Orthogonal Oscillator Representations, Laplace Equations and Intersections of Determinantal Varieties

TL;DR

This paper links representation theory, PDEs, and algebraic geometry by showing that associated varieties of certain Lie algebra representations are intersections of determinantal varieties, derived via Fourier transforms of oscillator representations.

Contribution

It explicitly characterizes associated varieties of infinite-dimensional irreducible representations as intersections of determinantal varieties, connecting multiple mathematical disciplines.

Findings

Associated varieties are intersections of determinantal varieties.

Fourier transforms relate oscillator representations to PDE solutions.

Provides explicit geometric descriptions of representation-theoretic objects.

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.