Jantzen Conjecture for singular characters

TL;DR

This paper proves that the Jantzen filtration matches the radical filtration for singular Verma modules, confirming the Jantzen Conjecture for all infinitesimal characters and extending previous results to parabolic Verma modules.

Contribution

It establishes the equivalence of Jantzen and radical filtrations for singular and parabolic Verma modules, confirming the conjecture in broader cases.

Findings

Jantzen filtration coincides with radical filtration for singular Verma modules.

The Jantzen Conjecture holds for all infinitesimal characters.

Extension of results to parabolic Verma modules.

Abstract

We show that the Jantzen filtration of a Verma module (possibly singular) coincides with its radical filtration. It implies that the Jantzen Conjecture on Verma modules holds for all infinitesimal characters, while the regular case was settled by Beilinson and Bernstein by geometric methods and reproved by Williamson by an algebraic approach. The coincidence between Jantzen filtration and radical filtration is also generalized to the case of parabolic Verma modules by Shan's results.