Projective modules in the category ${\scr O}_ S$: Loewy series

Abstract:

Let $\mathfrak {g}$ be a complex, semisimple Lie algebra with a parabolic subalgebra ${\mathfrak {p}_S}$. The Loewy lengths and Loewy series of generalized Verma modules and of their projective covers in ${\mathcal {O}_S}$ are studied with primary emphasis on the case in which ${\mathfrak {p}_S}$ is a Borel subalgebra and ${\mathcal {O}_S}$ is the category $\mathcal {O}$. An examination of the change in Loewy length of modules under translation leads to the calculation of Loewy length for Verma modules and for self-dual projectives in $\mathcal {O}$, assuming the Kazhdan-Lusztig conjecture (in an equivalent formulation due to Vogan). In turn, it is shown that the Loewy length results imply Vogan’s statement, and lead to the determination of Loewy length for the self-dual projectives and certain generalized Verma modules in ${\mathcal {O}_S}$. Under the stronger assumption of Jantzen’s conjecture, the radical and socle series are computed for self-dual projectives in $\mathcal {O}$. An analogous result is formulated for self-dual projectives in ${\mathcal {O}_S}$ and proved in certain cases.