The third in the "Loop groups and twisted K-theory" series by FHT treats compact Lie groups without any connectedness assumptions. I am trying to unwind what Theorem 2 of that paper (available [here](http://math.berkeley.edu/~teleman/math/finalDirac.pdf), page 5) says for a finite group.


Theorem 2: "For regular $\tau$, there is a natural isomorphism $R^\tau(LG\_s) \cong K\_G^\tau(G)$ ..."


For $G$ finite, I think that the regularity condition (page 7) is empty, as it has something to do with a maximal torus of $G$. So we can take $\tau = 0$. 


On the right-hand side, that means untwisted $K$-theory, which is naturally identified with $\bigoplus\_g R'(Z\_G(g))$, where $R'$ denotes the Grothendieck group of finite-dimensional complex representations, and $g$ runs over conjugacy class representatives. ($R'$ to avoid conflict with $R^\tau$, whose meaning I don't understand.)


What is on the left-hand side, when $\tau$ is zero and $G$ is finite? It is a Grothendieck group of representations of the group of free loops in $G$, modified somehow by $\tau$ ($\tau = 0$, so not modified at all). But I don't understand what it means. For finite $G$, loops are constant and the Lie algebra is trivial, so from the definitions on page 5, $LG = LG\_s = G$. But it can't simply be the Grothendieck group of complex representations of $G$, that is much smaller than the right-hand side.



What does Freed-Hopkins-Teleman say about finite groups?
Take a look at the Annals version of FHT III (2011). There, Theorem 2 is more carefully stated only for compact **connected** Lie $G$. I think what you want is Theorem 3 (whose statement and surrounding context are essentially the same as the '08 version of the paper you linked to):


**Theorem 3**. Let $G$ be a compact Lie group, $f \in G$. For a regular twisting $\tau$, there is a natural isomorphism $R^{\tau - \underline{\sigma}-\underline{d}}(L\_f G) \stackrel{\sim}{\to} K\_G^\tau([fG\_1])$.


If I'm not mistaken, for $G$ finite and $\tau = 0$, Theorem 3 reduces to an isomorphism $R(Z\_G(f))\cong K\_G(G/Z\_G(f))$. This isomorphism is, of course, an elementary observation in equivariant $K$-theory (cf. Example (ii), p. 132, [Segal 1968](http://archive.numdam.org/ARCHIVE/PMIHES/PMIHES_1968__34_/PMIHES_1968__34__129_0/PMIHES_1968__34__129_0.pdf)).


