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Victor Tan has a couple of papers on a regularized Siegel-Weil formula for U(2,2) and U(3). The papers I'm talking about are:

  1. "A Regularized Siegel-Weil Formula on U(2,2) and U(3)", Duke, 1998.
  2. "An Application of the Regularized Siegel-Weil Formula on Unitary Groups to a Theta Lifting Problem", Proceedings of the AMS, 1999.

One natural thing to look for after obtaining such a result is a Rallis inner product formula. Tan doesn't prove such a result in either of the two references mentioned above, though he seems to come close at the end of the second one.

Does anyone know if such a formula is written down anywhere?

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See recent preprint of Harris-Li, which is base on Ichino`s S-W formula for unitary groups

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As far as I know, in that paper they assume that m=n. So, the formula they give isn't valid for U(2,2) and U(3). – Neal Harris May 14 '10 at 15:11
@Neal: After Ichino's work, the formula of J-s Li which is based on the original work of Weil, should be extended routinely. Isn't it right? No one write it down maybe just because the value concerned is not at the central point, which is most interesting in arithmetic. – user4245 Jan 6 '11 at 14:30

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