Theta function transformation formulas and the Weil representation

A short proof is given that the theta functional is invariant under the Weil representation, and the explicit determination of the eighth root of unity which arises is also shown. Namely, the action of the Weil representation on the theta functional is described as the limit of integration against a specialization of the symplectic theta function as the symplectic variable approaches a cusp. The invariance is then a consequence of the automorphic nature of this theta function, coupled with the fact that in the limit it acts as the reproducing kernel for a certain lattice. Using results of Stark and Styer, this also allows one to determine the root of unity involved.