On the regularized Siegel–Weil formula (the second term identity) and non-vanishing of theta lifts from orthogonal groups
-
Wee Teck Gan
and
Shuichiro Takeda
Abstract
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 π be a cuspidal automorphic representation of an orthogonal group O(V) with dimV = m even and r + 1 ≦ m ≦ 2r. Assume further that there is a place ν such that πν ≅ πν ⊗ det. Then the global theta lift of π to Sp2r does not vanish up to twisting by automorphic determinant characters if the (incomplete) standard L-function LS(s, π) does not vanish at s = 1 + (2r – m)/2. Note that we impose no further condition on V or π. We also show analogous non-vanishing results when m > 2r (the “first term range”) in terms of poles of LS(s, π) and consider the “lowest occurrence” conjecture of the theta lift from the orthogonal group.
© Walter de Gruyter Berlin · New York 2011
Articles in the same Issue
- Period and index in the Brauer group of an arithmetic surface
- A characterization of freeness by invariance under quantum spreading
- Canonical bases and KLR-algebras
- Ranks of invariant subspaces of the Hardy space over the bidisk
- The Kähler–Ricci flow on Hirzebruch surfaces
- A correction to Hasse's version of the Grunwald–Hasse–Wang theorem
- On the regularized Siegel–Weil formula (the second term identity) and non-vanishing of theta lifts from orthogonal groups
Articles in the same Issue
- Period and index in the Brauer group of an arithmetic surface
- A characterization of freeness by invariance under quantum spreading
- Canonical bases and KLR-algebras
- Ranks of invariant subspaces of the Hardy space over the bidisk
- The Kähler–Ricci flow on Hirzebruch surfaces
- A correction to Hasse's version of the Grunwald–Hasse–Wang theorem
- On the regularized Siegel–Weil formula (the second term identity) and non-vanishing of theta lifts from orthogonal groups
