Classical Whittaker modules for the affine Kac-Moody algebras \(A_N^{(1)}\)
Let \(V_{\widehat{sl_{N+1}}}(\mathbf{\lambda}, \kappa)\) denotes the universal non-degenerate Whittaker module at level \(\kappa\) over affine Lie algebra \(\widehat{sl_{N+1}}\), (\(N\geq 2\)), where \(\mathbf{\lambda}\) denotes a \((N+1)\)-tuple of non-zero complex numbers.\N\NIn the work [Adv. Math. 289, 438--479 (2016 Zbl: 1369.17018)], \textit{D. Adamović} et al. determined the structure of classical Whittaker modules for the affine Kac-Moody algebra \(\widetilde{sl_{2}}\) and its derived algebra \(\widehat{sl_{2}}\) at arbitrary level. In the paper under review, the authors generalize these results to the case of Whittaker \(\widetilde{sl_{N+1}}\)-modules and \(\widehat{sl_{N+1}}\)-modules. In particular, they prove that at the noncritical level \(V_{\widehat{sl_{N+1}}}(\mathbf{\lambda}, \kappa)\) is irreducible. The main part in the classification of simple Whittaker modules at the critical level is the construction of a basis of \(V_{\widehat{sl_{N+1}}}(\mathbf{\lambda}, \kappa)\) at the noncritical level, which relies on Sugawara operators on the vacuum module (affine vertex algebra). They also give an explicit description on the structure of arbitrary non-degenerate Whittaker modules over these affine Lie algebras.
scientific article; zbMATH DE number 7900777
scientific article