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The McKay correspondence, the Coxeter element and representation theory. (English) Zbl 0605.22010
Élie Cartan et les mathématiques d’aujourd’hui. The mathematical heritage of Élie Cartan, Semin. Lyon 1984, Astérisque, No.Hors Sér. 1985, 209-255 (1985).
[For the entire collection see Zbl 0573.00010.]
Let \(\Gamma\) be a nontrivial finite subgroup of \(SU(2)\) and let \({\hat \Gamma}=\{\gamma_ 0,\gamma_ 1,...,\gamma_{\ell})\) be the set of equivalence classes of irreducible finite-dimensional complex representations of \(\Gamma\). Let \(\gamma: \Gamma\to SU(2)\) be the given 2-dimensional representation, and let \(A(\Gamma)\) denote the \((\ell+1)\times (\ell+1)\) matrix whose (0,j) entry \(A(\Gamma)_{ij}\) is the multiplicity of \(\gamma_ i\) in \(\gamma_ j\otimes \gamma\). It was observed by McKay that corresponding to \(\Gamma\) there exists a complex simple Lie algebra \({\mathfrak g}=\mu(\Gamma)\) of rank \(\ell\) such that \(2- A(\Gamma)\) is the Cartan matrix \(C(\tilde{\mathfrak g})\) of the affine Kac- Moody Lie algebra \(\tilde{\mathfrak g}\) associates with \({\mathfrak g}\), and that the correspondence \(\Gamma \mapsto \mu (\Gamma)={\mathfrak g}\) sets up a bijection between the set of isomorphism classes of finite subgroups \(\Gamma\) of \(SU(2)\) and the set of all isomorphism classes of complex simple Lie algebras of types A, D or E.
(Explicitly, the correspondence \(\mu\) is as follows. Let \(C_ n,\Delta_ n,A_ n,S_ n\) denote respectively the cyclic group of order n, the dihedral group of order 2n, the alternating group on n letters and the symmetric group on n letters. Then every finite subgroup F of \(SO(3)\) is isomorphic to one of \(C_ n,\Delta_ n,A_ 4,S_ 4\) or \(A_ 5\). Now let \(SU(2)\to SO(3)\) be the usual double covering, and for any finite subgroup F of \(SO(3)\) let \(F^*\) denote its inverse image in \(SU(2)\), so that \(| F^*| =2| F|\). Then every finite subgroup \(\Gamma\) of \(SU(2)\) that is not cyclic of odd order is of the form \(F^*\) for some finite subgroup F of \(SO(3)\), and the groups \(C^*_ n,\Delta^*_ n,A^*_ 4,S^*_ 4,A^*_ 5\) correspond respectively (in the correspondence \(\mu)\) to the Lie algebras of types \(A_{2n-1},D_{n+2},E_ 6,E_ 7,E_ 8.)\)
The Cartan matrix \(C(\tilde{\mathfrak g})\) is relative to an ordered set of simple roots \(\alpha_ i\in \tilde{\mathfrak h}'\) \((0\leq i\leq \ell)\), where \({\mathfrak h}\subset \tilde{\mathfrak h}\) are respectively Cartan subalgebras of \({\mathfrak g}\subset \tilde{\mathfrak g}\). The indexing may be chosen so that \(\gamma_ 0\) is the trivial representation of \(\Gamma\) and \(\alpha_ 0\in \tilde{\mathfrak h}'\) is the added simple root corresponding to the negative of the highest root \(\psi\in {\mathfrak h}'\) of \({\mathfrak g}.\)
Let \(\pi_ n\) be the representation of \(SU(2)\) on the nth symmetric power \(S^ n({\mathbb{C}}^ 2)\) \((n\geq 0)\); these are a complete set of irreducible unitary representations of the compact group SU(2). Consider the restriction of \(\pi_ n\) to the finite subgroup \(\Gamma\): if \(\pi_ n| \Gamma =\sum^{\ell}_{i=0}m_ i \gamma_ i\), we associate with \(\pi_ n\) the element \(v_ n\in \tilde {\mathfrak h}'\) in the root lattice of \((\tilde{\mathfrak g},\tilde{\mathfrak h})\) defined by \(v_ n=\sum^{\ell}_{i=0}m_ i \alpha_ i\). We may then form the generating function \(P_{\Gamma}(t)=\sum^{\infty}_{n=0}v_ n t^ n\), with coefficients in \(\tilde{\mathfrak h}'\). The problem to which this paper is devoted is the determination of \(P_{\Gamma}(t)\) in terms of data derived from the root structure of \({\mathfrak g}=\mu(\Gamma)\). The author assumes throughout that the Coxeter number h of \({\mathfrak g}\) is even, or equivalently that \(\Gamma =F^*\) for some finite subgroup F of \(SO(3)\). Thus g is not of type \(A_{\ell}\) with \(\ell\) even. The Dynkin diagram of \({\mathfrak g}\) then has a special node \(i_*\), which is the branch point if \({\mathfrak g}\) is of type D or E, and the middle node if \({\mathfrak g}\) is of type \(A_{2n-1}.\)
