Quantum affine Cartan matrices , Poincaré series of binary polyhedral groups , and reflection representations Ruedi Suter

One of the first results in Lie representation theory is that the symmetric powers S(C2) (for n ∈ Z>0) of the standard representation C of SU(2) are representatives for the list of equivalence classes of the irreducible complex representations of SU(2). Take a finite subgroup Γ ⊆ SU(2) and let i be an irreducible complex representation of Γ. It is a basic question to ask what is the multiplicity of i in the restriction of S(C2) to Γ. This question has been addressed and answered by Kostant [Ko1] in a beautiful way. A crucial ingredient in his approach is a Coxeter transformation caff of the affine Weyl group associated to Γ via the McKay correspondence [McK1, McK2, St]. Kostant writes caff as a product of two involutions r1 and r2 where r1 and r2 themselves are products of commuting simple reflections. This is only possible if the affine Coxeter-Dynkin diagram has no odd cycle. Hence type A2n must be omitted in this approach. In a somewhat different context Springer [Sp1] reproved Kostant’s results in his paper in the Mathematische Annalen volume dedicated to Hirzebruch on his sixtieth birthday. See also [Sp2]. Earlier papers by Gonzalez-Sprinberg and Verdier [G-SVe] as well as by Knörrer [Kn] also deal with the question stated above (but their main goal was something else, namely, the McKay correspondence and singularity theory). They used that Γ is an index 2 subgroup of a complex reflection group. As is often the case in well developed and elementary subjects, one can hardly avoid rediscovering previously known results. In the first few sections I shall show how one can derive some invariant theoretic results very easily and quickly by using quantum affine Cartan matrices. In this largely expository part I tried to avoid too much overlap with