Rings which are modules in the Bernstein-Gelfand-Gelfand \(\mathcal O\) category (Q1103037)

Let \(\mathfrak g\) be a complex semisimple Lie algebra. Let \(\mathcal O\) be the BGG-category of \(\mathfrak g\)-modules. The author describes, in this article, the structure of rings in the category \(\mathcal O\). Let \(\mathfrak g=\mathfrak n+\mathfrak h+\mathfrak n^-\) be a triangular decomposition of \(\mathfrak g\), \(\mathfrak b=\mathfrak h\oplus \mathfrak n\), \(R\) is the set of roots of the pair (\(\mathfrak g,\mathfrak h)\), \(R^+\) the subsystem of positive roots, defined by \(\mathfrak n\), \(\rho\) is the half sum of roots from \(R^+\). Let \(B\) be the basis of \(R^+\), \(B'\subset B\) any subset, \(\mu_{B'}\supset \mathfrak b\) the parabolic subalgebra \(\mathfrak g\), defined by \(B'\). Let \(V_{B'}(\mu)\) be the finite-dimensional irreducible \(\mu_{B'}\)-module with the highest weight \(\mu-\rho\). Set \[ M_{B'}(\mu)=\mathcal U(\mathfrak g)\otimes_{\mathcal U(\mu_{B'})}V_{B'}(\mu) \] the generalized Verma module. Denote by \(\delta M\) the dual of \(M\) in the category \(\mathcal O\). The author proves, that any ring \(A\) in the category \(\mathcal O\) such that \(A^n\) is completely prime and \(\dim (A^n)=1\), then \(A\cong \delta M_{B'}(\rho)\) for some B'\(\subset B\). The main result of the article is the description of semiprime \(\mathcal O\)-rings: every such ring is the finite direct sum of prime \(\mathcal O\)-rings, and every prime \(\mathcal O\)-ring is of the form \(\operatorname{End} E\otimes \delta M_{B'}(\rho)\) for some \(B'\subset B\) and some finite-dimensional \(\mu_{B'}\)-module \(E\) in which \(m_{B'}\) (the nil radical of \(\mu_{B'})\) acts trivially.