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

Abstract

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 $\mathcal {Y}_m$.