Invariants of Weyl Group Action and q-characters of Quantum Affine Algebras

Let W be theWeyl group corresponding to a finite dimensional simple Lie algebra g of rank l and let m > 1 be an integer. In Inoue (Lett. Math. Phys. 111(1):32, 2021), by applying cluster mutations, a W-action on Y-m was constructed. Here Y-m is the rational function field on cm(l) commuting variables, where c is an element of {1, 2, 3} depends on g. This was motivated by the q-character map chi(q) of the category of finite dimensional representations of quantum affine algebra Uq ((g) over cap). We showed in Inoue (Lett. Math. Phys. 111(1):32, 2021) that when q is a root of unity, Im.q is a subring of the W-invariant subfield Y-m(W) of Y-m. In this paper, we give more detailed study on Y-m(W); for each reflection r(i) is an element of W associated to the ith simple root, we describe the ri-invariant subfield Y-m(ri) of Y-m.