The degree of the Hilbert-Poincar\'e polynomial of PBW-graded modules

TL;DR

This paper investigates the degrees of Hilbert-Poincaré polynomials for PBW-graded simple modules of complex Lie algebras, providing explicit calculations for fundamental modules.

Contribution

It explicitly computes the degrees of Hilbert-Poincaré polynomials for PBW-graded modules, reducing the problem to fundamental modules.

Findings

Degrees are explicitly calculated for fundamental modules.

Reduction of the problem to modules with fundamental highest weight.

Provides a method for computing degrees of PBW-graded modules.

Abstract

In this note, we study the Hilbert-Poincar\'e polynomials for the PBW-graded of simple modules for a simple complex Lie algebra. The computation of their degree can be reduced to modules of fundamental highest weight. We provide these degrees explicitly.