# On the regularized Siegel-Weil formula the second term identity and non-vanishing of theta lifts from orthogonal groups - Mathematics > Number Theory

Abstract: We derive a weak second term identity for the regularized Siegel-Weilformula for the even orthogonal group, which is used to obtain a Rallis innerproduct formula in the -second term range-. As an application, we show thefollowing non-vanishing result of global theta lifts from orthogonal groups.Let $\pi$ be a cuspidal automorphic representation of an orthogonal group$OV$ with $\dim V=m$ even and $r+1\leq m\leq 2r$. Assume further that thereis a place $v$ such that $\pi v\cong\pi v\otimes\det$. Then the global thetalift of $\pi$ to $Sp {2r}$ does not vanish up to twisting by automorphicdeterminant characters if the incomplete standard $L$-function $L^Ss,\pi$does not vanish at $s=1+\frac{2r-m}{2}$. Note that we impose no furthercondition on $V$ or $\pi$. We also show analogous non-vanishing results when $m> 2r$ the -first term range- in terms of poles of $L^Ss,\pi$ and considerthe -lowest occurrence- conjecture of the theta lift from the orthogonal group.

Author: Wee Teck Gan, Shuichiro Takeda

Source: https://arxiv.org/