ON THE CHIEF FACTORS OF PARABOLIC MAXIMAL SUBGROUPS IN FINITE SIMPLE GROUPS OF NORMAL LIE TYPE

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1 Siberian Mathematical Journal, Vol. 55, No. 4, pp , 2014 Original Russian Text Copyright c 2014 Korableva V.V. ON THE CHIEF FACTORS OF PARABOLIC MAXIMAL SUBGROUPS IN FINITE SIMPLE GROUPS OF NORMAL LIE TYPE V. V. Korableva UDC Abstract: For every finite group of normal Lie type, we specify the description of the parabolic factors of a parabolic maximal subgroup occurring in the unipotent radical. For every module, we determine the highest weight and dimension. The results are presented in the tables where these parameters are given. DOI: /S Keywords: finite group of Lie type, parabolic subgroup, chief factor Introduction One of the fundamental problems of group theory is the study of the subgroup structure of a given group. In the postclassification theory of finite groups, investigating their subgroups and representations has become an actual problem. The important class of permutation representation of finite groups of Lie type is given by its parabolic representations; i.e., the representations on the cosets modulo parabolic subgroups. In our previous works, we obtained a description for the primitive parabolic permutation representations of all groups of (normal and twisted) Lie type. This article continues the study of the properties of these representations; namely, the chief factors are studied of parabolic maximal subgroups in finite simple groups of normal Lie type. Let G be a finite simple group of normal Lie type (a Chevalley group) over a finite field K of characteristic p and let P = UL be a parabolic maximal subgroup in G, where U is the unipotent radical and L is the Levi complement in P. Suppose that p 2 for the groups of type B l, C l, F 4 and p>3 for the groups of type G 2. Then the results of [1] imply that the factors of the lower central series of U are chief factors of P and irreducible KL-modules. The number of these factors is independent of the field K but depends on the Lie type of G. In the exceptional cases, the commutator relations influencing the structure of unipotent subgroups behave in a special way and require special consideration. In this article the author specifies the description of the factors of parabolic maximal subgroups P occurring in the unipotent radical under the above-indicated conditions. These chief factors are irreducible KL-modules. For every such module, we find the highest weight and dimension. 1. Notation and Auxiliary Results The modular representations are an important tool in the study of the subgroup structure of finite groups. The irreducible representations of finite groups of Lie type can be parametrized by highest weights as it was done in the theory of representations of complex semisimple Lie algebras (for example, see [2 5]). We will use the definitions and notation connected with the groups of Lie type from [6]. Let Φ be arootsystemforagroupg, let π be the set of simple roots in Φ, and let Φ + be the corresponding set of positive roots. It is known that G = x (t) t K, Φ, and for each root Φ, the root subgroup X = {x (t) t K} is isomorphic to the field K. Given a G-module M and a character χ of a Cartan The author was supported by the Russian Foundation for Basic Research (Grant ) and the Laboratory of Quantum Topology of Chelyabinsk State University (Grant 14.Z of the Government of the Russian Federation). Chelyabinsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 4, pp , July August, Original article submitted February 4, /14/ c 2014 Springer Science+Business Media, Inc.

2 subgroup H in G, the nonzero space M χ = {v M h v = χ(h)v for all h H} is called a weight space and χ is called a weight of the G-module M (or the weight of the corresponding representation). A G-module M can be decomposed into a direct sum of weight spaces. There exists a one-dimensional subspace in M invariant under the action of a Borel subgroup in G. A vector v generating such a space is called a maximal vector for M. This is equivalent to the fact that v 0 belongs to some weight space M χ and is fixed under the action of all root subgroups X with positive roots. For an irreducible G-module M, the weight χ is called its highest weight. Given a subset J in π, denotebyφ J the set of roots in Φ spanned by J and put Φ + J =Φ+ Φ J. Let P be the standard parabolic subgroup in G corresponding to the root system Φ J. A subgroup P is known to admit the Levi decomposition: P = UL, where U = X Φ + \ Φ + J is the unipotent radical and L = H, X Φ J is the Levi complement in P. Denote the jth term of the lower central series of U by U (j), where j 1andU = U (1). Given Φ +, write = J + J, where J = α J c αα and J = α π\j d αα (0 c α,d α Z). The number ht() = α J c α + α π\j d α is called the height of the root. Following [1], call the number level() = α π\j d α the level of the root and refer to shape() = J as the shape of. Given j 1, put U j = X Φ +, level() j. The Chevalley commutator formula [6, Theorem 5.2.2] implies that every subgroup U j is normal in P and the quotient group U j /U j+1 is isomorphic to X, where the product is taken in some fixed order over all positive roots for which level() =j. For all roots Φ + \ Φ + J with level() =j and shape() =S, put ( V S = X )U j+1 /U j+1. Then write shape()=s level()=j U j /U j+1 = V S, where S ranges over the set of different shapes of level j in Φ + \ Φ + J. For each j 1, the group L acts by conjugation on U j /U j+1. If Φ + \ Φ J and level() =j then put cx (t)u j+1 = x (ct)u j+1 for c, t K. Thus, U j /U j+1 becomes a KL-module. The Chevalley commutator formula easily yields Lemma 1. If S is the shape of the roots of level j in Φ + \ Φ + J then (1) the vector space V S over the field K is isomorphic to the external direct sum of root subgroups X for which shape() =S and level() =j, (2) V S is normalized by L and is a KL-submodule in U j /U j+1. If A and B are normal subgroups in P, while B is a subgroup in A, and the quotient group A/B is a minimal normal subgroup in P/B, then A/B is called a chief factor of P. The following result is essential in finding the chief factors of a parabolic subgroup and composing the corresponding tables. Lemma 2 [1, Theorem 2; 4, Theorem 17.6]. Suppose that G is a finite simple group of normal Lie type over a finite field K of characteristic p, p 2, for the groups of type B l, C l, F 4 and p>3 for the groups of type G 2 ; while P = UL is the parabolic subgroup in G corresponding to a root system Φ J with unipotent radical U, and Levi complement L in P ; the subgroups U j and U (j) for j 1 are as above. Then (1) U j = U (j) for each j 1; (2) for every shape S, among all roots Φ + \ Φ + J for which shape() =S, there exists a unique root S of maximal height, and the module V S is an irreducible KL-module of highest weight S ; (3) if K > 5, S and S are different shapes then V S and V S are nonequivalent KL-modules; 623

