Classical Whittaker modules for the affine Kac-Moody algebras AN(1)

Inspired by Sugawara operators, we introduce quasi Sugawara operators to construct several important operators on the universal non-degenerate Whittaker module of level κ over the affine Kac-Moody algebra of type $A_N^{\left(1\right)}$. As a result, we classify simple non-degenerate Whittaker modules for the affine algebras $\widehat{{\mathrm{sl}}_{N+1}}$ and $\widetilde{{\mathrm{sl}}_{N+1}}$ whether at the noncritical or critical level. In addition, we also give an explicit description on the structure of arbitrary non-degenerate Whittaker modules over these algebras. In particular, we recover the results on the classification of simple non-degenerate Whittaker $\widehat{{\mathrm{sl}}_2}$-modules ($\widetilde{{\mathrm{sl}}_2}$-modules) obtained by Adamović, Lü and Zhao.