We give details of the proof of the remark made in cite{G2} that the Chern characters of the canonical generators on the K homology of the quantum group $SU_q(2)$ are not invariant under the natural $SU_q(2)$ coaction. Furthermore, the conjecture made in cite{G2} about the nontriviality of the twisted Chern character coming from an odd equivariant spectral triple on $SU_q(2)$ is settled in the affirmative.
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