On the Siegel-Weil theorem for loop groups. I (Q533848)

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.