Equivariant K-theory of real vector spaces and real projective spaces

The computation of the equivariant K-theory K∗ G(V ) of the Thom space of a real vector bundle has been done successfully only under some spinoriality conditions [1], thanks to a clever use of the Atiyah–Singer index theorem (even if G is a finite group). One purpose of this paper is to fill this gap, at least for real vector spaces (considered as vector bundles over a point). For this purpose, we use at the same time the results in [7] (generalizing those of Atiyah) and the equivariant Chern character (Slominska [10], Baum and Connes [4]). The interest of such a computation comes from many sources. First, it answers a question raised recently by Le Gall and Monthubert [9] in their investigations on a suitable index theorem for manifolds with corners. They have to compute an “indicial K-theory” which is precisely the equivariant K-theory K∗ G(V ), where V = R and G=Sn the symmetric group of n letters acting naturally by permutation of the coordinates in R. Secondly, these topological computations are linked with two natural algebraic questions. One of them is the determination of the number of conjugacy classes of a subgroup of the orthogonal group O(V ) which split in the central extension induced by the pinorial group Pin(V ) (cf. Theorem 2.4). The second question is how to compute the number of simple factors in the crossed product algebra G C(V ), where C(V ) is the Clifford algebra of V (cf. Theorem 2.11, Corollary 3.6 and Remark 3.11). Finally, these methods enable us to determine completely the rank of the group K∗ G(P(V )), where P(V ) is the real projective space of V , in terms of the number of certain conjugacy classes of G (cf. Theorems 3.4 and 3.8). As we shall see, Algebra and Topology are deeply linked in these computations. The previous results have a pleasant formulation when G is the symmetric group Sn acting on V =Rn as above. In this case (cf. Corollary 1.9), the groups K0 G(V ) and K1 G(V )