3 (4) for each j 1, the module U j /U j+1 is a direct sum of irreducible modules V S, where the sum is taken in arbitrary order over all different shapes S of roots of level j in Φ + \ Φ + J. In particular, U j/u j+1 is a completely reducible KL-module; (5) V S is a chief factor for P. Put π = {p 1,...,p l }, k {1,...,l}, J =Π\{p k }, P J = P k and Φ J =Φ k. Then, for = l i=1 c ip i Φ + and j 1, we have level() =c k,shape() =c k p k, U j = X c k j = c k j X,and U j /U j+1 = V jpk = X. 2. The Exceptional Groups We will provide the tables of chief factors for a parabolic maximal subgroup in an exceptional group of normal Lie type. In the first column of the table for each parabolic maximal subgroup P k in G, we indicate all chief factors V S occurring in its unipotent radical. The module V S is isomorphic (in the additive notation) to a direct sum of root subgroups X, which are one-dimensional weight spaces of weight. In the second column, we write down all such weights. In the third column, we write down the highest weight S of the irreducible KL-module V S ; and, finally, in the last column, the dimension of V S. Denote an arbitrary root l i=1 c ip i inφbyc 1 c 2...c l. The unit subgroup and the integer unity will be designated as 1. Let K = GF (q), where q is a power of a prime p. For the group G 2 (q) with the Dynkin diagram c k =j the set of positive roots consists of the elements 10, 01, 11, 21, 31, 32. Up to conjugacy, G 2 (q) has two parabolic maximal subgroups. Write down the Levi decomposition for each of them: P 1 = X 10,X 11,X 21,X 31,X 32 H, X ±01, P 2 = X 01,X 11,X 21,X 31,X 32 H, X ±10. The chief factors in P 1 are given by V 1p1 = U 1 /U 2 = X10 X 11, V 2p1 = U 2 /U 3 = X21,andV 3p1 = U 3 /1 = X 31 X 32. Therefore, xu 2 is a maximal vector for the irreducible KL-module U 1 /U 2 for any nonunit element x X p1 +p 2. The element yu 3 is a maximal vector for the irreducible KL-module U 2 /U 3 for every nonunit element y X 2p1 +p 2. Similarly, for P 2 we get V 1p2 = U 1 /U 2 = X01 X 11 X 21 X 31, and V 2p2 = U 2 /1 = X 32. Put the so-obtained results in Table 1. Table 1. G = G 2 (q), p>3. V S = X S dim V S V 1p1 10, V 2p V 3p1 31, V 1p2 01, 11, 21, V 2p For the group F 4 (q) with the Dynkin diagram 624

