Abstract
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.
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© 2001 Springer-Verlag New York, Inc.
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Jorgenson, J., Lang, S. (2001). The Harish-Chandra Series and Spherical Inversion. In: Spherical Inversion on SL n (R). Springer Monographs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9302-3_8
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DOI: https://doi.org/10.1007/978-1-4684-9302-3_8
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-2883-2
Online ISBN: 978-1-4684-9302-3
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