Chern-Connes Character for the Invariant Dirac Operator in Odd   Dimensions
  In this paper we give a proof of the Lefschetz fixed point formula of Freed$^{\rm [1]}$ for an orientation-reversing involution on an odd dimensional spin manifold by using the direct geometric method introduced in [2] and then we generalize this formula under the noncommutative geometry framework. 
Yong Wang