<a href="https://arxiv.org/abs/0902.2832">Yoshinori Namikawa associates a Weyl group</a> $ W $ to any symplectic affine complex variety $ X $ with good $ \mathbb{C}^* $-action. He provides a semi-explicit description of $ W $, which requires finding a symplectic resolution $ π: Y \to X $ and then reading off a monodromy from $ π $ above every symplectic codimension-2 leaf. For determining this monodromy, does it suffice to take a “local resolution” around every leaf?

**Why I am asking**<br>
I am trying to determine the Namikawa-Weyl groups for Nakajima quiver varieties $ X = \mathcal{M} (Q, α) $. For many pairs $ (Q, α) $, a resolution is given by $ Y = \mathcal{M}_θ (Q, α) $ where $ θ \in \mathbb{R}^{Q_0} $ is a stability parameter. However, this procedure does not work for all $ (Q, α) $. I however succeed in defining an apparent local resolution of singularities on set-theoretic level (possibly even analytic or algebraic), and would like to show that my resolution suffices to determine the Namikawa-Weyl groups.

**The description of Namikawa-Weyl groups**<br>
While Namikawa initially constructs his Weyl group via Poisson deformation theory, in Theorem 1.1 he provides a semi-explicit description as follows. Choose a symplectic resolution $ π: Y \to X $. Then the Weyl group $ W $ is isomorphic to $ \prod\limits_{F \text{ codim-2 leaf}} W_F^{τ_F} $ where $ W_F $ is the classical ADE Weyl group of the leaf and $ τ_F $ is a Dynkin diagram automorphism. The automorphism $ τ_F $ is determined by reading off the monodromy of the $ \mathbb{P}^1 $ components of the exceptional fiber of $ π $ along the leaf $ F $. Basically, for every leaf $ F $ we need to find out whether different $ \mathbb{P}^1 $ are connected above $ F $ or stay separate.

**Why I expect local resolutions to suffice**<br>
My intuitition is that the symplectic resolution $ π $ only serves as a “magnifying glass” for determining the automorphism. In other words, I expect that the automorphism $ τ_F $ can already be read off from a neighborhood of the leaf $ F $. My best effort to formulate a concrete hypothesis is something like: Let $ F \subseteq X $ be a codimension-2 leaf and let $ U \subseteq X $ be an algebraic neighborhood of $ F $, then any symplectic resolution of $ U $ gives the same monodromy above $ F $ as if we took a symplectic resolution of the whole $ X $. I want to simplify things even further and show that it suffices to construct a local analytic resolution instead of a local algebraic resolution.

**My question**<br>
How to formalize and prove that the correct monodromy can already be read off from some kind of “local resolution”? How far can the requirements on the local resolution be weakened, such that we still compute the correct monodromy?

**Where I have already searched**<br>
I know some GAGA principles and have also looked into Stein spaces. I have tried to show that any two local analytic resolutions give the same monodromy, by reducing the question to automorphisms of Kleinian singularities. I have contemplated the option of lifting an analytic resolution to an algebraic one, attempting a singular GAGA. I have searched the internet with queries such as “local symplectic resolution”, and consulted Mathoverflow on the tag [symplectic-resolutions].