Irreducibility of G-varieties defined by quadrics

TL;DR

This paper proves that the zero locus of quadrics containing the orbit of a vector under a complex semisimple Lie group is always an irreducible variety, revealing fundamental geometric properties of these orbit closures.

Contribution

It establishes the irreducibility of the zero locus of quadrics containing G-orbits in projective space, a result not previously known in this generality.

Findings

Zero locus of quadrics containing G.y is irreducible.

Provides new insights into the geometry of G-orbits and their defining equations.

Enhances understanding of orbit closures in representation theory.

Abstract

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $G$ a simply connected and connected Lie group with Lie algebra $\mathfrak{g}$ and $V$ a finite dimensional representation. We prove that the zero locus of quadrics containing $G.y$ is an irreducible variety in $\PP V$.