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

TL;DR

This paper extends the parametrization of irreducible spin representations from Weyl groups to more general real reflection groups, linking them to special solvable points in the dual space.

Contribution

It generalizes Ciubotaru's parametrization to non-crystallographic root systems by introducing solvable points and establishing their connection to spin representations.

Findings

Defined solvable points in dual space for non-crystallographic root systems.

Established a correspondence between spin representations and solvable points.

Extended the parametrization framework beyond crystallographic cases.

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.