Heisenberg Group: Representation

This lecture introduces the Heisenberg group and its representation, starting with upper triangular matrices over real numbers. The group acts on functions by shifts and multiplications, showing almost commutativity. The lecture explores the center of the group, the action on L2 functions, and the construction of the representation using Stone-von Neumann theorem. It delves into the Fourier transform, the action of SL2R on the Heisenberg group, and the importance of Gaussian functions. The lecture also discusses the extension to finite fields, the Schrodinger representation, and the metaplectic group for actual representations. The construction of the representation on L2 of K and the uniqueness of the Fourier transform are highlighted.