On the canonical filtration of an irreducible representation

Abstract

The aim of this paper is to study the canonical filtration $L(\lambda {)}_l$ of an irreducible finite dimensional $\mathrm{SL}(V)$-module $L(\lambda )$ using the universal enveloping algebra $U(\mathfrak{sl}(V))$ and the annihilator ideal $ann(v)$ of a highest weight vector $v$ in $L(\lambda )$. We give a basis for $L(\lambda {)}_l$ and calculate the dimension of $L(\lambda {)}_l$ as a function of $l$. This is done in terms of the universal enveloping algebra of the nilpotent radical of an opposite parabolic sub algebra of the stabilizer Lie algebra of a flag $V_{\ast }$ in $V$ with respect to a choice of roots for $\mathfrak{sl}(V)$.

arXiv commentThis is a new and improved version of the preprint "On the annihilator ideal of a highest weight vector"