On jet bundles and generalized Verma modules

TL;DR

This paper explores the structure of jet bundles on Grassmannians using Lie theory and Verma modules, revealing their module structure and applications to the irreducibility of discriminants.

Contribution

It introduces a novel approach to studying jet bundles on Grassmannians via generalized Verma modules and Lie algebra techniques, providing explicit module descriptions.

Findings

Calculated the P-module of the dual jet bundle and identified it with the canonical filtration

Proved the irreducibility of the discriminant of any linear system on Grassmannians

Established connections between jet bundles, Verma modules, and geometric properties

Abstract

The aim of this paper is to initiate a study of the jet bundles on the grassmannian $X$ over a field of characteristic zero using higher direct images of $G$-linearized sheaves, Lie theoretic methods, enveloping algebra theoretic methods and generalized Verma modules. We calculate the $P$-module of the dual jet bundle $J^l(L)^*$ and prove it equals the $l$'th piece of the canonical filtration for $H^0(X,L)^*$. We use the results obtained to prove the discriminant of any linear system on any grassmannian is irreducible.