The author’s main results can now be stated. Theorem 1: The series \(P_{\Gamma}(t)\) is of the form \(z(t)/(1-t^ a)(1-t^ b)\), where a and b are even integers satisfying \(2\leq a\leq b\leq h\), and \(z(t)=\sum^{h}_{i=0}z_ i t^ i\) with \(z_ i\in \tilde {\mathfrak h}'\). Theorem 2: We have \(a=2d\), where d is the coefficient of \(\alpha_{i_*}\) in the highest root \(\psi\) ; moreover \(a+b=h+2\) and \(ab=2| \Gamma |.\)
Next let W be the (finite) Weyl group of (\({\mathfrak g},{\mathfrak h})\). There is a Coxeter element \(\sigma\in W\) corresponding to \(\Pi =\{\alpha_ 1,...,\alpha_{\ell}\}\) such that \(\sigma =\tau_ 2\tau_ 1\), where \(\tau_ 2,\tau_ 1\in W\) have order \(\leq 2\) and correspond to a decomposition \(\Pi =\Pi_ 1\cup \Pi_ 2\) into orthogonal subsets. The order may be chosen so that \(\tau_ 2\psi =\psi\). For \(n\geq 1\) let \(\tau_ n\) denote \(\tau_ 1\) or \(\tau_ 2\) according as n is odd or even, and let \(\tau^{(n)}=\tau_ n\tau_{n-1}...\tau_ 1\). Theorem 3: We have \(z_ 0=z_ h=\alpha_ 0\). For \(1\leq n\leq h-1\), we have \(z_ n\in {\mathfrak h}'\) (not just \(\tilde{\mathfrak h}')\) and indeed \(z_ n=(\tau^{(n-1)}-\tau^{(n)})\psi\). Moreover \(z_ g=2\alpha_{i_*}\), where \(g=h/2\), and \(z_{g+k}=z_{g-k}\) for \(1\leq k\leq g.\)
The Poincaré series \(P_{\Gamma}(t)_ i\) for each individual representation \(\gamma_ i\) is obtained by considering only the ith coefficient of the vectors \(v_ n\). By Th. 1, we have \(P_{\Gamma}(t)_ i=z(t)_ i/(1-t^ a)(1-t^ b)\), where \(z(t)_ i\) is the coefficient of \(\alpha_ i\) in z(t). When \(i=0\) we have \(z(t)_ 0=1+t^ h\), and hence Theorem 4: The Poincaré series \(P_{\Gamma}(t)_ 0\) of the algebra of invariants \(S({\mathbb{C}}^ 2)^{\Gamma}\) is \((1+t^ h)/(1-t^ a)(1-t^ b)\). Next, when \(i=i_*\), the representation \(\gamma_{i_*}\) is an irreducible representation of \(\Gamma\) of maximum dimension, and we have Theorem \(5: z(t)_{i_*}=\sum^{d-1}_{j=0}(t^{g-2j}+t^{g+2j}).\)
To determine \(z(t)_ i\) for the remaining nodes i, let \(\Phi\) be the set of positive roots of (\({\mathfrak g},{\mathfrak h})\) not orthogonal to the highest root \(\psi\). Let \(\Phi_ i\) be the intersection of \(\Phi\) with the orbit of \(\alpha_ i\) under the subgroup of W generated by the Coxeter element \(\sigma\). Then \(\Phi\) is the disjoint union of the \(\Phi_ i\), \(1\leq i\leq \ell\), and \(card(\Phi_ i)\) is equal to \(2d_ i\) if \(i\neq i_*\), and to \(2d_ i-1\) if \(i=i_*\), where \(d_ i\) is the coefficient of \(\alpha_ i\) in \(\psi\) (and is the degree of the representation \(\gamma_ i)\). For each \(n\geq 1\), let \(\Pi_ n\) denote \(\Pi_ 1\) or \(\Pi_ 2\) according as n is odd or even. Then for each positive root \(\phi\) of (\({\mathfrak g},{\mathfrak h})\) there is a unique integer \(n=n(\phi)\) such that \(1\leq n\leq h\) and \(\phi \in (\tau^{(n-1)})^{-1} \Pi_ n\), and we have Theorem 6: if \(i\neq 0\) or \(i_*\) then \(z(t)_ i=\sum_{\phi \in \Phi_ i}t^{n(\phi)}\), and all the coefficients of \(z(t)_ i\) are 1 or 0. Furthermore the coefficients of \(t^{g-k}\) and \(t^{g+k}\) (where \(g=h/2)\) are equal for \(k=1,...,g\) and vanish for \(k=0\).
Reviewer: I.G.Macdonald

22E46 Semisimple Lie groups and their representations
17B20 Simple, semisimple, reductive (super)algebras
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
20E07 Subgroup theorems; subgroup growth