Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
add comment

1 Answer

up vote 1 down vote accepted

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.