Parabolic BGG categories and their block decomposition for Lie superalgebras of Cartan type

In this paper, we study the parabolic BGG categories for graded Lie superalgebras of Cartan type over the field of complex numbers. The gradation of such a Lie superalgebra $\mathfrak{g}$ naturally arises, with the zero component $\mathfrak{g}_{0}$ being a reductive Lie algebra. We first show that there are only two proper parabolic subalgebras containing Levi subalgebra $\mathfrak{g}_{0}$: the “maximal one” $\mathsf{P}_{\max}$ and the “minimal one” $\mathsf{P}_{\min}$. Furthermore, the parabolic BGG category arising from $\mathsf{P}_{\max}$ essentially turns out to be a subcategory of the one arising from $\mathsf{P}_{\min}$. Such a priority of $\mathsf{P}_{\min}$ in the sense of representation theory reduces the question to the study of the “minimal parabolic” BGG category $\mathcal{O}^{\min}$ associated with $\mathsf{P}_{\min}$. We prove the existence of projective covers of simple objects in these categories, which enables us to establish a satisfactory block theory. Most notably, our main results are as follows. (1) We classify and obtain a precise description of the blocks of $\mathcal{O}^{\min}$. (2) We investigate indecomposable tilting and indecomposable projective modules in $\mathcal{O}^{\min}$, and compute their character formulas.