On jet bundles and generalized Verma modules II

TL;DR

This paper studies the structure and dimension of filtrations of generalized Verma modules associated with parabolic subgroups of SL(E), using geometric and algebraic tools like jet bundles and annihilator ideals.

Contribution

It provides a basis and dimension formula for the filtration U^k(g) of generalized Verma modules in the case of parabolic subgroups fixing a flag, extending previous work.

Findings

Explicit basis for U^k(g) constructed.

Dimension formulas for the filtrations derived.

Geometric interpretation via jet bundles established.

Abstract

Let G be a semi simple linear algebraic group over a field of characteristic zero and let V be a finite dimensional irreducible G-module with highest weight vector v. Let P in G be the parabolic subgroup fixing v and let g=Lie(G). We get a canonical filtration of V by P-modules U^k(g)v where U^k(g) is the filtration of the universal enveloping algebra U(g). This filtration was in a previous paper studied in the case where P in G=SL(E) is the subgroup fixing an m-dimensional subspace. The aim of this paper is to use higher direct images of G-linearized sheaves, filtrations of generalized Verma modules and annihilator ideals of highest weight vectors to give a basis for U^k(g) and to compute its dimension in the case where P in SL(E) is the parabolic group fixing a flag in E. We also interpret the filtration U^k(g) in terms of SL(E)-linearized jet bundles on SL(E)/P.