Three results on representations of Mackey Lie algebras

I. Penkov and V. Serganova have recently introduced, for any non-degenerate pairing $W\otimes V\to \mathbb{C}$ of vector spaces, the Lie algebra ${\mathfrak{g}\mathfrak{l}}^M={\mathfrak{g}\mathfrak{l}}^M(V,W)$ consisting of endomorphisms of $V$ whose duals preserve $W\subseteq V^{\ast }$. In their work, the category ${\mathbb{T}}_{{\mathfrak{g}\mathfrak{l}}^M}$ of ${\mathfrak{g}\mathfrak{l}}^M$-modules which are finite length subquotients of the tensor algebra $T(W\otimes V)$ is singled out and studied. In this note we solve three problems posed by these authors concerning the categories ${\mathbb{T}}_{{\mathfrak{g}\mathfrak{l}}^M}$. Denoting by ${\mathbb{T}}_{V\otimes W}$ the category with the same objects as ${\mathbb{T}}_{{\mathfrak{g}\mathfrak{l}}^M}$ but regarded as $V\otimes W$-modules, we first show that when $W$ and $V$ are paired by dual bases, the functor ${\mathbb{T}}_{{\mathfrak{g}\mathfrak{l}}^M}\to {\mathbb{T}}_{V\otimes W}$ taking a module to its largest weight submodule with respect to a sufficiently nice Cartan subalgebra of $V\otimes W$ is a tensor equivalence. Secondly, we prove that when $W$ and $V$ are countable-dimensional, the objects of ${\mathbb{T}}_{\mathrm{E}\mathrm{n}\mathrm{d}(V)}$ have finite length as ${\mathfrak{g}\mathfrak{l}}^M$-modules. Finally, under the same hypotheses, we compute the socle filtration of a simple object in ${\mathbb{T}}_{\mathrm{E}\mathrm{n}\mathrm{d}(V)}$ as a ${\mathfrak{g}\mathfrak{l}}^M$-module.