Gelfand-Kirillov dimensions of the ℤ-graded oscillator representations of 𝔬(n,ℂ) and 𝔰𝔭(2n,ℂ)

ABSTRACT

In this paper, we give a method to compute the Gelfand–Kirillov dimensions of some polynomial–type weight modules. These modules are infinite-dimensional irreducible 𝔬(n,ℂ)-modules and 𝔰𝔭(2n,ℂ)-modules that appeared in the ℤ-graded oscillator generalizations of the classical theorem on harmonic polynomials established by Luo and Xu. We also found that some of these modules have the secondly minimal GK-dimension, and some of them have the larger GK-dimension than the maximal GK-dimension apearing in unitary highest-weight modules.

KEYWORDS:

• Gelfand–Kirillov dimension
• oscillator representation
• weight module

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 17B10
• Secondary: 22E47

Acknowledgements

We would like to thank the referee for the comments on an earlier version of this paper.