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 g naturally arises, with the zero component g0 being a reductive Lie algebra. We first show that there are only two proper parabolic subalgebras containing Levi subalgebra g0: the “maximal one” Pmax and the “minimal one” P_min. Furthermore, the parabolic BGG category arising from Pmax essentially turns out to be a subcategory of the one arising from P_min. Such a priority of P_min in the sense of representation theory reduces the question to the study of the “minimal parabolic” BGG category O^min associated with 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 O^min. (2) We investigate indecomposable tilting and indecomposable projective modules in O^min, and compute their character formulas.