On the canonical filtration of an irreducible representation

Helge {\O}ystein Maakestad
Arxiv ID: 1003.3522Last 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 $\mathfrak{sl}(V)$.

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