I am learning Harmonic analysis on real reductive Lie groups. It seems to me that there are two kinds of treatment. One is through $(\mathfrak g, K)$ modules (e.g., Wallach, Real reductive groups), the other is through spherical functions (e.g., Gangolli and Varadarajan, Harmonic analysis of spherical functions on real reductive groups). Both of the books say that the theory is developed by Harsh-Chandra (and of course with others). I would like to understand the history and the relationship between these two approaches. Could any experts explain this here or suggest some references?
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