Equivariance of generalized Chern characters

In this note some generalization of the Chern character is discussed from the chromatic point of view. We construct a multiplicative G_{n+1}-equivariant natural transformation \Theta from some height (n+1) cohomology theory E^*(-) to the height n cohomology theory K^*(-)\hat{\otimes}_F L, where K^*(-) is essentially the n-th Morava K-theory. As a corollary, it is shown that the G_n-module K^*(X) can be recovered from the G_{n+1}-module E^*(X). We also construct a lift of \Theta to a natural transformation between characteristic zero cohomology theories.