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

Abstract

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” 𝖯_max and the “minimal one” 𝖯_min. Furthermore, the parabolic BGG category arising from 𝖯_max 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.