The Coxeter element and the branching law for the finite subgroups of SU(2)

TL;DR

This paper investigates the branching laws of finite subgroups of SU(2), explicitly determining certain polynomials related to representation multiplicities using Coxeter elements and the McKay correspondence, revealing connections to Lie algebra roots and Lusztig's work.

Contribution

It explicitly computes the polynomial z(t)_j for nontrivial representations of finite SU(2) subgroups using Coxeter orbits, linking representation theory, Lie algebras, and Lusztig's polynomials.

Findings

Explicit formulas for z(t)_j using Coxeter orbits

Connection between z(t)_j and Lusztig's polynomials

Identification of z(t)_j with characters of representations

Abstract

Let $\Gamma$ be a finite subgroup of SU(2) and let $\widetilde {\Gamma} = \{\gamma_i\mid i\in J\}$ be the unitary dual of $\Gamma$. The unitary dual of SU(2) may be written $\{\pi_n\mid n\in \Bbb Z_+\}$ where $dim \pi_n = n+1$. For $n\in \Bbb Z_+$ and $j\in J$ let $m_{n,j}$ be the multiplicity of $\gamma_j$ in $\pi_n|\Gamma$. Then we collect this branching data in the formal power series, $m(t)_j = \sum_{n=0}^{\infty}m_{n,j} t^n$. One shows that there exists a polynomial $z(t)_j$ and known positive integers $a,b$ (independent of $j$) such that $m(t)_j = {z(t)_j \over (1-t^a)(1-t^b)}$. The problem is the determination of the polynomial $z(t)_j$. If $o\in J$ is such that $\gamma_o$ is the trivial representation, then it is classical that $z(t)_o = 1 +t^h$ for a known integer $h$. The problem reduces to case where $\gamma_j$ is nontrivial. The McKay correspondence associates to $\Gamma$ a complex simple Lie algebra $\g$ of type A-D-E. We explicitly determine $z(t)_j$ for $j\in J-\{o\}$ using the orbits of a Coxeter element on the set of roots of $\frak{g}$. Mysteriously the polynomial $z(t)_j$ has arisen in a completely different context in some papers of Lusztig. Also Rossmann has recently shown that the polynomial $z(t)_j$ yields the character of $\gamma_j$.