On some explicit formulas in the theory of Weil representation

Abstract

The object of this paper is to derive some explicit formulae concerning the Weil representation that allow us to define this projective representation in a unique manner for each choice of symplectic basis. Let F be a self-dual locally compact field of char φ 2 and X a symplectic vector space over F. Let V, V* be two transversal Lagrangian subspaces. Then a classical construction due to Shale-SegalWeil gives a projective representation of the symplectic group Sp(JSΓ) in the Schwartz-space of V. The operators ξ(σ) corresponding to each σ e Sp(X) are determined uniquely only up to a scalar multiple. The starting point of this paper is an explicit integral formula for these operators ξ(σ), valid for all σ e Sp(X). In fact (see Lemma 3.2) we have for each σ e Sp(ΛΓ) ξ(σ)φ :x-+ fσ(x, x*)φ(xa + x*y) dμσ JV*/kevγ