

Tomasz Przebinda (University of Oklahoma)
Title:
Symmetry properties of the Fourier Transform and Howe Correspondence
Abstract:
The classical Fourier transform on the plane commutes with the action of rotations. This has
many consequences. We focus on the projection operator from L^{2}(ℝ) onto an O_{2}isotypic
component and explore it in terms of Weyl calculus. This leads us to a second group action,
specifically of SL_{2}(ℝ). These two groups are mutual centralizers in the symplectic group Sp_{4}(ℝ)
and the joint group actions happen to come from the Weil representation of the
metaplectic group S̃p_{4}(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 SL_{2}(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.


