Categories of induced modules for Lie algebras with triangular decomposition (Q2753158)

For a complex Lie algebra \(\mathfrak G\) with a triangular decomposition and for a fixed standard parabolic subalgebra \(\mathcal P\), a category \(\mathcal O(\mathcal P,\Lambda , H)\) analogous to the classical category \(\mathcal O\) is studied. Here \(\Lambda \) is a certain admissible category of \(\mathcal P\)-modules, and \(H\) is a finite subset of linear functions on the Cartan subalgebra of \(\mathfrak G\) generating weights of modules from \(\mathcal O(\mathcal P,\Lambda , H)\). For generalized Verma modules \(M(V)=U(\mathfrak G)\otimes _{U(\mathcal P)}V\), \(V\in \Lambda \) under certain limitations, an analogue of the BGG-reciprocity principle for such a category is proved. The decomposition of \(\mathcal O(\mathcal P,\Lambda , H)\) into blocks is constructed, and the relation to projectively stratified algebras is established. The induction from \(\mathcal P\) in the case when \(\mathcal P\) has a Levi factor \(sl(2, \mathbb C)\) is considered. For an indecomposable projective object \(V\in \Lambda \), the indecomposable tilting module \(T(V)\in \mathcal O(\mathcal P,\Lambda , H)\) is constructed. NEWLINENEWLINENEWLINEIn the concluding sections a category \(\mathcal O(\mathfrak B, \Lambda , H)\) of modules over an affine Lie algebra associated with a non-standard parabolic subalgebra \(\mathfrak B\) is studied, and the equivalence between \(\mathcal O(\mathfrak B, \Lambda , H)\) and a category \(\mathcal O(\mathcal P',\Lambda ', H')\) for a certain subalgebra in the case when the central element acts injectively on every object is established.