What is the [Categorical] Weil Representation?

The Weil representation is a central object in mathematics responsible for many important results. Given a symplectic vector space V over a finite field (of odd characteristic) one can construct a "quantum" Hilbert space H(L) attached to a Lagrangian subspace L in V. In addition, one can construct a Fourier Transform F(M,L): H(L)→H(M), for every pair of Lagrangians (L,M), such that F(N,M)F(M,L)=F(N,L), for every triples (L,M,N) of Lagrangians. This can be used to obtain a natural “quantum" space H(V) acted by the symplectic group Sp(V), obtaining the Weil representation. In the lecture I will give elementary introduction to the above constructions, and discuss the categorification of these Fourier transforms, what is the related sign problem, and what is its solution. The outcome is a natural category acted by the algebraic group G=Sp, obtaining the categorical Weil representation. The sign problem was worked together with Ofer Gabber (IHES).