Braid group actions, Baxter polynomials, and affine quantum groups

It is a classical result in representation theory that the braid group $\mathscr{B}_\mathfrak{g}$ of a simple Lie algebra $\mathfrak{g}$ acts on any integrable representation of $\mathfrak{g}$ via triple products of exponentials in its Chevalley generators. In this article, we show that a modification of this construction induces an action of $\mathscr{B}_\mathfrak{g}$ on the commutative subalgebra $Y_\hbar^0(\mathfrak{g})\subset Y_\hbar(\mathfrak{g})$ of the Yangian by Hopf algebra automorphisms, which gives rise to a representation of the Hecke algebra of type $\mathfrak{g}$ on a flat deformation of the Cartan subalgebra $\mathfrak{h}[t]\subset \mathfrak{g}[t]$. By dualizing, we recover a representation of $\mathscr{B}_\mathfrak{g}$ constructed in the works of Y. Tan and V. Chari, which was used to obtain sufficient conditions for the cyclicity of any tensor product of irreducible representations of $Y_\hbar(\mathfrak{g})$ and the quantum loop algebra $U_q(L\mathfrak{g})$. We apply this dual action to prove that the cyclicity conditions from the work of Tan are identical to those obtained in the recent work of the third author and S. Gautam. Finally, we study the $U_q(L\mathfrak{g})$-counterpart of the braid group action on $Y_\hbar^0(\mathfrak{g})$, which arises from Lusztig's braid group operators and recovers the aforementioned $\mathscr{B}_\mathfrak{g}$-action defined by Chari.