Groups from Class 2 Algebras and the Weil Character

We investigate the behaviour of the Weil character of the symplectic group on restriction to subgroups arising from commutative nilpotent algebras of class 2. We give explicit descriptions of the decomposition of the Weil character when restricted to the unipotent radical of the stabilizer of a maximal totally isotropic subspace and to its centralizer. Moreover, we show how these decompositions can be used to obtain alternative proofs for certain results concerning quadratic forms or Gauss sums