Theta correspondence of automorphic characters

✍ Scribed by Kobi Snitz

• Publisher: Elsevier Science
• Year: 2007
• Tongue: English
• Weight: 313 KB
• Volume: 122
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1016/j.jnt.2006.05.003

✦ Synopsis

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.