On Gelfand-Zetlin modules over orthogonal Lie algebras (Q2762294)

Let \(\mathfrak g=\mathfrak o(n,\mathbb C)\) and \(\mathcal U=\mathcal U(\mathfrak g)\) the universal enveloping algebra of \(\mathfrak g\). The Gelfand-Zetlin (GZ) subalgebra \(\Gamma\) of \(\mathcal U\) is the (commutative) subalgebra generated by the centers of the universal enveloping algebras of the subalgebras \(\mathfrak o(m,\mathbb C)\) of \(\mathfrak g\) (\(1\leq m\leq n\)), which can be thought of as subalgebras of \(\mathcal U\) via inclusion in the upper left hand corner. A \(\mathcal U\) module \(V\) is called a GZ module if for each \(\lambda\in\Gamma^*\), the generalized eigenspace \(V^{\lambda}\) corresponding to \(\lambda\) is finite-dimensional, and if \(V\) is the direct sum of the \(V^{\lambda}\). Each \(\lambda\) can be parametrized by a double indexed family in \(\mathbb C^{\frac{n(n+1)}2}\) called a tableau. NEWLINENEWLINENEWLINEThe author uses the GZ formulae for the orthogonal groups [\textit{I. M. Gelfand} and \textit{M. L. Zetlin}, Dokl. Akad. Nauk SSSR 71, 1017-1020 (1950; Zbl 0037.15302)] to construct a large family of (infinite-dimensional) simple GZ modules over \(\mathfrak g\) and studies their structure. Then he constructs two families of GZ modules having ``tableaux realization'', i.e., with one-dimensional eigenspaces \(V^{\lambda}\) and the tableaux corresponding to the weight spaces satisfying a certain condition he calls `good'. For this construction, he uses simplified GZ formulae. Again, he describes the structure of these modules. Finally, he constructs the orthogonal operator algebras associated with GZ formulae.