The General Noncompact Symmetric Space

The first four chapters of this tome considered various examples and applications of symmetric spacesXalong with harmonic analysis onX andX/Γ for discrete groups Γ of isometries ofX. Here we consider some aspects of analysis on a general noncompact symmetric spaceX =G/K. Our discussion will be very sketchy. The main goal is to lay the groundwork for extension of the results of the preceding chapters to other symmetric spaces which are of interest for applications; in particular, the Siegel upper half plane ℋn (which can be identified withSp(nℝ)/U(n)) and hyperbolic three space ℋc (which can be viewed asSL(2, ℂ)/SU(2)). We will also be interested in the fundamental domains ℋn/Sp(nℤ) for the Siegel modular group as well as the fundamental domain ℋc/SL(2, ℤ[i]) for the Picard group. It is possible to generalize just about everything we did in the earlier chapters for such examples; e.g., the Selberg trace formula. And our main motivations for doing so come from number theory. Because it is time consuming and sometimes not so enlightening to do each of these examples separately, we have decided to present some results on the general symmetric space. Those interested in number theoretic applications may find this equally tedious and attempt to jump to the next section. But I think it is useful to know what a general Iwasawa decomposition is, for example, in order to find the right coordinates to use in solving a given problem on the symmetric space. Of course, others will say that the discussion which follows is neither sufficiently general, detailed, or rigorous. We refer those characters to the texts of other authors which are listed below.