  Let G be a connected real reductive group. Orbit integrals define traces on
the group algebra of G. We introduce a construction of higher orbit integrals
in the direction of higher cyclic cocycles on the Harish-Chandra Schwartz
algebra of G. We analyze these higher orbit 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 orbit integrals and the K-theory of
the reduced group C*-algebra, and discuss their applications to representation
theory and K-theory.
