In the present paper we extend A. Weil's proof of the Siegel-Weil theorem to symplectic, nilpotent -modules (snt-modules), 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.
"On the Siegel-Weil theorem for loop groups, I." Duke Math. J. 157 (2) 283 - 336, 1 April 2011. https://doi.org/10.1215/00127094-2011-007