Reflection maps

TL;DR

This paper studies reflection maps arising from reflection groups acting on complex vector spaces, providing tools to analyze their singularities, stability, and finiteness properties, and relating these to existing conjectures.

Contribution

It introduces methods to study the singularities of reflection maps, identifies obstructions to their stability, and constructs infinite families of finite map-germs with arbitrary corank.

Findings

Identified obstructions to -stability of reflection maps.

Constructed infinite families of -finite map-germs of any corank.

Connected reflection map properties to conjectures of L, Mond, and Ruas.

Abstract

Given a reflection group $G$ acting on a complex vector space $V$, a reflection map is the composition of an embedding $X \hookrightarrow V$ with the orbit map $V\to\mathbb C^p$ that maps a $G$-orbit to a point. Reflection maps can be very singular, but we give tools to study them easily. We find obstructions to $\mathcal A$-stability of reflection maps and produce, in the unobstructed cases, infinite families of $\mathcal A$-finite map-germs of any corank. We also relate them to conjectures of L\^e, Mond and Ruas.