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

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 ${\mathfrak{p}}_{{}_M}(k)\le {\mathfrak{p}}_{M,M_0}(k)$ 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 $M_0$.