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

## TL;DR

This paper studies the structure of parabolic BGG categories for graded Lie superalgebras of Cartan type, classifies their blocks, and analyzes indecomposable modules and character formulas.

## Contribution

It classifies blocks of minimal parabolic BGG categories and analyzes indecomposable modules, providing new insights into their representation theory.

## Key findings

- Classification and description of blocks of the minimal parabolic BGG category
- Existence of projective covers for simple objects in these categories
- Character formulas for indecomposable tilting and projective modules

## Abstract

In this paper, we study the parabolic BGG categories for graded Lie superalgebras of Cartan type over complex numbers. The gradation of such a Lie superalgebra $\ggg$ naturally arises, with the zero component $\ggg_0$ being a reductive Lie algebra. We first show that there are only two proper parabolic subalgebras containing Levi subalgebra $\ggg_0$: the ``maximal one" $\sfp_\max$ and the ``minimal one" $\sfp_\min$. Furthermore, the parabolic BGG category arising from $\sfp_\max$, essentially turns out to be a subcategory of the one arising from $\sfp_\min$. Such a priority of $\sfp_\min$ in the sense of representation theory reduces the question to the study of the ``minimal parabolic" BGG category $\comi$ associated with $\sfp_\min$. 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 $\comi$. (2) We investigate indecomposable tilting and indecomposable projective modules in $\comi$, and compute their character formulas.