Branching Law for the Finite Subgroups of SL(4,C)

Abstract

In the framework of McKay correspondence we determine, for every finite subgroup $\Gamma$ of $\mathbf{SL}_4\mathbb{C}$, how the finite dimensional irreducible representations of $\mathbf{SL}_4\mathbb{C}$ decompose under the action of $\Gamma$. Let $\go{h}$ be a Cartan subalgebra of $\go{sl}_4\mathbb{C}$ and let $\varpi_1,\,\varpi_2,\,\varpi_3$ be the corresponding fundamental weights. For $(p,q,r)\in \mathbb{N}^3$, the restriction $\pi_{p,q,r}|_\Gamma$ of the irreducible representation $\pi_{p,q,r}$ of highest weight $p\varpi_1+q\varpi_2+r\varpi_3$ of $\mathbf{SL}_4\mathbb{C}$ decomposes as ${\pi_{p,q,r}}|_{\Gamma}=\bigoplus_{i=0}^l m_i(p,q,r)\gamma_i.$ We determine the multiplicities $m_i(p,q,r)$ and prove that the series $P_\Gamma(t,u,w)_i=\sum_{p=0}^\infty\sum_{q=0}^\infty\sum_{r=0}^\infty m_i(p,q,r)t^pu^qw^r$ are rational functions. This generalizes results from Kostant for $\mathbf{SL}_2\mathbb{C}$ and our preceding works about $\mathbf{SL}_3\mathbb{C}$.