On the canonical filtration of an irreducible representation

TL;DR

This paper investigates the structure of the canonical filtration of irreducible finite-dimensional SL(V)-modules, providing bases and dimension formulas using universal enveloping algebras and annihilator ideals.

Contribution

It introduces a basis for the canonical filtration layers and computes their dimensions via the universal enveloping algebra of specific Lie algebra substructures.

Findings

Provided explicit bases for the filtration layers.

Calculated the dimensions of filtration components.

Linked the filtration structure to the universal enveloping algebra of a nilpotent radical.

Abstract

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 $\mathfrak{sl}(V)$.