# Harmonic Analysis of Spherical Functions on Real Reductive by Ramesh Gangolli

By Ramesh Gangolli

Analysis on Symmetric areas, or extra typically, on homogeneous areas of semisimple Lie teams, is a topic that has gone through a full of life improvement in recent times, and has develop into a valuable a part of modern arithmetic. this is often in simple terms to be anticipated, because homogeneous areas and team representations come up obviously in diversified contexts starting from quantity conception and Geometry to Particle Physics and Polymer Chemistry. Its explosive development occasionally makes it tough to gain that it really is truly quite younger as mathematical theories pass. The early principles within the topic (as is the case with many others) return to Elie Cart an and Hermann Weyl who studied the compact symmetric areas within the 1930's. in spite of the fact that its complete improvement didn't commence until eventually the 1950's whilst Gel'fand and Harish Chandra dared to dream of a idea of representations that incorporated all semisimple Lie teams. Harish-Chandra's thought of round services was once primarily entire within the past due 1950's, and used to be to turn out to be the forerunner of his enormous paintings on harmonic research on reductive teams that has encouraged an entire new release of mathematicians. it's the harmonic research of round capabilities on symmetric areas, that's on the concentration of this ebook. the basic questions of harmonic research on symmetric areas contain an interaction of the geometric, analytical, and algebraic facets of those areas. they've got for this reason attracted loads of realization, and there were many glorious expositions of the topics which are attribute of this subject.

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**Example text**

9) we conclude that w = w'. Density of ~2 in L2(W). Suppose 1/1 E L2(W) and fEI/IAdw = 0 for A E m2 . In particular fEI/IA15dw = 0 for A E m2, D E p(m 2). 8), 15 E L2(W) and hence 1/115 E U(w). But then the vanishing of fEI/I15· A dw for all A E m2 implies that 1/115 = 0 w-almost everywhere. Hence 1/1 = 0 w-almost everywhere on In. 9) implies that 1/1 = 0 w-almost everywhere. 0 Let us now consider a Gel'fand pair (G,K). U(GjjK) is a commutative Banach algebra and is a subalgebra of U(G). 16) f KXK 1(k 1 )r(k 2)fdk 1 dk2 of L 1(G) onto L 1(G j j K), 1and r being respectively the left and right regular representations of G in L 1(G).

All vectors in VF are weakly analytic. This gives (i). To prove (ii) we start with the equation f/J(x) = EFn(x)E F and differentiate with respect to q, remembering that n(q) commutes with E F. 4. Suppose G is connected. Let b E ~(K) and let n be an irreducible representation of G in a complete locally convex space V such that dim(Vt)) < 00. Let f/J# = f/J:'o(G ---+ HomdW, W)) be the K-central elementary spherical function of type b associated with n. Then f/J# is an analytic function, the map ql--+r(q) = f/J#(l;q) = f/J#(q; 1) is an irreducible representation of U(gc)K in W, and (q E U(gJK.

We consider the commutative Banach algebra U(G//K) and use the Gel'fand transform theory to express linear functionals defined on Ll(G//K) or on dense ideals of it as integrals with respect to measures on the spectrum of L 1 (G//K). The Plancherel formula for G/K will follow from such an integral representation. We begin with a brief review of what is known as the abstract Plancharel theorem for commutative Banach algebras with involution. For more details and a general theory see for instance Loomis [1953].