# On jet bundles and generalized Verma modules II

Helge {\O}ystein Maakestad

Arxiv ID: 0903.3291•Last updated: 11/13/2020

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.

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