Theta correspondence of automorphic characters

This paper describes the lifting of automorphic characters of O(3)(A) to . It does so by matching the image of this lift with the lift of automorphic characters from O(1)(A) to . 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.