I'm not the person to understand everything in *Geometric Endoscopy and Mirror Symmetry*, but some parts of it are reasonably clear to me.

In particular, one of the main objects, mathematically speaking, is the category of coherent sheaves on an orbifold point $\mathrm{pt}/\Gamma$, where $\Gamma$ is a finite group of automorphisms of some ${}^LG$-local system. This category is well-known in algebraic geometry to be just $\mathrm{Rep}(\Gamma)$.

The main point of the paper is that some other, less obvious, additive category happens to be isomorphic to this well-known $\mathrm{Rep}(\Gamma)$. This means, in particular, that its objects are actually sums of things like $R\otimes V_R$ where $R$ goes over irreps of $\Gamma$.

But (9.5), (9.8) (numbers from the version 3) are different:

$${\mathcal F}_{\mathrm{Reg}(\Gamma)} = \bigoplus_{R \in \mathrm{Irrep}({}^LG)} R^* \otimes {\mathcal F}_R$$

Note the sum goes by $\mathrm{Irrep}({}^LG)$ where I thought $\mathrm{Irrep}(\Gamma)$ is appropriate. The source of the chain of equations seems to be on page 112, where, quote, the regular representation

$$\mathrm{Reg}(\Gamma) = \bigoplus_{R \in \mathrm{Irrep}({}^LG)} R^* \otimes R,$$

unquote. Thus I've decided it's a typo in the formula for the regular representation carried over for a next several pages. Yet I feel out of place until I'm completely sure there is no other explanation — theoretically there *could* be some relationship between the representations of $\Gamma$ and those of ${}^LG$, after all.

Question:do these two formulas have a typo or is there a meaning I miss?