4 the set of positive roots Φ + consists of the elements 1000, 0100, 0010, 0001, 1100, 0110, 0011, 1110, 0120, 0111, 1120, 1111, 0121, 1220, 1121, 0122, 1221, 1122, 1231, 1222, 1232, 1242, 1342, Up to conjugacy, F 4 (q) has four parabolic maximal subgroups P k = H, X Φ + Φ k,1 k 4. For each of these subgroups, put down in the table the chief factors, the weights of one-dimensional spaces contained in the corresponding irreducible module, the highest weight of the module, and its dimension (see Table 2). V S Table 2. G = F 4 (q), p 2. Roots with V S = X S dim V S V 1p1 1000, 1100, 1110, 1120, 1111, 1220, 1121, 1221, 1122, 1231, 1222, 1232, 1242, V 2p V 1p2 0100, 1100, 0110, 1110, 0120, 0111, 1120, 1111, 0121, 1121, 0122, V 2p2 1220, 1221, 1231, 1222, 1232, V 3p2 1342, V 1p3 0010, 0110, 0011, 1110, 0111, V 2p3 0120, 1120, 0121, 1220, 1121, 0122, 1221, 1122, V 3p3 1231, V 4p3 1242, 1342, V 1p4 0001, 0011, 0111, 1111, 0121, 1121, 1221, V 2p4 0122, 1122, 1222, 1232, 1242, 1342, In each of the groups E l (q), where l is equal to 6, 7, or 8, there are l parabolic maximal subgroups P k = H, X Φ + Φ k,1 k l. For these groups, compose the tables similar to those for G 2 (q) and F 4 (q) (see Tables 3 5). For E 6 (q) with the Dynkin diagram the set of positive roots Φ + consists of the elements , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

5 V S Table 3. G = E 6 (q). Roots with V S = X S dim V S V 1p , , , , , , , , , , , , , , , V 1p , , , , , , , , , , , , , , , , , , , V 2p V 1p , , , , , , , , , , , , , , , , , , , V 2p , , , , V 1p , , , , , , , , , , , , , , , , , V 2p , , , , , , , , V 3p , V 1p , , , , , , , , , , , , , , , , , , , V 2p , , , , V 1p , , , , , , , , , , , , , , , For E 7 (q) with the Dynkin diagram the set of positive roots Φ + consists of the elements , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

6 V S Table 4. G = E 7 (q). Roots with V S = X S dim V S V 1p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , V 2p V 1p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , V 2p , , , , , , V 1p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , V 2p , , , , , , , , , , , , , , V 3p , V 1p , , , , , , , , , , , , , , , , , , , , , , , V 2p , , , , , , , , , , , , , , , , , V 3p , , , , , , V 4p , , V 1p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , V 2p , , , , , , , , , , , , , ,

7 V S Table 4. G = E 7 (q) (continued). Roots with V S = X S dim V S V 3p , , , , V 1p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , V 2p , , , , , , , , , V 1p , , , , , , , , , , , , , , , , , , , , , , , , , , For E 8 (q) with the Dynkin diagram the set of positive roots Φ + of type E 8 consists of the elements , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

8 V S Table 5. G = E 8 (q). Roots with V S = X S dim V S V 1p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , V 2p , , , , , , , , , , , , , V 1p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , V 2p , , , , , , , , , , , , , , , , , , , , , , , , , , , V 3p , , , , , , , V 1p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

9 V S Table 5. G = E 8 (q) (continued). Roots with V S = X S dim V S V 2p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , V 3p , , , , , , , , , , , , , V 4p , , , , , , V 1p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , V 2p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , V 3p , , , , , , , , , , , , , , , , , , , V 4p , , , , , , , , , , , , , , V 5p , , , , , V 6p , , , , V 1p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

10 V S Table 5. G = E 8 (q) (continued). Roots with V S = X S dim V S V 2p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , V 3p , , , , , , , , , , , , , , , , , , , V 4p , , , , , , , , , V 5p , , , V 1p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , V 2p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , V 3p , , , , , , , , , , , , , , , V 4p , , V 1p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,

11 V S Table 5. G = E 8 (q) (continued). Roots with V S = X S dim V S V 2p , , , , , , , , , , , , , , , , , , , , , , , , , , V 3p , V 1p , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , V 2p Table 6. Positive Roots for Classical Groups. G Φ + A l (q) p 1,p 12,...,p 1l,p 2,p 23,...,p 2l,...,p l 1,p l 1l,p l B l (q) p 1,p 12,...,p 1l,p 2,p 23,...,p 2l,...,p l 1,p l 1l,p l ; p 1 +2p 2l,p 12 +2p 3l,...,p 1l 1 +2p l,p 2 +2p 3l,p 23 +2p 4l,...,p 2l 1 +2p l,..., p l 2 +2p l 1l,p l 2l 1 +2p l,p l 1 +2p l C l (q) p 1,p 12,...,p 1l,p 2,p 23,...,p 2l,...,p l 1,p l 1l,p l ; p 1 +2p 2l 1 + p l, p 12 +2p 3l 1 + p l,...,p 1l 2 +2p l 1 + p l,p 2 +2p 3l 1 + p l, p 23 +2p 4l 1 + p l,...,p 2l 2 +2p l 1 + p l,...,p l 2 +2p l 1 + p l ; 2p 1l 1 + p l, 2p 2l 1 + p l,...,2p l 2l 1 + p l, 2p l 1 + p l D l (q) p 1,p 12,...,p 1l,p 2,p 23,...,p 2l,...,p l 2,p l 2l 1,p l 2l,p l 1 ; p 1 +2p 2l 2 + p l 1l,p 12 +2p 3l 2 + p l 1l,...,p 1l 3 +2p l 2 + p l 1l, p 2 +2p 3l 2 + p l 1l,p 23 +2p 4l 2 + p l 1l,...,p 2l 3 +2p l 2 + p l 1l,..., p l 3 +2p l 2 + p l 1l ; p 1l 2 + p l,p 2l 2 + p l,...,p l 3l 2 + p l,p l 2 + p l,p l 3. The Classical Groups The classical groups of normal Lie are given by A l (q), B l (q), C l (q), and D l (q). For 1 i j l denote a positive root p i + p i p j by p ij and, in particular, p i = p ii. In this notation, write down all positive roots for each classical group (see Table 6). 632 Proposition. Suppose that G {A l (q),b l (q),c l (q),d l (q)}, K = GF (q) and P is the parabolic

