Weil Representation, Howe Duality, and the Theta correspondence

Theorem 1.3(Stone, von Neuman): The Heisenberg group H(W ) has aunique irreducible smooth representation on which k operates via the character ψ.(ρ ψ , S)We will denote this unique irreducible smooth representation of H(W ) by2 Metaplectic Group and the Weil RepresentationThe main theme of these lectures is not the representation (ρ ψ , S) of the Heisenberggroup constructed in the last section but rather a (projective)-representation ofthe symplectic group which is constructed using intertwining operators of thisrepresentation of the Heisenberg group. To define this, observe that the symplecticgroup Sp(W ) of W (i.e. the automorphisms of W preserving the alternating form) operates on H(W ) by g · (w, t) = (gw, t). Clearly this action is trivial onthe centre of H(W ) and therefore by the uniqueness theorem of Stone and vonNeumann, there is an operator ω ψ (g) (unique up to scaling) on S such thatNow defineThenρ ψ (gw, t) · ω ψ (g) = ω ψ (g) · ρ ψ (w, t), for all (w, t) ∈ H(W ) (∗).˜ Sp ψ (W ) = {(g, ω ψ (g)) such that (∗) holds}.˜ Sp ψ (W ) is a group under pointwise multiplication, called the metaplecticgroup, and fits in the following exact sequence:0 → C ∗ → ˜ Sp ψ (W ) p → Sp(W ) → 0.The metaplectic group comes equipped with a natural representation obtained byprojection on the second factor (g, ω ψ (g)) → ω ψ (g) ∈ Aut(S). This representationof the metaplectic group is called the Weil representation or the metaplecticrepresentation or the oscillator representation.Theorem 2.1: The map p restricted to the commutator subgroup Sp ˆ ψ (W ) =[ Sp ˜ ψ (W ), Sp ˜ ψ (W )] of Spψ ˜ (W ) is a surjection onto Sp(W ) with a kernel of order 2.In particular,Sp ˜ ψ (W ) = Sp ˆ ψ (W ) × Z/2 C ∗ .4