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Tomasz Przebinda (University of Oklahoma)


Symmetry properties of the Fourier Transform and Howe Correspondence


The classical Fourier transform on the plane commutes with the action of rotations. This has many consequences. We focus on the projection operator from L2(ℝ) onto an O2-isotypic component and explore it in terms of Weyl calculus. This leads us to a second group action, specifically of SL2(ℝ). These two groups are mutual centralizers in the symplectic group Sp4(ℝ) and the joint group actions happen to come from the Weil representation of the metaplectic group S̃p4(ℝ). This way, we enter Howe’s theory of dual pairs. We describe the Weyl symbol of the projection operator explicitly, relate it to the corresponding character of SL2(ℝ), and compute its wave front set. Then we extend this picture to the setting of Howe correspondence for dual pairs with one member compact.

This is joint work with Mark McKee and Angela Pasquale.