The classical Fourier transform F is essentially an element in the oscillator representation (ω_χ,L^2 (R^n)) of (Sp) ̃(2n,R) the two-fold cover of Sp(2n,R). Under the dual correspondence of the dual reductive pair (O(n),SL(2,R)), the space L^2 (R^n) is decomposed into multiplicity-free irreducible O(n)×(SL) ̃(2,R)-modules. When an operator T ̃=T∘F^(-1) commutes with O(n)×(SL) ̃(2,R)-actions, Schur's lemma implies that on each component: T ̃ acts as a scalar and thus T is just F up to the scalar. Then we obtain a family of operators T's that shares some important properties with F. A similar discussion applies to the dual pair (U(n),U(1)), as well as (U(p,q),U(1)) when the indefinite bilinear form is considered