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On the Siegel-Weil theorem for loop groups. I. (English) Zbl 1215.22010

A symplectic, nilpotent \(t\)-module is an \(F[t]\)-module, where \(F\) is a number field, annihilated by a sufficiently large power of \(t\) and equipped with a symplectic form for which \(t\) is self-dual. The main result of this paper asserts that in the context of symplectic, nilpotent \(t\)-modules the Siegel-Weil formula holds for symplectic and orthogonal groups over \(F[t]/(p(t))\) for an arbitrary polynomial \(p(t)\). This result is used in the sequel of the present paper to prove the Siegel-Weil theorem for arithmetic quotients of loop groups. Another possible application is a derivation of the Hasse principle.

MSC:

22E67 Loop groups and related constructions, group-theoretic treatment
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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References:

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