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?


It's a typo. R always denotes representations of Gamma, and representations of LG are named with a V.

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