[Paper Review] The Regularized Siegel-Weil Formula (The Second Term Identity) and the Rallis Inner Product Formula

Wee Teck Gan, Yannan Qiu+1 more|arXiv (Cornell University)|Jul 19, 2012

Advanced Algebra and Geometry47 references5 citations

TL;DR

This paper establishes the regularized Siegel-Weil formula's second term identity in full generality and derives the Rallis inner product formula for global theta lifts across all dual pairs. It resolves the non-vanishing problem of global theta lifts by proving a local-global criterion: the global theta lift is nonzero if and only if all local lifts are nonzero and the standard L-function of the representation is nonvanishing at a critical point.

ABSTRACT

In this paper, we establish the second term identity of the Siegel-Weil formula in full generality, and derive the Rallis inner product formula for global theta lifts for any dual pair. As a corollary, we resolve the non-vanishing problem of global theta lifts initiated by Steve Rallis.

Motivation & Objective

• To establish the second term identity of the regularized Siegel-Weil formula in full generality for all dual pairs.
• To derive the Rallis inner product formula for global theta lifts, linking the Petersson inner product of theta lifts to special values of L-functions.
• To resolve the non-vanishing problem of global theta lifts initiated by Steve Rallis, providing a local-global criterion for nonvanishing.
• To extend the theory of theta correspondence beyond the stable range and the first term range, including cases with non-convergent integrals via regularization.
• To prove that the nonvanishing of global theta lifts is equivalent to the nonvanishing of local lifts and the nonvanishing of the standard L-function at a critical point.

Proposed method

• Utilizes a see-saw duality diagram involving $ G(W_n) imes H(V_r) $ and $ G(U_n) imes G(U_n^-) $ to relate theta lifts to Eisenstein series and L-functions.
• Applies the regularized Siegel-Weil formula to interpret the inner integral in the theta lift as a special value or residue of a Siegel Eisenstein series.
• Employs the theory of doubling zeta integrals and the regularized Siegel-Weil formula to handle non-convergent integrals via analytic continuation and regularization.
• Uses the conservation relation for local theta correspondences to analyze the structure of induced representations and determine nonvanishing conditions.
• Applies the module diagram analysis of induced representations $ I_n^n(s, ho) $ to determine the nonvanishing of the zeta integral $ Z_v^*(s_{m,n}) $ at critical points.
• Combines the regularized Siegel-Weil formula with the Rallis inner product formula to relate the Petersson inner product of theta lifts to standard L-functions of automorphic representations.

Experimental results

Research questions

• RQ1 Under what conditions is the global theta lift of a cuspidal automorphic representation nonzero?
• RQ2 How can the second term identity of the regularized Siegel-Weil formula be established for all dual pairs, including in non-first-term-range cases?
• RQ3 What is the precise relationship between the nonvanishing of global theta lifts and the nonvanishing of standard L-functions at critical points?
• RQ4 How do local theta lifts and the conservation relation constrain the global nonvanishing behavior of theta lifts?
• RQ5 Can a local-global principle for nonvanishing of global theta lifts be established in full generality, including for non-split or non-stable-range dual pairs?

Key findings

• The second term identity of the regularized Siegel-Weil formula is established in full generality for all dual pairs $ G(U_n) imes H(V_r) $, extending beyond the first term range.
• The Rallis inner product formula is derived for all global theta lifts, showing that the Petersson inner product of theta lifts is proportional to the standard L-function $ L(s, ilde{ au} imes au) $ at a critical point.
• The nonvanishing of the global theta lift $ heta_{n,r}( au) $ is equivalent to the nonvanishing of all local lifts $ heta_{n,r}( au_v) $ and the nonvanishing of $ L(s_{m,n} + 1/2, au imes ilde{ au}) $.
• When $ au $ is cuspidal and $ heta_{n,j}( au) = 0 $ for $ j < r $, the global theta lift $ heta_{n,r}( au) $ is nonzero if and only if the standard L-function of $ au $ is nonvanishing at $ s_{m,n} + 1/2 $.
• In cases where $ heta_{n,r}( au) $ is nonzero, there exists a Hermitian space $ V' $ such that $ V' imes F_v o V_r imes F_v $ at all finite and complex places, and the global theta lift to $ H(V') $ is nonzero.
• Under specific conditions (e.g., $ ho_0 = -1 $, $ ho_0 = 0 $ with $ E_v = F_v imes F_v $ at archimedean places, $ ho_0 = 1 $ with $ F $ totally complex, or $ m = d(n) + 1 $), one may take $ V' = V_r $, so the original space realizes the nonvanishing lift.