On jet bundles and generalized Verma modules II

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.