The classical Fourier transform F is essentially an element in the oscillator representation (ω_x, L² (Rⁿ)) of S͠p(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 L²(Rⁿ) is decomposed into multiplicity-free irreducible O(n)xS͠L(2,R)–modules. When an operator T̃=ToF^-1 commutes with O(n)xS͠L(2,R)-actions, Schur's lemma implies that on each component: T̃ acts by scalar multiplication 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....[ Read more ]

The classical Fourier transform F is essentially an element in the oscillator representation (ω_x, L² (Rⁿ)) of S͠p(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 L²(Rⁿ) is decomposed into multiplicity-free irreducible O(n)xS͠L(2,R)–modules. When an operator T̃=ToF^-1 commutes with O(n)xS͠L(2,R)-actions, Schur's lemma implies that on each component: T̃ acts by scalar multiplication 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.