  In LM, we proved a family version of the famous Witten rigidity theorems and
several family vanishing theorems for elliptic genera. In this paper, we
gerenalize our theorems LM in two directions. First we establish a family
rigidity theorem for the Dirac operator on loop space twisted by general
positive energy loop group representations. Second we prove a family rigidity
theorem for spin^c-manifolds. Several vanishing theorems on both cases are also
obtained.
