J. Lond. Math. Soc. | 2019

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

 
 
 

Abstract


We introduce and study new categories T(g,k)of integrable sl(\\infty)-modules which depend on the choice of a certain reductive subalgebra k in g=sl(\\infty). The simple objects of these categories are tensor modules as in the previously studied category, however, the choice of k provides more flexibility of nonsimple modules. 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(\\infty)-action on the integral category O(m|n) 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), 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). We also conjecture that a natural tensor filtration on K(m|n) arises via the Duflo--Serganova functor sending the category O(m|n) to O(m-1|n-1). We prove this latter conjecture for a direct summand of K(m|n) corresponding to the finite-dimensional gl(m|n)-modules.

Volume 99
Pages 403-427
DOI 10.1112/jlms.12176
Language English
Journal J. Lond. Math. Soc.

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