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I will follow the definition of Coulomb branches of $3d$ $\mathcal{N}=4$ gauge theories from the paper by Braverman, Finkelberg and Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N=4 gauge theories, II

Physical considerations aside, the above paper gives us a recipe that, given a complex reductive group $G$ and a representation of contangent type $M = N \oplus N^*$, where $N$ is a finite dimensional representation of $G$, produces a normal Poisson variety $\mathcal{M}_C(G,N)$, which is symplectic on its smooth locus. The way the variety is constructued, the algebra of regular functions on the Coulomb branch has a natural quantization denoted by $\mathcal{A}_h(G,N)$, which is a finitely generated left and right Noetherian domain, and hence admits a skew field of fractions, denoted by $\operatorname{Frac}$.

In the above paper, Remark 5.23, it is explained that $\operatorname{Frac}\mathcal{A}_h(G,N)$ is the same as the division ring of fractions of the $W$-invariants of the algebra of $h$-difference operators on $\mathfrak{t}$, where $\mathfrak{t}$ is the Lie algebra of a maximal torus $T$ of $G$ and $W$ is its Weyl group.

I'm having some problem in figuring out what exactly this Remark says. I've seen different algebras given the name of algebra of difference operators on an affine space. So I would like to know exactly what algebra we are talking about in this case. Having setled this first question, I would like to know exactly how $W$ acts by algebra automorphisms on this algebra.

This topic is somewhat distant from my usual research, so I am aware that the answers to my questions are problably obvious for the specialists. Hence, I apologize in advance if my questions are too trivial.

Important Edit It seems that the previous question boils down to the following. Choose a maximal torus $T \subset G$, with associated Weyl group $W$. Let $N_T$ be the restriction of the $G$ representation $N$ to a $T$ representation.

What algebra is $\mathcal{A}_h(T,N_T)$, and how the Weyl group $W$ acts on it?

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The algebra of difference operators is very explicitly described at the beginning of the appendix (section A(i)) of Coulomb branches of 3d N=4 quiver gauge theories and slices in the affine Grassmannian. It is an algebra with generated by polynomial functions on $ \mathbf t$, and translations by elements of the root lattice.

The action of $ W $ comes from its natural action on $ \mathbf t $.

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  • $\begingroup$ Thank you very much Joel $\endgroup$
    – jg1896
    Commented Nov 26, 2023 at 16:19

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