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A regularized Siegel-Weil formula: The first term identity. (English) Zbl 0818.11024
In his analysis of Siegel’s mass formula Weil introduced a unified technique based on the ‘Weil’ representation. He showed that one can form a very general class of theta functions on $$G(k_ \mathbb{A})\times H(k_ \mathbb{A})$$ where $$G$$ is a symplectic group of the form $$\text{Sp} (2n)$$ and $$H$$ is the orthogonal group of a quadratic form of degree $$m$$ and Witt index $$r$$, where $$m$$ is taken here to be even. The integral of the theta function over $$H(k) \smallsetminus H(k_ \mathbb{A})$$ converges if either $$r=0$$ or $$m-r> n+1$$. Under somewhat stronger assumptions he proved that this integral is equal to an Eisenstein series.
In a previous paper the authors proved this equality, or a modified version precisely under the conditions $$r=0$$ or $$m-r> n+1$$. In this paper they investigate what happens when this condition is not fulfilled. They define a regularization of the integral over $$H(k) \smallsetminus H(k_ \mathbb{A})$$ by using an Eisenstein series. They also make use of the general theory of the analytic continuation of Eisenstein series and they obtain relations in this case which can be regarded as extensions of the Siegel- Weil formula. In the course of this investigation they uncover interesting relations with other aspects of the theory of automorphic forms.

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 11F46 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F27 Theta series; Weil representation; theta correspondences
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