Products of theta series and spectral analysis (Q1323863)

The authors give a characterization of the set of cuspidal automorphic forms on PGL(2) which arise as theta lifts from a Größencharakter of a quadratic extension of the ground field. The condition is of the form \(\int \varphi (g) \Theta (g) \overline \Theta^ \chi (g)\) \(dg\) where \(\Theta\) and \(\Theta^ \chi\) are one-dimensional theta functions and \(\varphi\) is the automorphic form in question. The proof uses the decomposition of the regular representation into irreducible representations and arguments rather akin to those used in the theory of the treatment of the trace formula but which bring in the Weil representation in an essential way. The authors expect that this method will be applicable in a wide number of cases.