The Weil Representation

This is my senior honors thesis done in my final year as an undergraduate at Stanford University, under the direction of Professor Akshay Venkatesh. We will construct the Weil representation of SL2(R) through a natural action of the Heisenberg group Heis(R) on the space of square-integrable complex-valued functions L 2(R), together with the celebrated Stone-von Neumann theorem of functional analysis. Our approach will be to explicitly construct the analogous representation of SL2(Fp) on L (Fq ), for p an odd prime, using the finite-field equivalents of the aforementioned ingredients. This will allow us to separate the functional analytic complications of unitary representation theory of Lie groups from the representation theoretic and purely algebraic ideas. As the final arc to our triptych of stories, we will construct the Lie algebra analogue of the Weil representation through the Heisenberg algebra. Throughout this paper, we will be as explicit as possible while simultaneously giving motivation to our computations, allowing us to speak of concepts concretely without sacrificing the overarching philosophy. In spite of these efforts to make this paper self-contained, some proofs are omitted to avoid straying from the main arc of the story of the Weil representation. We will draw upon ideas from homological algebra, group theory, algebraic topology, functional analysis, measure theory, Lie theory, and, of course, representation theory. What is remarkable is that this topic touches as many subject areas as this paper will use, including number theory, harmonic analysis, topology, and physics. Because of its place in the intersection of a multitude of different fields, the Weil representation is also known, in the literature, as the Segal-Shale Weil representation, the metaplectic representation, the harmonic representation, and the oscillator representation.