Siegel-Weil formula

introduction

• 틀:수학노트
• The Siegel-Weil formula was discovered by Siegel in the context of classical modular forms and then cast in the representation theoretic language and considerably extended in an influential paper of Weil
• It identifies the global theta lift of the trivial representation of $H(V_r)$ to $G(W_n)$ as an Eisenstein series
• Roughly speaking, the Siegel-Weil formula says that the theta integral associated to a vector space (quadratic or Hermitian) is the special value of some Eisenstein series at certain point when both the theta integral and Eisenstein series (at the point) are both absolutely convergent.
• Kudla and Rallis pushed the results to non-convergent regions in a series of 3 papers [KR1-3]
• Their work culminates in a regularized Siegel-Weil formula, and they established what is now known as the first term identity, at least when $G(W_{n})$ is symplectic and $H(V_{r})$ orthogonal.
• Their work was subsequently refined and extended to other dual pairs by others, especially in the work of Ikeda, Ichino and Yamana

Rallis inner product formula

• regularised Rallis inner product formula that relates the pairing of theta lifts to the central value of the Langlands $L$-function of $\pi$

related items

• Classical theta lift and Shimura correspondence
• Howe duality
• Volume of a compact Lie group

expositions

• Gan, Wee Teck. “The Regularized Siegel-Weil Formula (the Second Term Identity) (Automorphic Representations and Related Topics),” December 2013. http://repository.kulib.kyoto-u.ac.jp/dspace/handle/2433/195472.
• Garrett, Paul. The Siegel-Weil formula in the convergent range, Aug 2005. http://www.math.umn.edu/~garrett/m/v/siegel_weil.pdf
• Kudla, Some extensions of the Siegel-Weil formula, Jan 2002. http://www.math.toronto.edu/skudla/kyoto.pdf

articles

• Chenyan Wu, On a Critical Case of Rallis Inner Product Formula, http://arxiv.org/abs/1603.04123v1
• Gan, Wee Teck, Yannan Qiu, and Shuichiro Takeda. “The Regularized Siegel–Weil Formula (the Second Term Identity) and the Rallis Inner Product Formula.” Inventiones Mathematicae 198, no. 3 (March 29, 2014): 739–831. doi:10.1007/s00222-014-0509-0.
• Gan, Wee Teck, and Shuichiro Takeda. 2009. “On the Regularized Siegel-Weil Formula (the Second Term Identity) and Non-Vanishing of Theta Lifts from Orthogonal Groups.” arXiv:0902.0419 [math], February. http://arxiv.org/abs/0902.0419.
• Kudla, Stephen S., and Stephen Rallis. 1994. “A Regularized Siegel-Weil Formula: The First Term Identity.” The Annals of Mathematics 140 (1): 1. doi:10.2307/2118540.