The equivariant Atiyah-Jänich theorem is an isomorphism
$$
[X,F]_G \cong K_G^0(X),
$$
where $G$ is a compact Lie group, $X$ is a compact $G$-manifold, $F$ is the space of Fredholm operators on a certain $G$-Hilbert space, $[-,-]_G$ denotes $G$-homotopy classes of $G$-maps, and $K_G^0(X)$ is the $G$-equivariant K-theory of $X$, defined as the Grothendieck group of $G$-equivariant vector bundles over $X$. 


The standard reference for the equivariant Atiyah-Jänich theorem (see, e.g. [this question][1]) seems to be Matumoto, T., [Equivariant K-theory and Fredholm operators][2]. J. Fac. Sci. Univ. Tokyo Sect. I A Math. 18 1971 109–125. 

The proof given there proceeds like Atiyah's proof of the original theorem, it constructs an index map $[X,F]_G \to K_G^0(X)$, shows surjectivity, identifies the kernel with a space $[X,GL]_G$ of $G$-homotopy classes, and then proves the $G$-equivariant contractibility of $GL$, so that the kernel vanishes. 

Now, while the $G$-equivariant contractibility of $GL$ is treated carefully in all details, the construction of the index map doesn't seem to be treated at all (,,...will be expected to give an element...``). In particular, none of the difficulties of Atiyah's proof is mentioned, e.g. finding a closed subspace complementary to the kernels, etc. 


Also, a proof of surjectivity of the index map is only sketched in one sentence on page 3. 

EDIT: Another problem is that the paper claims that the regular representation of a compact Lie group $G$ is norm continuous, which is in fact only true if $G$ is discrete.

> So, my question is: is there a reference for the $G$-equivariant index map?

I have looked into Segal's paper on equivariant K-theory and have also checked the literature given [in the answers to this question][3]. 

EDIT: I noticed that [Freed-Hopkins-Teleman][4] also were aware of a missing proof; they wrote "While this is certainly well-known, we were unable to find an explicit statement in the literature."

  [1]: https://mathoverflow.net/questions/219916/equivariant-fredholm-operators-classify-equivariant-k-theory
  [2]: http://jairo.nii.ac.jp/0021/00002586/en
  [3]: https://mathoverflow.net/questions/5986/references-for-equivariant-k-theory
  [4]: https://arxiv.org/abs/0711.1906