12 maximal subgroup in G obtained by removing the kth vertex from the Dynkin diagram in the standard ordering of the vertices: Assume that q is odd for the groups of type B l and C l, while U is the unipotent radical of P, and the subgroups U j and V jpk, j 1, are as above. Then (1) If G = A l (q) then, for each 1 k l, (a) U = U 1 is an abelian group; (b) dim V 1pk = k(l k +1). (2) If G = B l (q) then (a) U = U 1 is an abelian group for k =1; (b) the fragment of the chief series of P lying in U has the form U = U 1 >U 2 > 1 for 2 k l; (c) dim V 1pk = k(2l 2k +1)for 1 k l; (d) dim V 2pk = k(k 1)/2 for 2 k l. (3) If G = C l (q) then (a) the fragment of the chief series of P lying in U has the form U = U 1 >U 2 > 1 for 1 k l 1; (b) U = U 1 is an abelian group k = l; (c) dim V 1pk =2k(l k) for 1 k l 1 and dim V 1pl = l(l +1)/2; (d) dim V 2pk = k(k +1)/2 for 1 k l 1. (4) If G = D l (q) then (a) the fragment of the chief series of P lying in U has the form U = U 1 >U 2 > 1 for 2 k l 2; (b) U = U 1 is an abelian group k {1,l 1,l}; (C) dim V 1pk =2k(l k) for 1 k l 2 and dim V 1pk = l(l 1)/2 for k {l 1,l}; (d) dim V 2pk = k(k 1)/2 for 2 k l 2. Proof. Case G = A l (q). If 1 k l then Φ + \ Φ + k = {p 1k,p 1k+1,...,p 1l,p 2k,p 2k+1,...,p 2l,...,p k 1k,p k 1k+1,...,p k 1l, p k,p kk+1,p kk+2,...,p kl }. For Φ + \ Φ + k,wehaveshape() =1p k and U = U 1. Therefore, U is an abelian group and there exists only one chief factor V 1pk = U 1 /1ofP included in U. For every t K, the element x p1l (t) U is a maximal vector and the root p 1l is the highest weight of the KL-module V 1pk. Since dim V 1pk = Φ + \ Φ + k = k(l k + 1), Case (1) of the proposition is proved. Case G = B l (q). If 1 k l then Φ + \ Φ + k = {p 1k,p 1k+1,...,p 1l,p 2k,p 2k+1,...,p 2l,...,p k 1k,p k 1k+1,..., p k 1l,p k,p kk+1,...,p kl ; p 1 +2p 2l,p 12 +2p 3l,...,p 1l 1 +2p l, p 2 +2p 3l,p 23 +2p 4l,...,p 2l 1 +2p l,...,p k 1 +2p kl,p k 1k +2p k+1l,...,p k 1l 1 +2p l,p k +2p k+1l,p kk+1 +2p k+2l,...,p kl 1 +2p l }; 633

