The Weil representation gives an algebraic method to classify the irreducible representations of symplectic groups Sp(2n,Fq). In this talk, we will work out this method on Sp(2,Fq) = SL(2,Fq), and show how in this case, it associates to each irreducible representation of SL(2,Fq) a maximal torus. We will first construct the Heisenberg group corresponding to a symplectic vector space and classify its irreducible representations via the finite Stone-Von-Neumann theorem. We then use an action of Sp(4,Fq) on the irreducible representations to construct the Weil representation of Sp(4,Fq). Finally, we will decompose the Weil representation into irreducible representations of Sp(2,Fq) via Howe duality. Speaker(s): Calvin Yost-Wolff (UM)