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 gl^M = gl^M(V,W) consisting of endomorphisms of V whose duals preserve W ⊆ V*. In their work, the category T_gl^M of gl^M-modules, which are finite length subquotients of the tensor algebra T(W ⊗ V), is singled out and studied. Denoting by T_{V ⊗ W} the category with the same objects as T_gl^M but regarded as V ⊗ W-modules, we first show that when W and V are paired by dual bases, the functor T_gl^M → T_V⊗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 T_End(V) have finite-length as gl^M-modules. Finally, under the same hypotheses, we compute the socle filtration of a simple object in T_End(V) as agl^M-module.