This work continues the ideas presented in the author's book, The Universality of the Radon Transform (Oxford, 2003), which deals with the group SL(2,R). The complication that arises for G = SL(3,R) comes from the fact that there are now two fundamental representations. This has the consequence that the wave operator, which plays a central role in our work on SL(2,R) is replaced by an overdetermined system of partial differential equations. The analog of the wave operator is defined using an MN invariant orbit of G acting on the direct sum of the symmetric squares of the fundamental representations. The relation of orbits, or, in general, of any algebraic variety, to a system of partial differential equations comes via the Fundamental Principle, which shows how Fourier transforms of functions or measures on an algebraic variety correspond to solutions of the system of partial differential equations defined by the equations of the variety. In particular, we can start with the sum T of the delta functions of the orbit of the group Γ = SL(2,Z) on the light cone. We then take its Fourier transform, using a suitable quadratic form. We then decompose the Fourier transform under the commuting group of G. In this way, we obtain a Γ invariant distribution which has a natural restriction to the orbit G/K, which is the symmetric space of G. This restriction is (essentially) the nonanalytic Eisenstein series. We can compute the periods of the Eisenstein series over various orbits of subgroups of G by means of the Euclidean Planchere formula. A more complicated form of these ideas is needed to define Poincaré series.
Fast fourier transform method for partial differential equations, case study: The 2-D diffusion equation
