2 5 Ju n 20 09 On the Siegel - Weil Theorem for Loop Groups ( II ) Howard Garland

This is the second of our two papers on the Siegel-Weil theorem for loop groups. In the first paper [3] we proved the Siegel-Weil theorem for (finite dimensional) snt-modules ([3], Theorem 8.1). In the present paper we use this result to obtain the Siegel-Weil theorem for loop groups, Theorem 7.5, below. In addition to the corresponding result for snt-modules, our proof depends on a convergence condition for certain Eisenstein series on loop groups (Theorem 5.3, below). We note that this convergence criterion is used for the convergence criterion for Eisenstein series associated with snt-modules (Theorem 6.6, below). The uniform convergence obtained in Theorem 6.6 is crucial for applying the abstract lemma in Weil [8] (see [8], Proposition 2, page 7). The Siegel-Weil theorem for snt-modules does not immediately give the result for loop groups. The failure to do so is measured by the terms on the right hand side of (7.8). However, in §8, we show that in fact, these ”error terms” vanish! We now describe briefly our main result. Let F be a number field, F 2n be the standard symplectic space over F , and let (V, (, )) be a finite dimensional F -space with an anisotropic non-degenerate symmetric bilinear form (, ) with corresponding orthogonal group G. The space F 2n⊗V is naturally a symplectic space with isometry group Sp2N (where 2N = 2ndimV ). The groups Sp2n and G are commuting subgroups in Sp2N . The Weil representation can be generalized to loop symplectic groups [9]. Let S((t−1F [t−1]2n ⊗ V )A)