Branching Law for the Finite Subgroups of SL4C

In the framework of McKay correspondence we determine, for every finite subgroup Γ of SL4C, how the finite dimensional irreducible representations of SL4C decompose under the action of Γ. Let h be a Cartan subalgebra of sl4C and let ̟1, ̟2, ̟3 be the corresponding fundamental weights. For (p, q, r) ∈ N, the restriction πp,q,r|Γ of the irreducible representation πp,q,r of highest weight p̟1 + q̟2 + r̟3 of SL4C decomposes as πp,q,r|Γ = ⊕l i=0 mi(p, q, r)γi. We determine the multiplicities mi(p, q, r) and prove that the series PΓ(t, u, w)i = ∑ ∞ p=0 ∑ ∞ q=0 ∑ ∞ r=0 mi(p, q, r)t uw are rational functions. This generalizes results from Kostant for SL2C and our preceding works about SL3C.