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【ESI会议04/22】David Soudry - New types of Siegel-Weil formula
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This talk was part of the Workshop on "Minimal Representations and Theta Correspondence" held at the ESI April 11 to 15, 2022. I will present several conjectural identities which bear a striking similarity with the Kudla-Rallis regularized Siegel-Weil formula. The key players, in the role of theta series, are the residues of Eisenstein series on symplectic groups, induced from Speh representations on Siegel parabolic subgroups. I will focus on a special case, already highly nontrivial, of the residual Eisenstein series on Sp(4m), Theta(\tau,4m), induced from a Speh representation on GL(2m) corresponding to a cuspidal representation \tau on GL(2), with trivial central character and non-vanishing L-function at 1/2, the pole of the Eisenstein series being taken at s=m/2. I will examine a regularized "Theta lift" of Theta(\tau,4l) to Sp(4m), m bigger l-1, where the "Theta kernel" is Theta(\tau,4(m+l)), restricted to Sp(4l) x Sp(4m). The result is a residual Eisenstein series induced from the tensor product of the Speh representation of GL(2l) corresponding to \tau and Theta(\tau, 4(m-l)). For a good choice of data, this is equal to a special value of an Eisenstein series on Sp(4m) induced from the Speh representation of GL(2m) corresponding to \tau. This is an ongoing joint work with David Ginzburg.
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