11
$\begingroup$

For a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$, the fundamental group of $\mathfrak{h}_\text{reg}/W$ (where $\mathfrak{h}$ is the Cartan subalgebra, $\mathfrak{h}_\text{reg}$ is the subset of its regular elements, and $W$ the Weyl group) is the Artin braid group of type $\mathfrak{g}$.

Similarly, if we take a simple and simply-connected Lie group $G$ over $\mathbb{C}$ and consider the orbifold fundamental group of $T_\text{reg}/W$ (where $T$ is the Cartan torus of $G$), it is the affine Artin braid group of the corresponding type.

The references to the proofs to these statements can be found e.g. in Charney-Davis.

Now, take the moduli space of $G$-bundles on an elliptic curve $E$, which is known to be given by $(E\otimes_{\mathbb{Z}}\Lambda) /W$ (cf. Laszlo ) where $\Lambda$ is the coweight lattice of $T$. I believe the fundamental group of this space (minus 'non-regular' elements) is more or less given by the double affine braid group of the corresponding type, but I don't find any explicit reference. Could you give me one?

$\endgroup$
4
$\begingroup$

After a day of hard literature searching myself, apparently the statement goes back to Cherednik's original paper on DAHA in 1992, and also to van den Lek's article "Extended Artin Groups" in 1983.

A modern reference with all the detailed proofs can be found in this paper by B. Ion. If I understand the paper correctly, the $\pi_1$ of the regular points of the moduli space of flat G-bundles on $T^2$ is the quotient of the double affine Artin group by the central subgroup generated by an element usually denoted by $X_\delta$.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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