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

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