Irreducibility conjecture for generalized Whittaker modules of \widehat{\mathfrak{sl}}_2

Let $M_1(\bm{\lambda},\bm{\mu})$ and $N_1(\bm{\eta})$ be the Weyl and Heisenberg modules described in the source, with $\mu_M\ne0$, $\lambda_N\ne0$, and $\eta\ne0$. Let $\psi$ be a Whittaker function satisfying the condition defining the Whittaker function associated with the vector $w=v_{\bm{\lambda},\bm{\mu}}\otimes v_{\bm{\eta}}$, and let $\widehat{R}(\psi)$ denote the corresponding generalized Whittaker module for $V^{\kappa}(\mathfrak{sl}_2)$. Generalized Whittaker irreducibility conjecture. If $k\ne-2$, then $\widehat{R}(\psi)$ is an irreducible $V^{\kappa}(\mathfrak{sl}_2)$-module and

$$\widehat{R}(\psi)\cong M_1(\bm{\lambda},\bm{\mu})\otimes N_1(\bm{\eta}).$$

The preceding calculation shows that the tensor-product generator is a Whittaker vector and identifies its Whittaker function. The conjecture asserts both irreducibility and the expected identification of the resulting generalized Whittaker module away from the critical level $k=-2$.