13 in particular, Φ + \ Φ + 1 = {p 1,p 12,...,p 1l ; p 1 +2p 2l,p 12 +2p 3l,...,p 1l 1 +2p l }. For Φ + \ Φ + 1,wehave shape() =1p 1, U = U 1 is an abelian group, there exists only one chief factor V 1p1 = U 1 /1forP included in U, and dim V 1p1 =2l 1. For every t K, the element x p1 +2p 2l (t) U is a maximal vector and the root p 1 +2p 2l is the highest weight of the KL-module V 1p1. If Φ + \ Φ + k and 2 k l 1 then shape() =1p k or shape() =2p k. We have the series U = U 1 >U 2 > 1 and two chief factors V 1pk = U 1 /U 2 and V 2pk = U 2 /1ofP.Therootp 1k +2p k+1l is the highest weight of the KL-module V 1pk and p 1 +2p 2l is the highest weight of V 2pk. If k = l then Φ + \ Φ + l = {p 1l,p 2l,...,p l 1l,p l ; p 1 +2p 2l,p 12 +2p 3l,...,p 1l 1 +2p l,p 2 +2p 3l, p 23 +2p 4l,...,p 2l 1 +2p l,...,p l 1 +2p l }. The roots p 1l,p 2l,...,p l 1l,p l have shape 1p l, and the remaining roots in Φ + \ Φ + l have shape 2p l ; therefore, we obtain a fragment U = U 1 >U 2 > 1 of the central series of P with two factors V 1pl = U 1 /U 2 and V 2pl = U 2 /1. The root p 1l is the highest weight of the first KL-module, whereas p 1 +2p 2l is the highest weight of the second. Now, it is easy to compute the dimensions of V 1pk and V 2pk. Case G = C l (q). If 1 k l 2 then Φ + \ Φ + k = {p 1k,p 1k+1,...,p 1l,p 2k,p 2k+1,...,p 2l,...,p k 1k,p k 1k+1,..., p k 1l,p k,p kk+1,...,p kl ; p 1 +2p 2l 1 + p l,p 12 +2p 3l 1 + p l,...,p 1l 2 +2p l 1 + p l,p 2 +2p 3l 1 + p l,p 23 +2p 4l 1 + p l,..., p 2l 2 +2p l 1 + p l,...,p k 1 +2p kl 1 + p l,p k 1k +2p k+1l 1 + p l,...,p k 1l 2 +2p l 1 + p l,p k +2p k+1l 1 + p l,p kk+1 +2p k+2l 1 + p l,...,p kl 2 +2p l 1 + p l ;2p 1l 1 + p l, 2p 2l 1 + p l,...,2p kl 1 + p l }, Φ + \ Φ + l 1 = {p 1l 1,p 1l,p 2l 1,p 2l,...,p l 2l 1,p l 2l,p l 1,p l 1l ; p 1 +2p 2l 1 + p l, p 12 +2p 3l 1 + p l,...,p 1l 2 +2p l 1 + p l,p 2 +2p 3l 1 + p l, p 23 +2p 4l 1 + p l,...,p 2l 2 +2p l 1 + p l,...,p l 2 +2p l 1 + p l ; 2p 1l 1 + p l, 2p 2l 1 + p l,...,2p l 2l 1 + p l, 2p l 1 + p l }, Φ + \ Φ + l = {p 1l,p 2l,...,p l 1l,p l ; p 1 +2p 2l 1 + p l,p 12 +2p 3l 1 + p l,..., p 1l 2 +2p l 1 + p l,p 2 +2p 3l 1 + p l,...,p 2l 2 +2p l 1 + p l,..., p l 3 +2p l 2l 1 + p l,p l 3l 2 +2p l 1 + p l,p l 2 +2p l 1 + p l ; 2p 1l 1 + p l, 2p 2l 1 + p l,...,2p l 2l 1 + p l, 2p l 1 + p l }. If Φ + \ Φ + k and 1 k l 2 then shape() = 1p k or shape() = 2p k. We have the series U = U 1 >U 2 > 1 and two chief factors V 1pk and V 2pk of P. The root p 1k +2p k+1l 1 + p l is the highest weight of the first KL-module V 1pk, whereas 2p 1l 1 + p l is the highest weight of the second. For k = l 1 the situation is similar excluding the highest weight of V 1pl 1. If Φ + \ Φ + l then shape() =1p l,andu is an abelian group. For every t K, the element x 2p1l 1 +p l (t) U is a maximal vector and the root 2p 1l 1 + p l is the highest weight of V 1pl. Calculate the number of roots for each shape and thus find the dimensions of V 1pk and V 2pk,1 k l. Case G = D l (q). Given a root, from Φ + \ Φ + 1 = {p 1,p 12,...,p 1l ; p 1l 2 + p l ; p 1 +2p 2l 2 + p l 1l,p 12 +2p 3l 2 + p l 1l,...,p 1l 3 +2p l 2 + p l 1l } 634

