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The isomorphism for $K_G^1$ in the question is right. see

Freed, Daniel S.; Hopkins, Michael J.; Teleman, Constantin, http://jtopol.oxfordjournals.org/content/4/4/737.full.pdf+htmlLoop groups and twisted K-theory I Section. J. Topol. 4 (2011), no. 4, 737–798.

Section 3.5.4 and Appendix A.5

The isomorphism for $K_G^1$ in the question is right. see http://jtopol.oxfordjournals.org/content/4/4/737.full.pdf+html Section 3.5.4 and Appendix A.5

The isomorphism for $K_G^1$ in the question is right.

Freed, Daniel S.; Hopkins, Michael J.; Teleman, Constantin, Loop groups and twisted K-theory I. J. Topol. 4 (2011), no. 4, 737–798.

Section 3.5.4 and Appendix A.5

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Bo Liu
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The isomorphism for $K_G^1$ in the question is right. see http://jtopol.oxfordjournals.org/content/4/4/737.full.pdf+html Section 3.5.4 and Appendix A.5