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On two geometric theta lifts. (English) Zbl 1088.11030
In this article the authors study a connection between Borcherds lifts from a certain space \(H_k\) of automorphic forms of negative weight \(k\) on the upper half plane to automorphic forms on orthogonal groups on one side and the Kudla-Millson lift carrying certain differential forms on a locally symetric space \(X\) of an orthogonal group to meromorphic modular forms of weight \(2-k\) on the upper half plane on the other side. Since both liftings are defined in terms of (regularized) theta liftings, the idea is to compare the theta kernel \(\theta(\tau,z,\varphi_0)\) attached to the standard gaussian that is used in the Borcherds lifting with the theta kernel \(\theta(\tau,z,\varphi_{KM})\) attached to a certain Schwartz function \(\varphi_{KM}\) with values in differential forms on \(X\) that has been introduced and studied by Kudla and Millson. As a result the authors obtain (first in the hermitian case of signature \((p,2)\)) an adjointness result between both liftings. They generalize the result to arbitrary signature \((p,q)\), in which case the Borcherds lift so far had received little attention. The authors also show that their result gives new proofs of some of the geometric properties of the Borcherds lift.

MSC:
11F27 Theta series; Weil representation; theta correspondences
11F55 Other groups and their modular and automorphic forms (several variables)
14C25 Algebraic cycles
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