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, 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.