Three results on representations of mackey lie algebras

I. Penkov and V. Serganova have recently introduced, for any nondegenerate pairing W ⊗ V → C of vector spaces, the Lie algebra glM = glM(V,W) consisting of endomorphisms of V whose duals preserve W ⊆ V ∗. In their work, the category T glM of glM-modules, which are finite length subquotients of the tensor algebra T (W V), is singled out and studied. Denoting by TV⊗W the category with the same objects as T glM but regarded as V ⊗ W-modules, we first show that when W and V are paired by dual bases, the functor T glM → TV⊗W taking a module to its largest weight submodule with respect to a sufficiently nice Cartan subalgebra of V ⊗W is a tensor equivalence. Secondly, we prove that when W and V are countable-dimensional, the objects of TEnd(V) have finite-length as glM-modules. Finally, under the same hypotheses, we compute the socle filtration of a simple object in TEnd(V) as a glM-module.