Download e-book for kindle: Harmonic Analysis of Spherical Functions on Real Reductive by Ramesh Gangolli

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By Ramesh Gangolli

ISBN-10: 3642729568

ISBN-13: 9783642729560

ISBN-10: 3642729584

ISBN-13: 9783642729584

Analysis on Symmetric areas, or extra regularly, on homogeneous areas of semisimple Lie teams, is a topic that has gone through a energetic improvement in recent times, and has develop into a critical a part of modern arithmetic. this is often in simple terms to be anticipated, on the grounds that homogeneous areas and staff representations come up certainly in different contexts starting from quantity idea and Geometry to Particle Physics and Polymer Chemistry. Its explosive progress occasionally makes it tough to gain that it truly is really fairly younger as mathematical theories move. The early rules 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. even if its complete improvement didn't commence until eventually the 1950's while Gel'fand and Harish­ Chandra dared to dream of a concept of representations that integrated all semisimple Lie teams. Harish-Chandra's thought of round services used to be primarily whole within the overdue 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 a complete new release of mathematicians. it's the harmonic research of round capabilities on symmetric areas, that's on the concentration of this e-book. the basic questions of harmonic research on symmetric areas contain an interaction of the geometric, analytical, and algebraic points of those areas. they've got as a result attracted loads of awareness, and there were many fantastic expositions of the topics which are attribute of this subject.

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Additional resources for Harmonic Analysis of Spherical Functions on Real Reductive Groups

Sample text

10. Let

and is nontrivial, and we can choose an irreducible Frechet representation n of class 1 such that

is one-one from the set of all elementary spherical functions to the set of all nontrivial homomorphisms of CAGjjK) that are continuous.

24 1. The Concept of a Spherical Function If (i), and hence (iv) is true, for f E Cc,F(G), y E G, we have Jf(x) I['(xy) dx = JJ f(xk- G 1 )eF(k) I['(xy) dx dk GxK = JJ f(x)eF(k) I['(xky) dx dk GxK = Jf(x) I['(x) dx' I['(y) . G Hence proving (i) => (ii). Similarly we prove (i) => (iii). In both cases, the representation involved is (', 1['). No other representation is possible. For, if fV * I[' = p(f) 1[', evaluating both sides at 1 we get (f, 1[') = p(f); similarly, in (iii), (J, 1[') = p'(f). This remark also proves the implications (ii) => (i) and (iii) => (i).

E G) as well as y(f)(fE Cc,F(G)) span the same Proof. Let H = HomdU, U). The linear functional f - t Y(~F * f * ~F) on Cc(G) is continuous and so there is a unique H-valued measure fJ. 3. * ~F) for all I, we must have ~F * J1. * ~F = J1.. Select g E Ce,F(G) such that y(g) = 1. ). But g being an element of Ce(G), gV * J1. = h is a continuous function on G. Hence y(f) =

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

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