SOME LOCAL-GLOBAL NON-VANISHING RESULTS FOR THETA LIFTS FROM ORTHOGONAL GROUPS

We, first, improve a theorem of B, Roberts which characterizes non-vanishing of a global theta lift from O(X) to Sp(n) in terms of non-vanishing of local theta lifts. In particular, we will remove all the Archimedean conditions imposed upon his theorem. Secondly, following Roberts, we will apply our theorem to theta lifting of low rank similitude groups. Namely we characterize the non-vanishing condition of a global theta lift from GO(4) to GSp(2) in our improved setting. Also we consider non-vanishing conditions of a global theta lift from GO(4) to GSp(1) and explicitly compute the lift when it exists.