2015.12.10 On the order of the component group of a reductive group Yuichiro Taguchi For an algebraic group G over a field, we denote by Go the identity component of G. Theorem. Let n ≥ 1 be an integer. Then there exists an integer J = J(n) ≥ 1 depending only on n such that, for any field F of charac- teristic 0 and any split reductive subgroup G of GLn over F , its group of connected components G=Go has an abelian normal subgroup M of index ≤ J that is the image of a finite abelian subgroup M of G. As in [6], we use: Lemma. There exists an integer N = N(n) ≥ 1 depending only on n o such that, for any split torus Z in GLn, if H denotes its centralizer in o GLn, there exists an immersive morphism H=Z ,! GLN of algebraic groups over F . o Proof. If Z is a split torus in GLn, its centralizer H in GLn is a con- nected split reductive group (cf. [1], Exp. XIX, x1.6) of rank n. Indeed, An o if V = F is the affine n-space over F on which GLn (and hence Z ) acts ⊕ canonically and V = χ2X(Zo)Vχ is the weight-spaceQ decomposition of o V as a representation of Z , then we have H = χ2X(Zo) GL(Vχ). The quotient H := H=Zo is also a connected split reductive group. Let W be the Weyl group of H with respect to a maximal torus T of H. The representation theory of the split connected reductive group H may be identified with the \W -symmetric part" of that of T (e.g. [4], x3.6, Th. 4). Thus any finite subset S of the character group X(T ) of T which contains a basis of X(T ) and is stable under the action of W gives rise to a faithful representation of H of dimension jSj (i.e. immersive morphism H ! GLjSj of algebraic groups). One can take such an S of cardinality at most rankZX(T ) · jW j, which is bounded by an integer N depending only on n. □ Proof of the Theorem. Let Z be the center of the identity component Go of G, and Zo its identity component. Then Zo is a torus in Go and o G=Go ' G=G with G := G=Zo o Note that G = Go=Zo is semisimple ([5], Prop. 7.3.1, (i)). Let G act o on G by conjugation x 7! gxg−1. Then we have a morphism o (∗) G ! Aut(G ) 1 2 of algebraic groups over F , where o o ad Aut(G ) = (G ) o A0 o with the notation of [5], x16.3. Here, (G )ad is the adjoint group of o G ([5], x16.3.5), and A0 is a certain subgroup of the automorphism o group of the Dynkin diagram defined by a basis of a root system of G relative to a maximal torus of it (The group A0 may be thought of as o \Out(G )"). In particular, jA0j is bounded in terms of n. If C denotes o the centralizer of G in G, the morphism (∗) induces an exact sequence o 1 ! C=Z ! G=G ! A0 o o of algebraic groups over F , where Z = G \ C is the center of G o and it contains the identity component C of C. Note that Z is finite o since G is semisimple. Hence C is also a finite subgroup of G. Let o o H be the centralizer of Z in GLn and set H = H=Z . Then G is an algebraic subgroup of H. With N = N(n) as in the Lemma, H, and hence C, may be identified with a subgroup of GLN . By a theorem of Jordan ([3]), there exists an integer j depending only on N such that any finite subgroup of GLN has an abelian normal subgroup of index ≤ j. Let M be such a subgroup of C; thus jC=Mj ≤ j, and hence o G=Go ' G=G has an abelian normal subgroup M=(Z \ M) of index ≤ J(n) := j × jA0j. Being a finite abelian subgroup of the connected reductive group H in characteristic zero, C is contained in a torus T of H. Let T be the inverse images of T in H. Then T is also a torus. Since Zo and T are tori, the exact sequence 1 ! Zo ! T ! T ! 1 splits modulo isogeny, and hence there is a subtorus T 0 of T such that the projection T ! T induces a surjective morphism f : T 0 ! T with finite kernel. The inverse image M of M by f is in fact contained in G since M ⊂ G and G is the inverse image by the projection H ! H, and has the property claimed in the Theorem. □ Remark. Possible orders of the abelian subgroup M depend on the nature of the base field F . With the same notations as above, assume further that the group G(F ) of F -rational points of G is Zariski dense in G. Then each connected component has an F -rational point, and hence (the group of geometric points of) G may be identified with G(F ). Then G (resp. M) may be identified with a finite subgroup of GLN (F ) (resp. the F -rational points of a subtorus of GLN ). Thus j j ≤ N N ≤ 1 we have M µF , where µF is the maximum ( ) of the order × × · · · × × of the torsion subgroup of F1 Fr , where (F1;:::;Fr) varies throughP the set of r-tuples (r also varies) of finite extensions Fi of F r ≤ with i=1[Fi : F ] N. 3 References [1] M. Demazure and A. Grothendieck, Sch´emasen Groupes, III, Lect. Notes in Math. 153, Springer-Verlag, 1970 [2] S. Friedland, The maximal orders of finite subgroups in GLn(Q), Proc. A. M. S. 125 (1997), 3519{3526 [3] C. Jordan, M´emoire sur les ´equations differentielles lin´eaires ´a int´egrale alg´ebrique, J. f¨urdie reine und angew. Math. 84 (1878), 89{215 [4] J.-P. Serre, Groupes de Grothendieck des sch´emas en groupes r´eductifs d´eploy´es, Publ. Math. I.H.E.S. 34 (1968), 37{52 [5] T. A. Springer, Linear Algebraic Groups, Second ed., Progress in Math. 9, Birkh¨auserBoston, Inc., Boston, MA, 1998, xiv+334 pp. [6] Y. Taguchi, On potentially Abelian geometric representations, Ramanujan J. 7 (2003), 477{483.