14 V S for A l (q) Table 7. Roots with V S = V 1pk p 1k,p 1k+1,...,p 1l,p 2k,p 2k+1,...,p 2l,...,p k 1k,p k 1k+1,..., k is arbitrary p k 1l,p k,p kk+1,...,p kl X V S for B l (q) V 1p1 p 1,p 12,...,p 1l ; p 1 +2p 2l,p 12 +2p 3l,...,p 1l 1 +2p l V 1pk p 1k,p 1k+1,...,p 1l,p 2k,p 2k+1,...,p 2l,...,p k,p kk+1,...,p kl ; 2 k l 1 p 1k +2p k+1l,p 1k+1 +2p k+2l,...,p 1l 1 +2p l,p 2k +2p k+1l, p 2k+1 +2p k+2l,...,p 2l 1 +2p l,...,p k +2p k+1l, p kk+1 +2p k+2l,...,p kl 1 +2p l V 2pk p 1 +2p 2l,p 12 +2p 3l,...,p 1k 1 +2p kl,p 2 +2p 3l, 2 k l 1 p 23 +2p 4l,...,p 2k 1 +2p kl,...,p k 2 +2p k 1l, V 1pl p k 2k 1 +2p kl,p k 1 +2p kl p 1l,p 2l,...,p l 1l,p l V 2pl p 1 +2p 2l,p 12 +2p 3l,...,p 1l 1 +2p l, V S for C l (q) p 2 +2p 3l,p 23 +2p 4l,...,p 2l 1 +2p l,...,p l 1 +2p l V 1pk p 1k,p 1k+1,...,p 1l,p 2k,p 2k+1,...,p 2l,...,p k 1k,p k 1k+1, 1 k l 2...,p k 1l,p k,p kk+1,...,p kl ; p 1k +2p k+1l 1 + p l, p 1k+1 +2p k+2l 1 + p l,...,p 1l 2 +2p l 1 + p l, p 2k +2p k+1l 1 + p l,...,p 2l 2 +2p l 1 + p l,...,p k 1k +2p k+1l 1 + p l,p k 1k+1 +2p k+2l 1 + p l,..., p k 1l 2 +2p l 1 + p l,p k +2p k+1l 1 + p l, p kk+1 +2p k+2l 1 + p l,...,p kl 2 +2p l 1 + p l V 2pk p 1 +2p 2l 1 + p l,p 12 +2p 3l 1 + p l,...,p 1k 1 +2p kl 1 + p l, 1 k l 2 p 2 +2p 3l 1 + p l,...,p 2k 1 +2p kl 1 + p l,..., V 1pl 1 p k 1 +2p kl 1 + p l ;2p 1l 1 + p l, 2p 2l 1 + p l,...,2p kl 1 + p l p 1l 1,p 1l,p 2l 1,p 2l,...,p l 2l 1,p l 2l,p l 1,p l 1l V 2pl 1 p 1 +2p 2l 1 + p l,p 12 +2p 3l 1 + p l,...,p 1l 2 +2p l 1 + p l, p 2 +2p 3l 1 + p l,p 23 +2p 4l 1 + p l,...,p 2l 2 +2p l 1 + p l,...,p l 2 +2p l 1 + p l ; 2p 1l 1 + p l, 2p 2l 1 + p l,...,2p l 2l 1 + p l, 2p l 1 + p l V 1pl p 1l,p 2l,...,p l 1l,p l ; p 1 +2p 2l 1 + p l,p 12 +2p 3l 1 + p l,..., p 1l 2 +2p l 1 + p l,p 2 +2p 3l 1 + p l,p 23 +2p 4l 1 + p l,..., p 2l 2 +2p l 1 + p l,...,p l 2 +2p l 1 + p l ; 2p 1l 1 + p l, 2p 2l 1 + p l,...,2p l 2l 1 + p l, 2p l 1 + p l we have shape() =1p 1 ; hence, U is an abelian group. For every t K, the element x p1 +2p 2l 2 +p l 1l (t) U is a maximal vector and the root p 1 +2p 2l 2 +p l 1l is the highest weight of the KL-module V 1p1. Since dim V 1p1 = Φ + \ Φ + 1,wehavedimV 1p k =2l

