Representations With a Reduced Null Cone

TL;DR

This paper investigates whether the null cone associated with a complex reductive group action on a module is reduced, providing complete results for specific groups and irreducible modules.

Contribution

The paper offers a comprehensive analysis of the reducedness of null cones for various groups and modules, including new classifications for SL_2, SL_3, and adjoint types.

Findings

Null cone is reduced for SL_2, SL_3, and simple adjoint groups.

Complete characterization of reduced null cones for irreducible modules of semisimple adjoint groups.

Identification of cases where the null cone is not reduced.

Abstract

Let G be a complex reductive group and V a G-module. Let \pi: V \to V//G be the quotient morphism and set N(V) = \pi^{-1}(\pi(0)). We consider the following question. Is the null cone N(V) reduced, i.e., is the ideal of N(V) generated by G-invariant polynomials? We have complete results when G is SL_2, SL_3 or a simple group of adjoint type, and also when G is semisimple of adjoint type and the G-module V is irreducible.