Branching Law for the Finite Subgroups of $SL_4ℂ$ and the Related Generalized Poincaré Polynomials

Abstract

Within the framework of McKay correspondence, we determine, for every finite subgroup $Γ$ of $SL_4ℂ$, how the finite-dimensional irreducible representations of $SL_4ℂ$ decompose under the action of $Γ$. Let $\mathfrak{h}$ be a Cartan subalgebra of $\mathfrak{sl}_4ℂ$ and let $ϖ_1, ϖ_2$, and $ϖ_3$ be the corresponding fundamental weights. For $(p, q, r) ∈ ℕ^3$, the restriction $π_{p,q,r} | Γ$ of the irreducible representation $π_{p,q,r}$ of the highest weight $pϖ_1 + qϖ_2 + rϖ_3$ of $SL_4ℂ$ decomposes as $π_{p, q, r} | Γ  = ⊕_{i = 0}^l m_i (p, q, r)γ_i$, where $\{γ_0,…, γ_l\}$ is the set of equivalence classes of irreducible finite-dimensional complex representations of $Γ$. We determine the multiplicities $m_i (p, q, r)$ and prove that the series $${P}_{\varGamma }{\left(t,u,w\right)}_i={\displaystyle \sum_{p=0}^{\infty }{\displaystyle \sum_{q=0}^{\infty }{\displaystyle \sum_{r=0}^{\infty }{m}_i\left(p,q,r\right){t}^p{u}^q{w}^r}}}$$ are rational functions. This generalizes the results of Kostant for $SL_2ℂ$ and the results of our preceding works for $SL_3ℂ$.