Theta correspondence of automorphic characters

Abstract

This paper describes the lifting of automorphic characters of O ( 3 ) ( A ) to SL ˜ 2 ( A ) . It does so by matching the image of this lift with the lift of automorphic characters from O ( 1 ) ( A ) to SL ˜ 2 ( 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 ( χ , φ ) determined by the Schwartz function φ ∈ S ( V ( A ) ) in the lift of the character χ we match an automorphic form I U ( μ , φ 0 ) determined by the Schwartz function φ 0 ∈ 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 φ 0 is given by an orbital integral involving φ.