Spin representations of real reflection groups of non-crystallographic root systems

Abstract

A uniform parametrization for the irreducible spin representations of Weyl groups in terms of nilpotent orbits is recently achieved by Ciubotaru (2011). This paper is a generalization of this result to other real reflection groups. Let $(V_0, R, V_0^{\vee}, R^{\vee})$ be a root system with the real reflection group $W$. We define a special subset of points in $V_0^{\vee}$ which will be called solvable points. Those solvable points, in the case $R$ crystallographic, correspond to the nilpotent orbits whose elements have a solvable centralizer in the corresponding Lie algebra. Then a connection between the irreducible spin representations of $W$ and those solvable points in $V_0^{\vee}$ is established.