Shuichiro Takeda

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

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.

ON SHALIKA PERIODS AND A THEOREM OF JACQUET-MARTIN

Let

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

\pi

Some local-global non-vanishing results of theta lifts for symplectic-orthogonal dual pairs

be a cuspidal automorphic representation of

Hecke algebra correspondences for the metaplectic group

GL_4(\Bbb A)

The local Langlands conjecture for GSp(4)

with central character

A proof of the Howe duality conjecture

\mu^2

The regularized Siegel–Weil formula (the second term identity) and the Rallis inner product formula

. It is known that

The Local Langlands Conjecture for Sp(4)

\pi

Theta correspondences for GSp(4)

has Shalika period with respect to