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 IV(χ,φ) determined by the Schwartz function φ ∈ S(V(A)) in the lift of the character χ we match an automorphic form IU(μ,φ0) determined by the Schwartz function φ0 ∈ S(U(A)) in the lift of the character μ. Our work shows that, the space U is explicitly determined by the character χ. The character μ is explicitly determined by the space V and the function φ0 is given by an orbital integral involving φ.