Scalar generalized Verma modules

TL;DR

This paper investigates scalar generalized Verma modules for SL(E), exploring their structure, simple quotients, and geometric properties of jet bundles, extending classical formulas to algebraically closed fields of characteristic zero.

Contribution

It generalizes Smoke's classical formula on jet bundles from complex analysis to algebraically closed fields of characteristic zero using algebraic and geometric methods.

Findings

Characterization of scalar generalized Verma modules and their simple quotients.

Extension of Smoke's jet bundle formula to new algebraic settings.

Use of annihilator ideals and geometric techniques in module analysis.

Abstract

In this paper we study the scalar generalized Verma module $M$ associated to a character of a parabolic subgroup of $\operatorname{SL}(E)$. Here $E$ is a finite dimensional vector space over an algebraically closed field $K$ of characteristic zero. The Verma module $M$ has a canonical simple quotient $L$ with a canonical filtration $F$. In the case when the quotient $L$ is finite dimensional we use left annihilator ideals in $U(\mathfrak{sl}(E))$ and geometric results on jet bundles to generalize to an algebraically closed field of characteristic zero a classical formula of W. Smoke on the structure of the jet bundle of a line bundle on an arbitrary quotient $\operatorname{SL}(E)/P$ where $P$ is a parabolic subgroup of $\operatorname{SL}(E)$. This formula was originally proved by Smoke in 1967 using analytic techniques.