Gelfand-Kirillov Dimensions of the Z-graded Oscillator Representations of $\mathfrak{o}(n,\mathbb{C})$ and $\mathfrak{sp}(2n,\mathbb{C})$

TL;DR

This paper calculates the Gelfand-Kirillov dimensions of certain infinite-dimensional modules related to orthogonal and symplectic Lie algebras, revealing new insights into their size and structure compared to known modules.

Contribution

It provides explicit Gelfand-Kirillov dimension computations for oscillator modules of $rak{o}(n,bC)$ and $rak{sp}(2n,bC)$, including identifying modules with minimal and larger GK-dimensions.

Findings

Some modules have the second minimal GK-dimension.

Some modules have GK-dimension larger than the maximal for unitary highest-weight modules.

Explicit GK-dimension values are computed for these modules.

Abstract

In this paper, we compute the Gelfand-Kirillov dimensions for the infinite-dimensional explicit irreducible $\mathfrak{o}(n, \mathbb{C})$-modules and $\mathfrak{sp}(2n, \mathbb{C})$-modules that appeared in the $\mathbb{Z}$-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. We also found that some of the modules have the secondly minimal GK-dimension, and some of them have the larger GK-dimension than the maximal GK-dimension of unitary highest-weight modules.