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

Abstract

Let $\Gamma$ be a finite subgroup of SU(2) and let $\tilde{\Gamma }=\{{\gamma }_i\mid i\in J\}$ be the unitary dual of $\Gamma$. The unitary dual of SU(2) may be written $\{{\pi }_n\mid n\in {\mathbb{Z}}_{+}\}$ where $dim{\pi }_n=n+1$. For $n\in {\mathbb{Z}}_{+}$ 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 $m(t{)}_j=\frac{z(t{)}_j}{(1-t^a)(1-t^b)}$. 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 $\mathfrak{g}$. 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 ${\gamma }_j$.

13 pages, plain.tex