Integral Bases and Invariant Vectors for Weil Representations

Abstract

We show that the Weil representation associated with any discriminant form admits a basis in which the action of the representation involves algebraic integers. The action of a general element of ${\mathrm{SL}}_2(\mathbb{Z})$ on many parts of these bases is simple an explicit, a fact that we use for determining the dimension of the space of invariants for some families of discriminant forms.

29 pages