Long time no post. Boalch proved in some of his papers that the wild character variety on the punctured $\mathbb A^1$ comes from a quasi-Hamiltonian reduction of a “higher fission space” $_G A_H$ by actions of $G$ and $H$ (some Levi subgroup in $G$). The quotient $_G A_H / (G \times H)$ thus carries a Poisson structure, and if one looks at the quasi-Hamiltonian reduction instead, i.e. looks only at the preimage under the moment maps of conjugacy classes $\mathcal C_G \subseteq G, \mathcal C_H \subseteq H$ then we get a symplectic leaf of this space (which furthermore is hyperkähler etc etc.)

In our language, since Boalch is looking at meromorphic singularities, he’s only considering the case where our knot around the puncture is a torus link made of full twists. In the regular case, $H$ is a maximal torus, and all the strands of the link have microlocal rank 1. The map $\mu_G$ corresponds to taking the global monodromy around the puncture, and $\mu_H$ is taking the microlocal monodromy. So looking at $\mu_G^{-1}(1)/G$ corresponds to closing up into a $\mathbb P^1$ and specifying $\mathcal C_H$ corresponds to prescribing the microlocal monodromy along each strand. Note that in this case every strand goes back to itself.

So I’ve been trying to generalize this to more general cases, and to see whether the quasi-Hamiltonian conditions still hold. When the knot around the puncture is not made of full twists, we can’t use the fission spaces, but instead we need to use something like open Bott-Samelson varieties instead. These already appeared in Lusztig’s early work on character sheaves.

For a particular braid $s_1…s_k$ fix a Borel B and consider the open BS variety given by $Z^o_s = Bs_1B \times^B … \times^B Bs_kB$ (B superscript indicates quotient by twisted diagonal action of B). This has adjoint actions of $B$, $H$ and $U=B/H$. Lusztig then defines two varieties associated with this braid:
$$ Y_s = (G \times Z^o_s)/B, \dot{Y}_s = (G \times Z^o_s)/U $$
where B and U act on the right on G, and by the adjoint action on $Z^o_s$. There’s obviously a map $\dot{Y}_s \to Y_s$, which sort of corresponds to forgetting the microlocal monodromy. I believe that the space $\dot{Y}_s$ is the correct generalization of Boalch’s varieties, because in the case where the braid is made of $r$ full twists, this is isomorphic to $_GA^r_H = G \times H \times (U \times U_-)^r$. And this also has and action of, and a map to $G \times H$ which agrees with the previous case.

We’d like these to also be quasi-Hamiltonian. This is not a problem in the $G$ case because it’s entirely analogous, but the map $\mu_H$ has a problem: it is not equivariant wrt the conjugation action $H \curvearrowright H$. Instead it’s equivariant wrt a twisted conjugation action:
$$ h\cdot h_0 = (\dot{w}^{-1} h \dot{w})h_0 h^{-1} $$
where $w$ is the corresponding element $s_1…s_k$ in the Weyl group $N(H)/H$.

So I need to look at a “twisted quasi-Hamiltonian” action, and check that this still has some nice properties. Moreover, even in the usual case where $w = 1$ I’d like to find another proof of the quasi-Hamiltonian conditions, by building the action crossing by crossing (start with the braid $s_1$, add another crossing $s_1 s_2$ etc.), and this requires considering twisted conjugation actions in the middle steps.

One of the hopes for this is that it’s possible to define symplectic reduction with other targets other than $G$ with conjugation G-action. This is described by Safronov here where we have a 1-shifted symplectic stack $X$ and a Lagrangian $L \to X$. Then, given an action $G \curvearrowright M$ and a $M/G \to G/G$ we can take the intersection of Lagrangians in $G/G$ and this agrees with the notion of (quasi-)Hamiltonian quotient in the case $pt/G \to \mathfrak g^*/G$ ($pt/G \to G/G$.)

So this maybe works for the twisted case, we have to prove that this twisted conjugacy class is “Lagrangian”. What’s missing is the corresponding notion of a fusion product between compatible twisted actions, since the usual notion of fusion requires the usual conjugation action. Has anyone here read/understood better the Safronov papers (or even better understood PTVV), and do you think this is a feasible thing to expect?