The General Noncompact Symmetric Space

Audrey Terras
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Audrey Terras: University of California at San Diego, Department of Mathematics

Chapter Chapter V in Harmonic Analysis on Symmetric Spaces and Applications II, 1988, pp 258-350 from Springer

Abstract: Abstract 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.

Keywords: Symmetric Space; Modular Form; Fundamental Domain; Eisenstein Series; Iwasawa Decomposition (search for similar items in EconPapers)
Date: 1988
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DOI: 10.1007/978-1-4612-3820-1_2

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