On the Siegel-Weil theorem for loop groups, I

Abstract

In the present paper we extend A. Weil's proof of the Siegel-Weil theorem to symplectic, nilpotent $t$-modules (snt-modules), $t$ being a nilpotent endomorphism of a finite-dimensional, symplectic, vector space satisfying a certain consistency condition with respect to the symplectic structure. This extension is key for our proof of the Siegel-Weil theorem for loop groups in the sequel to this paper.