Principal $\widehat{sl}(3)$ subspaces and quantum Toda Hamiltonian

We study a class of representations of the Lie algebra of Laurent polynomials with values in the nilpotent subalgebra of 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 $U_{v}sl(3)$. We also obtain a fermionic formula for an eigenfunction of the sl(3) quantum Toda Hamiltonian.