Reflection maps

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.