Weil Representation, Howe Duality, and the Theta correspondence

Let W be a finite dimensional vector space over k with a non-degenerate alternatingform . Such a vector space will be called a symplectic space. Itsdimension is even and we denote it by 2n. The Heisenberg group H(W ) associatedto the symplectic space W is a non-trivial central extension of W by k and isdefined to be the group of pairswith the law of multiplication{(w, t)|w ∈ W, t ∈ k},(w 1 , t 1 )(w 2 , t 2 ) = (w 1 + w 2 , t 1 + t 2 + 1 2 < w 1, w 2 >)The Heisenberg group H(W) clearly sits in the exact sequence0 → k → H(W ) → W → 0.The commutator subgroup of H(W ) is k and all the one dimensional representationsof H(W ) factor through W . We construct below an infinite dimensionalsmooth representation of H(W ). For this fix a decomposition of W as W = W 1 ⊕W 2where W 1 and W 2 are maximal totally isotropic subspaces of W (ie, subspaces onwhich the alternating form is identically zero). Such a decomposition will be calleda complete polarisation of W .The representation depends on the choice of anadditive character ψ of k which will be fixed in all these lectures.Define the representation ρ ψ of the Heisenberg group H(W ) on S(W 1 ) as follows:ρ ψ (w 1 )f(x) = f(x + w 1 ) for all x, w 1 ∈ W 1 ,ρ ψ (w 2 )f(x) = ψ(< x, w 2 >)f(x) for all x ∈ W 1 , w 2 ∈ W 2ρ ψ (t)f(x) = ψ(t)f(x) for all t ∈ k, x ∈ W 1 .It can be easily checked that this gives a smooth representation of H(W ). Thisrepresentation of H(W ) is called the Schrödinger representation.Remark 1.1: If ¯ψ denotes the character ¯ψ(x) = ψ(−x), then ρ ¯ψ is the smoothdual of ρ ψ . To see this observe that∫(f 1 , f 2 ) → f 1 f 2 dw, f 1 , f 2 ∈ S(W 1 )W 1is a non-degenerate invariant form.There is another model of this representation, called the lattice model whichis also quite useful. In fact we define a representation which is more general thanboth of these.2