  Recently, two of the authors of this paper constructed cyclic cocycles on
Harish-Chandra's Schwartz algebra of linear reductive Lie groups that detect
all information in the $K$-theory of the corresponding group $C^*$-algebra. The
main result in this paper is an index formula for the pairings of these
cocycles with equivariant indices of elliptic operators for proper, cocompact
actions. This index formula completely determines such equivariant indices via
topological expressions.
