Orthogonal Oscillator Representations, Laplace Equations and Intersections of Determinantal Varieties

TL;DR

This paper identifies the associated varieties of a broad family of infinite-dimensional irreducible $sl(n)$-modules ${\mathscr H}_{\langle\ell_1,\ell_2\rangle}$, arising from orthogonal oscillator representations and spaces of Laplace equation solutions, as explicit intersections of determinantal varieties. The authors develop a concrete filtration via a degree function ${\mathfrak d}$ and a transformation $T$ to obtain a spanning description, then prove linear independence of key products of alternating polynomials modulo determinantal relations. They compute the annihilator of the associated graded module to derive exact determinantal descriptions of the associated varieties, including special cases such as the Serge variety and standard determinantal varieties, and provide GK-dimension equalities. The work bridges representation theory, partial differential equations, and algebraic geometry, giving explicit geometric realizations of primitive ideals and their varieties for this class of $sl(n)$-modules.

Abstract

Associated varieties are geometric objects appearing in infinite-dimensional representations of semisimple Lie algebras (groups). By applying Fourier transformations to the natural orthogonal oscillator representations of special linear Lie algebras, Luo and the second author (2013) obtained a big family of infinite-dimensional irreducible representations of the algebras on certain spaces of homogeneous solutions of the Laplace equation. In this paper, we prove that the associated varieties of these irreducible representations are the intersections of explicitly given determinantal varieties. This provides an explicit connection among representation theory, partial differential equations and algebraic geometry.

Figures (3)

• Figure 1: The case $i_1=i_1'$
• Figure 2: The cases $i_1=i_2'$ and $i_1=i_3'$
• Figure 3: If $y_{t_{\ell_3}}\not|f_{\iota}$

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Corollary 2.1
• proof
• Proposition 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof