Structure of Semisimple Lie Groups and Differential Operators on Them

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The theory of spherical functions attains its most beautiful development in the setting of a pair (G, K) where G is a connected semisimple Lie group with finite center and K is a maximal compact subgroup of G. In this chapter we shall describe the structure theory of these groups as well as the differential operators on them. Actually we shall treat a much wider class of groups, for technical reasons. For instance, a basic tool of the theory is that of descent: properties of spherical functions associated with the pair (G, K) are deduced from properties of spherical functions associated with pairs (M, K M ) where M is a suitable reductive subgroup of the group G, and K M is a maximal compact subgroup of M. In fact, M will be the reductive component of a parabolic subgroup of G. Now, even if G is connected and semi- simple, M will often have neither of these properties. Thus, in order to have a smooth inductive procedure, it is convenient to work with a class of groups wide enough to be closed under the descent from G to M. This is possible if we start with groups G that belong to the so-called Harish-Chandra class, first introduced by Harish-Chandra in [1975] for substantially the above reasons. We shall call them groups of class ℋ.