Hilbert Polynomials of Noncanonical Orthogonal Oscillator Representations of $sl(n)$

TL;DR

This paper investigates the Hilbert polynomials of associated varieties of certain infinite-dimensional irreducible representations of sl(n), derived from Fourier-transformed oscillator representations, revealing their structure and growth properties.

Contribution

It computes the Hilbert polynomial of these associated varieties and establishes conditions for equality with the polynomial of the module, including explicit leading term determination.

Findings

Hilbert polynomial of associated varieties is explicitly computed.

Conditions for equality of Hilbert polynomials are established.

Leading term of the Hilbert polynomial is explicitly determined.

Abstract

By applying Fourier transformations to the natural orthogonal oscillator representations of special linear Lie algebras, Luo and the second author (2013) obtained a large family of infinite-dimensional irreducible representations of the algebras on the homogeneous solutions of the Laplace equation. In our earlier work, we proved that the associated varieties of these irreducible representations are the intersection of determinantal varieties. In this paper, we find the Hilbert polynomial $\mathfrak p_{ M}(k)$ of these associated varieties. Moreover, we show that the Hilbert polynomial $\mathfrak p_{ M, {M_0}}(k)$ of such an irreducible module $M$ with respect to any generating subspace $M_0$ satisfies $\mathfrak p_{_M}(k)\leq \mathfrak p_{ M, {M_0}}(k)$ for sufficiently large positive integer $k$ and find a necessary and sufficient condition that the equality holds. Furthermore, we explicitly determine the leading term of $\mathfrak p_{ M, {M_0}}(k)$, which is independent of the choice of $M_0$.