Integrable sl(∞)-modules and category O for gl(m|n)

We introduce and study new categories T _g,k of integrable g = sl(∞)-modules which depend on the choice of a certain reductive in g subalgebra k ⊂ g. The simple objects of T _g,k are tensor modules as in the previously studied category T _g [Dan-Cohen, Penkov and Serganova, Adv. Math. 289 (2016) 250–278]; however, the choice of k provides for more flexibility of nonsimple modules in T _g,k compared to T _g . We then choose k to have two infinite-dimensional diagonal blocks, and show that a certain injective object K _m|n in T _g,k realizes a categorical sl(∞)-action on the category O _m|n ^Z , the integral category O of the Lie superalgebra gl(m|n). We show that the socle of K _m|n is generated by the projective modules in O _m|n ^Z , and compute the socle filtration of K _m|n explicitly. We conjecture that the socle filtration of K _m|n reflects a ‘degree of atypicality filtration’ on the category O _m|n ^Z . We also conjecture that a natural tensor filtration on K _m|n arises via the Duflo–Serganova functor sending the category O _m|n ^Z to O _m-1|n-1 ^Z . We prove a weaker version of this latter conjecture for the direct summand of K _m|n corresponding to finite-dimensional gl(m|n)-modules.