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The Mckay Correspondence, the Coxeter Element and Representation Theory Astérisque, Tome S131 (1985), P Astérisque BERTRAM KOSTANT The McKay correspondence, the Coxeter element and representation theory Astérisque, tome S131 (1985), p. 209-255 <http://www.numdam.org/item?id=AST_1985__S131__209_0> © Société mathématique de France, 1985, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Société Mathématique de France Astérisque, hors série, 1985, p. 209-255 THE McKAY CORRESPONDENCE, THE COXETER ELEMENT AND REPRESENTATION THEORY BY Bertram KOSTANT* 1.1. Introduction. — A fundamental question in harmonie analysis is : given a locally compact group £?, a closed subgroup H, and an irreducible representation n of G, how does the restriction ir\H decompose. Consider this question in the following very basic situation : G = SU(2) and H — T is any finite subgroup of SU(2). The classification of all finite subgroups T of SU(2) 2 2 is classical and well known. See e.g. [12] or [13]. Let S(C ) = n=0 (c ) 2 be the symmetric algebra over C and let nn be the representation of SU(2) n 2 2 on S (C ) induced by its action on C . One knows nn is irreducible for all n and the set of equivalence classes {TTn}, n = 0,..., defines the unitary dual of SU(2). Now let r C 517(2) be any finite group. The question then in this case is : how does rn| T decompose for any n E Z+. The question is particularly interesting in the light of the MCKAY correspondence. The latter sets up a bijection between T, the unitary dual of T, with the nodes of the extended Dynkin diagram of a simple Lie algebra g, of type A, D or E. Thus, the question becomes : what is the multiplicity of a particular node for any n ez+. The problem above has arisen recently in connection with the resolution of certain algebraic singularities. One way of dealing with it is by the use of the theory of complex reflection groups. The latter enables one to decompose polynomials into a sum of products of "invariants" and "harmonics". This is an old technique and its appropriateness in the present context was pointed out by P. SLODOWY. This approach has been carried out by G. GONZALEZ- SPRINBERG and J.-L. VERDIER in [2] and H. KNORRER in [5]. An effective * Supported in part by NSF grant MCS 8105633. 210 B. KOSTANT method for computing these multiplicities is achieved and consequently one obtains an explanation of the MCKAY correspondence. However the multiplicities are not significantly related to g itself. In this paper the whole matter is viewed in a completely different way. We accept the MCKAY correspondence; thereby setting up a bijection of all such r and all simple complex Lie algebras g = fi(T) of type A, D and E. The question we ask is : in what way does V and the multiplicities in rn |T "see" the structure of g and vice versa. What we show is that these multiplicities come in a beautiful way from the root structure of g. More explicitly they come from the orbit structure of the Coxeter element a on the set of roots of g. In fact the "harmonics" come from intersecting the orbits of a with the roots of a distinguished Heisenberg subalgebra n of g and the "invariants" come from the scalar product of those roots in that orbit of a which contains the highest root if). (The degree of the minimal "invariant" appears then as twice the coefficient of y at the branch point.) Besides making connection with 7rn|r, new results in the orbit structure of a are obtained. We remark that all the results are obtained using Lie theoretic principles and there is no reliance on empirical observations. Also, the decomposition itself into "invariants" times "harmonics" is seen in the root structure and there is no need for recourse to reflection group theory. 1.2. — Let r Ç SU(2) be a non-trivial finite subgroup of SU (2) and let {70,...,li} = f be the set of equivalence classes of irreducible finite dimensional complex representations of T. Then if 7 : r -+ SU(2) is the given 2-dimensional representation one defines an (Z + 1) X (Z + 1) matrix Air) with entries in Z+ by decomposing the tensor product 7? ® 7 = iA (r)fj7.