THESIS
2017
Abstract
The classical Fourier transform F is essentially an element in the oscillator
representation (ω
_{x}, L
^{2} (R
^{n})) 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
L
^{2}(R
^{n}) 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....[
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The classical Fourier transform F is essentially an element in the oscillator
representation (ω
_{x}, L
^{2} (R
^{n})) 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
L
^{2}(R
^{n}) 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.
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