Hilbert Polynomials of Noncanonical Orthogonal Oscillator Representations of $sl(n)$
Hengjia Zhang, Xiaoping Xu
TL;DR
This work computes and relates Hilbert invariants for associated varieties of infinite-dimensional irreducible $sl(n)$-modules arising from noncanonical orthogonal oscillator representations. By leveraging a Fourier-transformed, $\ obreak{\mathbb Z}^2$-graded model of the representation, the authors derive explicit Hilbert series and polynomials for the coordinate rings of the associated determinantal varieties, establishing that the arithmetic genus is $1$ and detailing degree formulas across several index-weight regimes. They prove a fundamental inequality ${\mathfrak p}_{M}(k)\le {\mathfrak p}_{M,M_0}(k)$ for large $k$ and give a sharp, necessary-and-sufficient condition for equality, namely that the generating subspace $M_0$ be the one-dimensional subspace $V_0$ (uniquely determined up to the stated cases). Furthermore, they show the leading coefficient of ${\mathfrak p}_{M,M_0}$ is intrinsic to $M$, independent of the choice of $M_0$, and provide explicit formulas for this leading term in several parameter regimes, clarifying how the representation-theoretic data $(n_1,n_2,\ell_1,\ell_2)$ influence the algebraic geometry of the associated varieties. These results deepen the link between noncanonical oscillator realizations of $sl(n)$ and determinantal geometry, with invariant information captured by the leading Hilbert-term across filtrations and generating subspaces.
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 $M_0$.