What does Freed-Hopkins-Teleman say about finite groups? 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.
 A: 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).
A: This is maybe more of a comment. Some time ago there was a paper by Simon Willerton titled "The twisted Drinfeld double of a finite group via gerbes and finite groupoids" which tried to make sense of the statement of Freed-Hopkins-Teleman in the case of finite groups (i.e. Dijkgraaf-Witten theory).
