Principal $\widehat{\mathfrak{sl}_3}$ subspaces and quantum Toda Hamiltonian

Abstract

We study a class of representations of the Lie algebra $\mathfrak{n} \otimes \mathbb{C} [t, t^{-1}]$, where $\mathfrak{n}$ is a nilpotent subalgebra of $\mathfrak{sl}_3$. We derive Weyl-type (bosonic) character formulas for these representations. We establish a connection between the bosonic formulas and the Whittaker vector in the Verma module for the quantum group $U_v (\mathfrak{sl}_3)$. We also obtain a fermionic formula for an eigenfunction of the $\mathfrak{sl}_3$ quantum Toda Hamiltonian.