  Let G be a finite group acting on a finite dimensional real vector space V.
We denote by P(V) the projective space associated to V. In this paper we
compute in a very explicit way the rank of the equivariant complex K-theory of
V and P(V), using previous results by Atiyah and the author. The interest of
this computation comes from explicit formulas given by the
Baum-Connes-Slominska Chern character and the basic fact that the equivariant
K-theory of V is free. We use these topological computations to prove algebraic
results like computing the number of conjugacy classes of G which split in a
central extension. Our main example is the case where V = R^n and G = the
symmetric group of n letters acting on V by permutation of the coordinates.
This example is related to the famous pentagonal identity of Euler and
(ironically) the Euler-Poincare characteristic of the equivariant K-theory of
V.
