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We construct vertex algebraic intertwining operators among certaingeneralized Verma modules for $\widehat{\mathfrak{sl}(2,\mathbb{C})}$ andcalculate the corresponding fusion rules. Additionally, we show that under someconditions these intertwining operators descend to intertwining operators amongone generalized Verma module and two (generally non-standard) irreduciblemodules. Our construction relies on the irreducibility of the maximal propersubmodules of generalized Verma modules appearing in the Garland-Lepowskyresolutions of standard $\widehat{\mathfrak{sl}(2,\mathbb{C})}$-modules. Weprove this irreducibility using the composition factor multiplicities ofirreducible modules in Verma modules for symmetrizable Kac-Moody Lie algebrasof rank $2$, given by Rocha-Caridi and Wallach.

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