PBW filtration and bases for symplectic Lie algebras

TL;DR

This paper investigates the PBW filtration on symplectic Lie algebra representations, providing a detailed description of the associated graded modules, constructing bases, and deriving combinatorial character formulas.

Contribution

It offers a new explicit description of the graded modules, constructs bases, and derives combinatorial formulas for symplectic Lie algebra representations.

Findings

Explicit description of the associated graded module

Construction of new bases for modules

Graded combinatorial formula for characters

Abstract

We study the PBW filtration on the highest weight representations $V(\la)$ of $\msp_{2n}$. This filtration is induced by the standard degree filtration on $U(\n^-)$. We give a description of the associated graded $S(\n^-)$-module $gr V(\la)$ in terms of generators and relations. We also construct a basis of $gr V(\la)$. As an application we derive a graded combinatorial formula for the character of $V(\la)$ and obtain a new class of bases of the modules $V(\la)$.