# On the canonical filtration of an irreducible representation

Helge {\O}ystein Maakestad

Arxiv ID: 1003.3522•Last updated: 11/13/2020

The aim of this paper is to study the canonical filtration $L(\lambda)_l$ of
an irreducible finite dimensional $\operatorname{SL}(V)$-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 $V_*$ in $V$ with respect to a choice of roots for
$\mathfrak{sl}(V)$.

#### PaperStudio AI Chat

I'm your research assistant! Ask me anything about this paper.