Weil Representation, Howe Duality, and the Theta correspondence

of Sp(W ⊗ V ) to Sp(W ) is isomorphic to the tensor product of the metaplecticrepresentations of Sp(V ) associated to the characters ψ αi .Remark 2.9: For V and W as in the previous remark, O(V ) ⊂ Sp(W ⊗ V )in the obvious way. Let W = W 1 ⊕ W 2 be a complete polarisation of W . ThenW ⊗ V = W 1 ⊗ V ⊕ W 2 ⊗ V is a complete polarisation for W ⊗ V . Recall nowthat the Schrödinger model of the metaplectic representation of Sp(W ˜ ⊗ V ) isobtained on the compactly supported locally constant functions on W 1 ⊗ V . NowO(V ) operates on S(W 1 ⊗ V ) in a natural way, and it is easy to see from theexplicit formulae for the representation of the Heisenberg group that this action isan intertwining operator; therefore we conclude that(i) The metaplectic covering of Sp(W ⊗ V ) splits on the subgroup O(V ).(ii) The metaplectic representation of Sp(W ⊗ V ) restricted to O(V ) is thenatural representation of O(V ) on functions on W 1 ⊗ V (at least, up to a characterof O(V )).Remark 2.10: Similarly if V = V 1 ⊕ V 2 where V 1 and V 2 are maximal totallyisotropic subspaces of the quadratic space V , W ⊗ V = W ⊗ V 1 ⊕ W ⊗ V 2 is acomplete polarisation of W ⊗ V .We conclude as in the previous remark thatthe metaplectic covering of Sp(W ⊗ V ) splits on Sp(W ), and the metaplecticrepresentation of Sp(W ⊗ V ) restricted to Sp(W ) is the natural representation ofSp(W ) on S(W ⊗ V 1 ).Remark 2.11 : By remark 1.1, the dual of the metaplectic representationω ψ of Sp(W ˜ ) associated to the character ψ(x) is the metaplectic representation ofSp(W ˜ ) associated to the character ψ(−x). Therefore by remarks 2.8 and 2.10,ω ψ ⊗ ω ∗ ψ ∼ = S(W ),where Sp(W ) operates on S(W ) in the natural way (observe that ω ψ ⊗ ω ∗ ψ is a representationof Sp(W )). This relation is specially useful for the Weil representationof Sp(n, F q ) where the Weil representation is self-dual if −1 is a square in F q (seeremark 2.4), i.e. for q ≡ 1 mod 4, and the above relation can be used to calculatethe character of the Weil representation up to sign.8