On the regularized Siegel-Weil formula (the second term identity) and non-vanishing of theta lifts from orthogonal groups

We derive a (weak) second term identity for the regularized Siegel-Weil formula for the even orthogonal group, which is used to obtain a Rallis inner product formula in the "second term range''. As an application, we show the following non-vanishing result of global theta lifts from orthogonal groups. Let pi be a cuspidal automorphic representation of an orthogonal group O(V) with dim V = m even and r + 1 <= m <= 2r. Assume further that there is a place v such that pi(upsilon) congruent to pi(upsilon) circle times det. Then the global theta lift of p to Sp(2r) does not vanish up to twisting by automorphic determinant characters if the (incomplete) standard L-function L(S)(s, pi) does not vanish at s = 1 + (2r - m)/2. Note that we impose no further condition on V or pi. We also show analogous non-vanishing results when m > 2r (the "first term range'') in terms of poles of L(S)(s, pi) and consider the "lowest occurrence'' conjecture of the theta lift from the orthogonal group.