Fourier transform and minimal representation

The Fourier transform on a vector space possesses many distinguished properties 

making it one of the central figures in many fields of mathematics. 

From the point of view of representation theory, it is not alone. It belongs

to an infinite family of operators forming a distinguished (minimal) representation

of the metaplectic group. We shall explore how minimal representations

of other (p-adic) groups give rise to generalized Fourier transforms on cones.