Plancherel Theorem on the Symplectic Group SP(4,R)

TL;DR

This paper develops the Fourier transform and establishes the Plancherel theorem for the semisimple Lie group H and its inhomogeneous extension G, with applications to the symplectic group and its nilpotent subgroup.

Contribution

It introduces a Fourier transform framework and proves the Plancherel theorem for the group H and the inhomogeneous group G, including the symplectic subgroup.

Findings

Plancherel theorem established for H and G

Fourier transform defined for these groups

Results obtained for the nilpotent symplectic subgroup

Abstract

Let H be the 15- dimensional connected semisimple Lie group with its Iwasawa decomposition of H. Let G be the group of the semi direct product of H and the four dimensional real vector group . The goal of this paper is to define the Fourier transform in order to obtain the Plancherel theorem on H and so on G. Since its symplectic group of dimension 10 is a subgroup of H, then it will be easy to get the Plancherel theorem on the symplectic group and so on its inhomogeneous group. To this end, we obtain some interesting results on its nilpotent symplectic group