Higher Orbital Integrals, Cyclic Cocycles and Noncommutative Geometry

Let G be a linear real reductive Lie group. Orbital integrals define traces on the group algebra of G. We introduce a construction of higher orbital integrals in the direction of higher cyclic cocycles on the Harish-Chandra Schwartz algebra of G. We analyze these higher orbital integrals via Fourier transform by expressing them as integrals on the tempered dual of G. We obtain explicit formulas for the pairing between the higher orbital integrals and the K-theory of the reduced group -algebra, and we discuss their application to K-theory.