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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) seems to be Matumoto, T., Equivariant K-theory and Fredholm operators. 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.

EDIT: I noticed that Freed-Hopkins-Teleman 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."

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    $\begingroup$ As far as I can see the construction is basically the same as the non-equivariant case: given an equivariant map $X \to F$, equivariantly perturb the it so that $\ker(F_x)$ and $\text{coker}(F_x)$ are equivariant vector bundles - this can be achieved by $G$-averaging the perturbations from Atiyah's original argument, for instance. The index is then the class $\ker(F_x) - \text{coker}(F_x)$ in $K_G(X)$. $\endgroup$
    – Paul Siegel
    Commented Jan 5, 2017 at 13:16
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    $\begingroup$ @Paul: thanks! I was guessing that this is possible, but I was wondering if in the last 45 years somebody has bothered to carry it out. But yes, it seems to me now that perturbing the map so that $ker(F_x)$ and $coker(F_x)$ are finite-dimensional is easer than what I was looking for. I was looking for the proof Atiyah gives in the Appendix of his book "K-Theory", which goes without perturbing. It rather finds a closed finite-codimension subspace of $H$ orthogonal to the kernel of all $F_x$. I was wondering if such subspace can be found in the equivariant setting... $\endgroup$
    – Konrad Waldorf
    Commented Jan 5, 2017 at 14:04

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