The Coxeter element and the branching law pdf

The Coxeter element and the branching law

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Let $Gamma$ be a finite subgroup of SU(2) and let $widetilde {Gamma} = {gamma_imid iin J}$ be the unitary dual of $Gamma$. The unitary dual of SU(2) may be written ${pi_nmid nin Bbb Z_+}$ where $dim pi_n = n+1$. For $nin Bbb Z_+$ and $jin J$ 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 $oin J$ 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 $jin J-{o}$ 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

• Creator/s: Bertram Kostant
• Date: 11/7/2004

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