  In a previous work the authors gave a conceptual explanation for the
linearity of the Weil representation over a finite field k of odd
characteristic: There exists a canonical system of intertwining operators
between the Lagrangian models of the Heisenberg representation. This defines a
canonical vector space H(V) associated with a symplectic vector space V over k.
In this paper we prove a general theorem about idempotents in categories, and
we use it to solve the sign problem, formulated by Bernstein and Deligne, on
the compatibility between the associativity constraint and the convolution
structure of the l-adic sheaf of canonical intertwining kernels. This sheaf
governs---via the sheaf-to-function correspondence---the function theoretic
system of intertwiners. As an application we define a canonical category C(V)
associated with the symplectic vector space variety V, and we obtain the
canonical model of the categorical Weil representation.

Representation Theory;Algebraic Geometry