Freed, Hopkins and Teleman constructed an isomorphism (we call it FHT isomorphism) between twisted equivariant K-theory of compact Lie group
and the "Verlinde ring" of the loop group of
([FHT1, FHT2, FHT3]).
However, naturality of the isomorphism
with respect to group homomorphisms has not been verified.
We construct induced homomorphisms
whose tangent map is injective.
is a positive central extension of the loop group of
, so that FHT isomorphism is a natural transformation between two objects.
In fact, we construct another object char and verify that three objects are naturally isomorphic with respect to
Moreover, we extend these constructions for K-theory and
to compact connected Lie group with torsion-free
satisfying the "decomposable condition", and verify that they are isomorphic. This is a generalization of naturality of
verified in [FHT1].