process

Twisted K-theory – old and new 139set of conjugacy classes hi, but as a function f on the full group with certain extraproperties which we shall explain now.If we replace by 0 such that 0 D 1 , we have a canonical isomorphism W H .X 0 / C.0/ ! H .X / C./ induced by x 7! x. This map exchanges f./and f. 0 / and we have the relation f./D .f . 0 //, which says that f is essentiallya function on the set of conjugacy classes.On the other hand, if the action of zG is of linear type, we have an extra relation, aneasy consequence of the formula in [11], which is f ./ D f ./ when is an n throot of unity. To summarize, we get the following theorem.Theorem 6.6. Let G be a finite group and A D M n .C/ with a G-action. Thenthe ungraded twisted equivariant K-theory K .A/G.X/ is a subgroup of the equivariantK-theory KG z .X/, where zG is the pull-back diagramzGGSU.n/ PU.n/.More precisely, K .A/G.X/ ˝Z C may be identified with the C-vector space of functionsf on D zG with f./ in H even .X g / C./ , ./ D g, such that the following twoidentities hold:1. If 0 D 1 , one has f./D .f . 0 //, according to the formula above.2. f ./ D f ./ if is an n th root of unity.In particular, if X is reduced to a point, we have D Id and K .A/G.X/ is free withrank the number of conjugacy classes of G which split into n conjugacy classes in zG(as seen in 6.5.)Remark 6.7. This theorem is not really new. In a closely related context, one findssimilar results in [1] and [42]. We should also notice that the same ideas have beenused in [31] for representations of “linear type”. Finally, the theorem easily extendsto locally compact spaces if we consider cohomology with compact supports on theright-hand side.Theorem 6.8. Let A be any finite-dimensional graded semi-simple complex algebrawith a graded action of a finite group G. Then the graded K-theory GrK 0 .A 0 / ˚GrK 1 .A 0 / of the semi-direct product A 0 D G Ë A is a non trivial free Z-module. Inparticular, if V is a real finite-dimensional vector space with a G action, the groupKG A.V / ˚ KA G.V ˚ 1/ is free non trivial thanks to the Thom isomorphism.Proof. The algebra G ËA is graded semi-simple over the complex numbers. Therefore,it is a direct sum of graded algebras Morita equivalent to M n .C/ M n .C/ or M 2n .C/.In both cases, the graded K-theory is non trivial. The last part of the theorem followsfrom 4.2.