Gelfand-Tsetlin modules for \(\mathfrak{gl}(m|n)\) (Q2166374)

Let the underlying field be the set of complex numbers. Let \(\mathfrak{g}\) be the Lie superalgebra \(\mathfrak{gl}(m|n)\). A chain \(\mathfrak{g}=\mathfrak{g}^{1}\supset \mathfrak{g}^{2}\supset \ldots \supset \mathfrak{g}^{m+n}\) of subalgebras of \(\mathfrak{g}\) is such that \(\mathfrak{g}^k\) is isomorphic to \(\mathfrak{gl}(p|q)\) with \(p+q=m+n-k+1 \), which forms a complete flag in \(\mathfrak{g}\) if \(\mathfrak h^k:=\mathfrak h\cap\mathfrak{g}^k\) is a Cartan subalgebra of \(\mathfrak{g}^k\). A complete flag induces a chain of Cartan subalgebras \(\mathfrak h=\mathfrak h^{1}\supset \mathfrak h^{2}\supset \ldots \supset \mathfrak h^{m+n}\). Every complete flag \(\mathcal C\) in \(\mathfrak{g}\) defines the commutative subalgebra \(\Gamma_{\mathcal C}\) in the universal enveloping algebra \(U(\mathfrak{g})\) generated by the centers of the members of the chain. Here, \(\Gamma_{\mathcal C}\) is the Gelfand-Tsetlin subalgebra of \(U(\mathfrak{g})\) associated with the flag \(\mathcal{C}\). Letting \(\Gamma=\Gamma_{\mathcal C}\) be a Gelfand-Tsetlin subalgebra of \(U(\mathfrak{g})\), a finitely generated module \(M\) over \(\mathfrak{g}\) is called a Gelfand-Tsetlin module with respect to \(\Gamma\) if \(M\) decomposes as the direct sum of \(M(\textbf{m})\) as a \(\Gamma\)-module, with \(\textbf{m}\) \(\in\) \(\text{Specm}(\Gamma)\), where \(M(\textbf{m}) = \{ x\in M : \textbf{m}^k x = 0 \text{ for some } k\geq 0\}\) and \(\text{Specm} (\Gamma)\) is the set of maximal ideals of \(\Gamma\). V. Futorny, V. Serganova, and J. Zhang address the problem of classifying irreducible Gelfand-Tsetlin modules for \(\mathfrak{g}\) and show that it reduces to the classification of Gelfand-Tsetlin modules for the even part. They also give an explicit tableaux construction and the irreducibility criterion for the class of quasi typical and quasi covariant Gelfand-Tsetlin modules, which includes all essentially typical and covariant tensor finite-dimensional modules. In the quasi typical case, new irreducible representations are infinite-dimensional \(\mathfrak{g}\)-modules which are isomorphic to the parabolically induced Kac modules.