Generalised Jantzen filtration of Lie superalgebras I

TL;DR

This paper introduces a generalized Jantzen filtration for Lie superalgebras, establishing its uniqueness, calculating decomposition numbers via Kazhdan-Lusztig polynomials, and explicitly determining its length based on atypicality.

Contribution

It extends the Jantzen filtration concept to Lie superalgebras, proving its uniqueness and linking it to Kazhdan-Lusztig polynomials, with explicit length formulas.

Findings

The generalized Jantzen filtration is the unique Loewy filtration for type I Lie superalgebras.

Decomposition numbers are determined by inverse Kazhdan-Lusztig polynomial coefficients.

The length of the filtration depends explicitly on the atypicality degree of the highest weight.

Abstract

A Jantzen type filtration for generalised Varma modules of Lie superalgebras is introduced. In the case of type I Lie superalgebras, it is shown that the generalised Jantzen filtration for any Kac module is the unique Loewy filtration, and the decomposition numbers of the layers of the filtration are determined by the coefficients of inverse Kazhdan-Lusztig polynomials. Furthermore, the length of the Jantzen filtration for any Kac module is determined explicitly in terms of the degree of atypicality of the highest weight. These results are applied to obtain a detailed description of the submodule lattices of Kac modules.