Vertex algebraic intertwining operators among generalized Verma modules for $\widehat{\mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$

We construct vertex algebraic intertwining operators among certain generalized Verma modules for $\widehat{\mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$ and calculate the corresponding fusion rules. Additionally, we show that under some conditions these intertwining operators descend to intertwining operators among one generalized Verma module and two (generally non-standard) irreducible modules. Our construction relies on the irreducibility of the maximal proper submodules of generalized Verma modules appearing in the Garland-Lepowsky resolutions of standard $\widehat{\mathfrak{s}\mathfrak{l}(2,\mathbb{C})}$-modules. We prove this irreducibility using the composition factor multiplicities of irreducible modules in Verma modules for symmetrizable Kac-Moody Lie algebras of rank $2$, given by Rocha-Caridi and Wallach.