It is an important property of usual equivariant $K$-theory that $K_G(X)\cong K(X/G)$ whenever $G$ acts freely on $X$.
What can be said about $KR(X)$ when the action of $C_2$ on $X$ is free? In the compact case one can, from a Real vectorbundle on $X$, produce an $\mathbb{R}$ vectorbundle on $Y:=X/C_2$ by qoutienting out the antilinear involution. I am interested in what kind of structure/property characterises vectorbundles arising in this way.
If $X\to Y$ is a principal bundle which admits a section, then this can be refined to an isomorphism $KR(X)\cong K(Y)$, since then $X\cong Y\coprod Y$ and a Real vectorbundle on $X$ is the disjoint union of a complex vectorbundle on $Y$ and its conjugate.
In the case of a general $C_2$-principal bundle $X\to Y$, let $w\in H^1(Y,C_2)$ denote the corresponding class in cohomology. I suspect that the $\mathbb{R}$ vectorbundles on Y arising in the above way have the following property:
Let $E\to X$ be a Real vectorbundle (of rank $N$), and denote $F=E/C_2\to Y$ the associated $\mathbb{R}$ bundle. In a standard way, for a suitable cover $U$ of $Y$, F defines a class in $\check{H}^1(Y,O(2N),U)$ . I think this has a refinement to a class with coefficients in $U^R(N)$, the subgroup of orthogonal matrices which act either linear or antilinear wrt the standard complex structure on $\mathbb{R}^{2N}$ (for this to happen I guess both $F$ and $X$ should trivialize over $U$). This group has a surjection to $C_2$ tracking whether a matrix acts linear or antilinear, and the corresponding change of group morphism should carry the class of $F$ to (a Cech version of) $w$.
The construction I describe above is very adhoc and seems unnatural to me, and doesn't generalize to general even rank $\mathbb{R}$ vectorbundles anyway. But maybe one can find a formulation which does work for general vectorbundles? It seems that way, since somehow $w$ is the obstruction to choosing a complex structure on $F$ globally, while locally it has a canonical one (up to conjugation).
Also, more conceptual answers about the nature of $KR$ as an $C_2$ equivariant cohomology are welcome.
