M. Brou e and J. Michel, Sur certains elements reguliers des groupes de Weyl et les varietes de Deligne-Lusztig associees, in Finite reductive groupes, Progress in Mathematics, Birkhauser (1996).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....characters of a relative Weyl group, which in general is a non real reflection group. A possible interpretation of this result was subsequently proposed by Brou e and the author [2] in terms of the so called cyclotomic Hecke algebra attached to a finite complex reflection group (see also [6]) The results of Lusztig show that the set of unipotent characters of a finite group of Lie type only depends on the Weyl group of the associated algebraic group, together with the action of the Frobenius endomorphism on it. In the course of this classification Lusztig observed that similar sets ....

.... of the principal series unipotent characters are the generic degrees of H(W;x) This is precisely the formula one would obtain if the specialization of the cyclotomic algebra H(W G (L; u) of the relative Weyl group were the endomorphism algebra of R G(q) L(q) L q ( see [2] and also [6] for a conjectural explanation) 4.9. Unipotent degrees. Let now G = V; W OE) be an arbitrary spets, with spetsial reflection group W over k. Assume given for any Levi subspets L = V; WLwOE) of G a set E(L) with an action of NW (WL ) a degree map deg : E(L) k[x] and for any Phi and any ....

M. Brou'e and J. Michel, Sur certains 'el'ements r'eguliers des groupes de Weyl et les vari'et'es de Deligne-Lusztig associ'ees, Progress in Mathematics (M. Cabanes, ed.), vol. 141, Birkhauser, 1997, pp. 73--140.

....general setting. For example, generalizing a result of Deligne and Brieskorn Saito valid for Coxeter groups, we check here, in most cases, that the centers of braid groups associated to irreducible complex reflection groups are cyclic. 1 see for example [AlLu] Ari] ArKo] BreMa] BrMa] [BrMi], BMM] Lu] Ma1] Ma2] Complex Reflection Groups, Braid Groups, Hecke algebras 3 Also (at least in the case of the infinite series) the pure braid group of an r dimensional irreducible complex reflection group has a natural structure as an r fold iterated semidirect product of free ....

....[0; 1] M defined by fi : t 7 x 0 exp(2it=jZ(W )j) The following result is a consequence of Corollary 2.25. Notice that it generalizes a result of Deligne [De1] 4.21) see also [BrSa] from which it follows that if W is a Coxeter group, then ( 2N . It was noticed experimentally in [BrMi], 4.8) 2.20. Corollary. We have (fi) N N ) jZ(W )j and ( N N : From now on, we assume that W acts irreducibly on V . Note that, since W is irreducible on V , it results from Schur s lemma that Z(W ) fexp(2ik=jZ(W )j) j (k 2 Z)g ; and so in particular fi defines an element ....

M. Brou'e and J. Michel, Sur certains 'el'ements r'eguliers des groupes de Weyl et les vari'et'es de Deligne-Lusztig associ'ees, Finite Reductive Groups : Related Structures and Representations (M. Cabanes, eds.), Progress in Mathematics, vol. 141, Birkhauser, 1997, pp. 73--140.

....be seen as a generalization of the well known fact that, if C is the class such that Cmin is the set of Coxeter elements and h is the Coxeter number, then w h = w 2 I for w 2 Cmin , and if h is even then there exists an element w 2 Cmin such that w h=2 = w I . They also generalize results of [5] on the class of regular elements of order d when the length of elements of Cmin is 2N=d (where N is the number of reflections in W ) These elements are characterized by the fact that the relation in Theorem 1.1 holds with r = 1, and all elements of Cmin are good. However, they are not ....

....field of K) are of the form i e i d Y s2I v f s;i s for i = 1; m: Moreover, the exponents f s;i can be determined explicitly from the ordinary character table of W , various parabolic subgroups of W , and the corresponding induce restrict matrices. This again generalizes results of [5] for regular elements. A formula for the exponents f s;i will be given in x 4. The problem of determining the exponents e i seems to be more subtle, and is not solved in general. This information on eigenvalues of irreducible representations of KH will serve as the missing tool to compute the ....

M. Brou' e and J. Michel, Sur certains 'el'ements r'eguliers des groupes de Weyl et les vari'et'es de Deligne-Lusztig associ'ees, to appear in Progress in Math., Birkhauser.

....words, we have, for all integer k, s k H ) j=eC Gamma1 X j=0 m C;j det j V (s k H ) We denote by the complex conjugate of the character , which is then the character of the contragredient representation of a representation defining . The following properties may be found in [BrMi], 4.1 and 4.2. Towards Spetses I 7 1.11. Proposition. Whenever 2 Irr(W ) we have (1) N( P C2A=W P j=eC Gamma1 j=0 j N C m C;j ; 2) N( N( P C2A=W P j=eC Gamma1 j=1 N C e C m C;j ; 3) 1) N N ) Gamma (N( N( P C2A=W N C e C m C;0 : ....

....for j = j C ) C2A=W we have j = Y C2A=W (det C;jC ) t : Central morphisms associated with irreducible characters. More generally, let 2 Irr(W ) Recall that, for C 2 A=W , we denote by m C;j the multiplicity of det j V in the restriction of to the cyclic group WH . It results from [BrMi], 4.17, that m C;j N C e C (1) 2 N : Since K(t) is a splitting field for the algebra K(t)H(W;u) the irreducible character t defines an algebra morphism from the center of K(t)H(W;u) onto K(t) which we denote by t : Z(K(t)H(W;u) K(t) It results from [BrMi] prop. 4.16, that t ....

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M. Brou'e and J. Michel, Sur certains 'el'ements r'eguliers des groupes de Weyl et les vari'et'es de Deligne-Lusztig associ'ees, Finite Reductive Groups : Related Structures and Representations (M. Cabanes, eds.), Progress in Mathematics, vol. 141, Birkhauser, 1997, pp. 73--140.

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M. Brou e and J. Michel, Sur certains elements reguliers des groupes de Weyl et les varietes de Deligne-Lusztig associees, in Finite reductive groupes, Progress in Mathematics, Birkhauser (1996).

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M. Broue and J. Michel, Sur certains elements reguliers des groupes de Weyl et les varietes de Deligne-Lusztig associees, in "Finite reductive groups", Birkhauser, 73--139, 1997.

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