15 Table 7 (continued). V S for D l (q) V 1p1 p 1,p 12,...,p 1l ; p 1l 2 + p l,p 1 +2p 2l 2 + p l 1l, p 12 +2p 3l 2 + p l 1l,...,p 1l 3 +2p l 2 + p l 1l V 1pk p 1k,p 1k+1,...,p 1l,p 2k,p 2k+1,...,p 2l...,p k 1k,p k 1k+1,..., 2 k l 3 p k 1l,p k,p kk+1,...,p kl ; p 1l 2 + p l,p 2l 2 + p l,...,p kl 2 + p l ; p 1k +2p k+1l 2 + p l 1l,p 1k+1 +2p k+2l 2 + p l 1l,..., p 1l 3 +2p l 2 + p l 1l,p 2k +2p k+1l 2 + p l 1l, p 2k+1 +2p k+2l 2 + p l 1l,...,p 2l 3 +2p l 2 + p l 1l,..., p k 1k +2p k+1l 2 + p l 1l,p k 1k+1 +2p k+2l 2 + p l 1l,..., p k 1l 3 +2p l 2 + p l 1l,p k +2p k+1l 2 + p l 1l, p kk+1 +2p k+2l 2 + p l 1l,...,p kl 3 +2p l 2 + p l 1l V 2pk p 1 +2p 2l 2 + p l 1l,p 12 +2p 3l 2 + p l 1l,..., 2 k l 3 p 1k 1 +2p kl 2 + p l 1l,p 2 +2p 3l 2 + p l 1l,..., p 2k 1 +2p kl 2 + p l 1l,...,p k 1 +2p kl 2 + p l 1l V 1pl 2 p 1l 2,p 1l 1,p 1l,p 2l 2,p 2l 1,p 2l,...,p l 2,p l 2l 1,p l 2l ; p 1l 2 + p l,p 2l 2 + p l,...,p l 2 + p l V 2pl 2 p 1 +2p 2l 2 + p l 1l,p 12 +2p 3l 2 + p l 1l,..., p 1l 3 +2p l 2 + p l 1l,p 2 +2p 3l 2 + p l 1l,p 23 +2p 4l 2 + p l 1l,...,p 2l 3 +2p l 2 + p l 1l,...,p l 4 +2p l 3l 2 + p l 1l, p l 4,l 3 +2p l 2 + p l 1l,p l 3 +2p l 2 + p l 1l V 1pl 1 p 1l 1,p 1l,p 2l 1,p 2l,...,p l 2l 1,p l 2l,p l 1 ; p 1 +2p 2l 2 + p l 1l,p 12 +2p 3l 2 + p l 1l,..., p 1l 3 +2p l 2 + p l 1l,p 2 +2p 3l 2 + p l 1l,..., p 2l 3 +2p l 2 + p l 1l,...,p l 3 +2p l 2 + p l 1l V 1pl p 1l,p 2l,...,p l 2l ; p 1 +2p 2l 2 + p l 1l,p 12 +2p 3l 2 + p l 1l,...,p 1l 3 +2p l 2 + p l 1l,p 2 +2p 3l 2 + p l 1l, p 23 +2p 4l 2 + p l 1l,...,p 2l 3 +2p l 2 + p l 1l,...,p l 3 +2p l 2 + p l 1l ; p 1l 2 + p l,p 2l 2 + p l,...,p l 3l 2 + p l,p l 2 + p l,p l For 2 k l 3, we infer Φ + \ Φ + k = {p 1k,p 1k+1,...,p 1l,p 2k,p 2k+1,...,p 2l,...,p k,p kk+1,...,p kl ; p 1l 2 + p l,p 2l 2 + p l,...,p kl 2 + p l ; p 1 +2p 2l 2 + p l 1l, p 12 +2p 3l 2 + p l 1l,...,p 1l 3 +2p l 2 + p l 1l,p 2 +2p 3l 2 + p l 1l,...,p 2l 3 +2p l 2 + p l 1l,...,p k 1 +2p kl 2 + p l 1l, p k 1k +2p k+1l 2 + p l 1l,...,p k 1l 3 +2p l 2 + p l 1l, p k +2p k+1l 2 + p l 1l,...,p kl 3 +2p l 2 + p l 1l }. The roots p 1k,p 1k+1,...,p 1l,p 2k,p 2k+1,...,p 2l,...,p k,p kk+1,...,p kl ; p 1l 2 + p l, p 2l 2 + p l,...,p kl 2 + p l ; p 1k +2p k+1l 2 + p l 1l,p 1k+1 +2p k+2l 2 + p l 1l,..., p 1l 3 +2p l 2 + p l 1l,p 2k +2p k+1l 2 + p l 1l,p 2k+1 +2p k+2l 2 + p l 1l,..., p 2l 3 +2p l 2 + p l 1l,...,p k 1k +2p k+1l 2 + p l 1l,p k 1k+1 +2p k+2l 2 + p l 1l,...,p k 1l 3 +2p l 2 + p l 1l,p k +2p k+1l 2 + p l 1l,...,p kl 3 +2p l 2 + p l 1l 636

