The Harish-Chandra Series and Spherical Inversion
This chapter is fundamentally based on [Har 58a]. We define the Harish-Chandra series for eigenfunctions of Casimir, prove its basic properties, and show how the spherical functions can be expressed in terms of this series. We incorporate from the start a crucial estimate by Gangolli to insure the possibility of term by term differentiation [Gan 71]. The need for such an estimate had arisen in Helgason’s approach to getting the inversion theorem on the Paley-Wiener space [Hel 66]. The applications to the inversion problem will come in the next chapter.
KeywordsHaar Measure Spherical Function Polynomial Growth Chapter Versus Iwasawa Decomposition
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