- into irreducible components. John MCKAY has made the remarkable observa­ tion (see [6] and [7]) that there exists a complex simple Lie algebra g = Mr of rank Z such that A(T) = 2-C(%), where g is the affine Kac-Moody Lie algebra associated to g and C(g) is a Cart an matrix of g. Moreover the correspondence r - /X r = g sets up a bijection between the set of all isomorphism classes of such subgroups r C SU(2) and the set of all isomorphism classes of complex Lie algebras of type A, D and E. The Cartan matrix C(g) is with respect to an ordered set of simple roots ai G h', i = 0,... ,Z, where h Ç h are, respectively, Cartan subalgebras of g Q g. The indexing may be chosen so that 70 is the trivial representation THE McKAY CORRESPONDENCE 211 G is the added simple root corresponding to the negative of of r and a0 h' the highest root Y E h' of g. Now if i rn r = mai, i=0 we associate to rn |T the element E h' in the root lattice defined by vn putting I vn = irti oci. i'=0 The problem we set for ourselves is then the determination of the generating function Pr{t), with coefficients in h', defined by putting v tn. M*) = n 71 = 0 We restrict our attention to the cases where the Coxeter number h = 2g of g is even (so that g G Z+). This excludes only the case where T is a cyclic group of odd order. (The latter case is readily dealt with by considering first the cyclic group T X Z2 of even order where z2 = {±/}.) We can now speak of the special node i* of the Dynkin diagram of g. If g = Ai then i* is the branch point. If g ¥ A2m-1 (Aom has been excluded) then i* is the midpoint. Our main results will be stated in the following sequence of six theorems. Remark — In the light of Slodowy's observation (see § 1.1) the following result (THEOREM 1.3) is not new. However, the product decomposition in THEOREM 1.3 arises not from reflection group theory but from a study of the Coxeter element. As a consequence the numbers involved are directly related to the root structure of g = fi[T). But then if one proceeds to make the connection with the reflection group theory one obtains as a theorem (not just an observation) that the numbers of reflecting hyperplanes is given in terms of the Coxeter number and the lesser degree of the two fundamental invariants is given in terms of the highest coefficient of the maximal root of g. ; THEOREM 1.3. There exist zl• G h , % = 0,..., h, and even integers 2 < a < b < h such that h x Z{t (1.3.1) Pr(t) = i=0 (1 -la){\ -tb) 212 B. KOSTANT The integers a and b are determined by the next result. The table of these values is given in THEOREM 5.17. THEOREM 1.4. One has a — 2d where d — d{. is the coefficient of the highest root ip corresponding to the special node i* (1 for Ah 2 for Dh 3 for Eq, 4 for Ej and 6 for Eg). Furthermore b is given by (1.4.1) b = h + 2 - a. In addition one has (1.4.2) ab = 2 ITI. If W is the (finite) Weyl group of (h,g) then one knows that there is a Coxeter element a G W corresponding to II = {ai,...,a/} such that o = r2t 1 where r2, t1 G W have order at most 2 and correspond to a decomposition n = ili un2 into orthogonal sets. The order may be chosen so that r2ip = rp. For n G Z_i_ let rn — Ti if n is odd and Tn = r2 if n is even. (n) Also let r(") be the alternating decreasing product T = l~n1~n-l • • • 7*1. Let c = 6/2. The vectors Z{ are determined in THEOREM 1.5. For n = 1,..., h — 1, One has = Oi0. one ¿o = zh f mm zneh (not just h' ) and zn is given by (n-l) (n) Y (1.5.1) zn = T W — T where ip is the highest root of (h,g). Furthermore (1.5.2) Zg = 2a{. and in general one has the symmetry (1.5.3) Zg + k — Zg-k for fc = l,...,flf. Finally if there exists distinct G n = l,....ûf-l OL{1 , . , Q¿ir ny,yG{l,2}, where j = n mod 2 snc/i that (1.5.4) 2n = ai1 H h a;tr. Moreover 1 IF1< n < d - 1 : (1.5.5.) r = 2 if d < n < e — 1; 3 •if e < n < g — 1; THE McKAY CORRESPONDENCE 213 and in fact r — 1,2, or 3 according as to whether (r (n) Y, r (n-1) Y) is positive.
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