16 Table 8. The Chief Factors, the Highest Weights, and the Powers for Classical Groups. G k V S S dim V S A l (q) is arbitrary V 1pk p 1l k(l k +1) B l (q), p 2 1 k l 1 V 1pk p 1k +2p k+1l k(2l 2k +1) B l (q), p 2 2 k l V 2pk p 1 +2p 2l k(k 1)/2 B l (q), p 2 l V 1pl p 1l l C l (q), p 2 1 k l 2 V 1pk p 1k +2p k+1l 1 + p l 2k(l k) C l (q), p 2 1 k l 2 V 2pk 2p 1l 1 + p l k(k +1)/2 C l (q), p 2 l 1 V 1pl 1 p 1l 2(l 1) C l (q), p 2 l 1 V 2pl 1 2p 1l 1 + p l l(l 1)/2 C l (q), p 2 l V 1pl 2p 1l 1 + p l l(l +1)/2 D l (q) 1 k l 3 V 1pk p 1k +2p k+1l 2 + p l 1l 2k(l k) D l (q) l 2 V 1pl 2 p 1l 4(l 2) D l (q) 2 k l 2 V 2pk p 1 +2p 2l 2 + p l 1l k(k 1)/2 D l (q) l 1 V 1pl 1 p 1 +2p 2l 2 + p l 1l l(l 1)/2 D l (q) l V 1pl p 1 +2p 2l 2 + p l 1l l(l 1)/2 have shape 1p k, whereas the remaining roots in Φ + \ Φ + k have shape 2p k; therefore, for 2 k l 3 we get the chief series U = U 1 >U 2 > 1. The root p 1k +2p k+1l 2 + p l 1l is the highest weight of the KL-module V 1pk = U 1 /U 2,andp 1 +2p 2l 2 + p l 1l is the highest weight of V 2pk = U 2 /1. Next, Φ + \ Φ + l 2 = {p 1l 2,p 1l 1,p 1l,p 2l 2,p 2l 1,p 2l,...,p l 2,p l 2l 1,p l 2l ; p 1 +2p 2l 2 + p l 1l,p 12 +2p 3l 2 + p l 1l,...,p 1l 3 +2p l 2 + p l 1l, p 2 +2p 3l 2 + p l 1l,p 23 +2p 4l 2 + p l 1l,...,p 2l 3 +2p l 2 + p l 1l,...,p l 4 +2p l 3l 2 + p l 1l,p l 4l 3 +2p l 2 + p l 1l, p l 3 +2p l 2 + p l 1l ; p 1l 2 + p l,p 2l 2 + p l,...,p l 2 + p l }. The roots p 1l 2,p 1l 1,p 1l,p 2l 2,p 2l 1,p 2l,...,p l 2,p l 2l 1,p l 2l ; p 1l 2 + p l,p 2l 2 + p l,...,p l 2 + p l have shape 1p l 2, and the remaining roots in Φ + \ Φ + l 2 have shape 2p l 2; therefore, for k = l 2, we have the series U = U 1 >U 2 > 1, and two chief factors V 1pl 2 = U 1 /U 2 and V 2pl 2 = U 2 /1ofP. The root p 1l is the highest weight of the KL-module V 1pl 2,andp 1 +2p 2l 2 + p l 1l is the highest wight of V 2pl 2. Consider k {l 1,l}. We infer Φ + \ Φ + l 1 = {p 1l 1,p 1l,p 2l 1,p 2l,...,p l 2l 1,p l 2l,p l 1 ; p 1 +2p 2l 2 + p l 1l, p 12 +2p 3l 2 + p l 1l,...,p 1l 3 +2p l 2 + p l 1l,p 2 +2p 3l 2 + p l 1l, p 23 +2p 4l 2 + p l 1l,...,p 2l 3 +2p l 2 + p l 1l,...,p l 3 +2p l 2 + p l 1l }, Φ + \ Φ + l = {p 1l,p 2l,...,p l 2l ; p 1 +2p 2l 2 + p l 1l,p 12 +2p 3l 2 + p l 1l,..., p 1l 3 +2p l 2 + p l 1l,p 2 +2p 3l 2 + p l 1l,p 23 +2p 4l 2 + p l 1l,..., p 2l 3 +2p l 2 + p l 1l,...,p l 3 +2p l 2 + p l 1l ; p 1l 2 + p l,p 2l 2 + p l,...,p l 3l 2 + p l,p l 2 + p l,p l }. For all Φ + \ Φ + l 1 or Φ+ \ Φ + l,wehaveshape() =1p l 1 or shape() =1p l respectively, and hence U is an abelian group and there is only one chief factor. The dimensions of the modules are now easily computable. The proposition is proved. 637

17 We now compile Table 7. In the first column of the table, indicate the chief factors V S = V jpk occurring in the unipotent radical of each parabolic maximal subgroup P k in a classical group. Recall that the module V S is isomorphic to the direct sum of root subgroups X, which are one-dimensional weight spaces of weight. In the second column, we write down all these weights. We give all remaining information about the chief factors of parabolic maximal subgroups in the classical groups in the last combined Table 8. In the first column we indicate G {A l (q),b l (q),c l (q),d l (q)}; in the second column, write down the number of the vertex k that is removed from the Dynkin diagram for obtaining a parabolic maximal subgroup P = P k in G; in the third column of the table, for P k,we point out all chief factors V S = V jpk occurring in its unipotent radical. In the fourth column, we write down the highest weight of the irreducible KL-module V S ; and, finally, the last column will contain the dimension of V S. References 1. Azad H., Barry M., and Seitz G., On the structure of parabolic subgroups, Comm. Algebra, 18, No. 2, (1990). 2. Bourbaki N., Groups and Lie Algebras. Chapters 7 and 8 [Russian translation], Mir, Moscow (1978). 3. Humphreys J., Modular Representation of Finite Groups of Lie Type, Cambridge Univ. Press, Cambridge (2006). 4. Malle G. and Testerman D., Linear Algebraic Groups and Finite Groups of Lie Type, Cambridge Univ. Press, Cambridge (2011). 5. Steinberg R., Lectures on Chevalley Groups, Yale University, New Haven (1968). 6. Carter R. W., Simple Groups of Lie Type, Wiley, London (1972). V. V. Korableva Chelyabinsk State University, Chelyabinsk, Russia address: vvk@csu.ru 638

arxiv: v1 [math.rt] 14 Nov 2007

arxiv: v1 [math.rt] 14 Nov 2007 arxiv:0711.2128v1 [math.rt] 14 Nov 2007 SUPPORT VARIETIES OF NON-RESTRICTED MODULES OVER LIE ALGEBRAS OF REDUCTIVE GROUPS: CORRIGENDA AND ADDENDA ALEXANDER PREMET J. C. Jantzen informed me that the proof