Intersection of conjugacy classes with Bruhat cells in Chevalley groups

Ellers, Erich W.; Gordeev, Nikolai

doi:10.2140/pjm.2004.214.245

Cited by 23 publications

(25 citation statements)

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“…The case where m = 2, which includes, for example, semisimple long root elements of the special linear and the symplectic groups and weight elements of type 1 in orthogonal groups, was considered in [13,31]. In the recent paper [48] by Erich Ellers and Nikolai Gordeev, one can find similar results for the elements of large residue.…”

Section: Introductionmentioning

confidence: 89%

Weight elements of Chevalley groups

Vavilov

2008

St. Petersburg Math. J.

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Abstract. The paper is devoted to a detailed study of some remarkable semisimple elements of (extended) Chevalley groups that are diagonalizable over the ground field -the weight elements. These are the conjugates of certain semisimple elements h ω (ε) of extended Chevalley groups G = G(Φ, K), where ω is a weight of the dual root system Φ ∨ and ε ∈ K * . In the adjoint case the h ω (ε)'s were defined by Chevalley himself and in the simply connected case they were constructed by Berman and Moody. The conjugates of h ω (ε) are called weight elements of type ω. Various constructions of weight elements are discussed in the paper, in particular, their action in irreducible rational representations and weight elements induced on a regularly embedded Chevalley subgroup by the conjugation action of a larger Chevalley group. It is proved that for a given x ∈ G all elements x(ε) = xh ω (ε)x −1 , ε ∈ K * , apart maybe from a finite number of them, lie in the same Bruhat coset BwB, where w is an involution of the Weyl group W = W (Φ). The elements h ω (ε) are particularly important when ω = i is a microweight of Φ ∨ . The main result of the paper is a calculation of the factors of the Bruhat decomposition of microweight elements x(ε) for the case where ω = i . It turns out that all nontrivial x(ε)'s lie in the same Bruhat coset BwB, where w is a product of reflections in pairwise strictly orthogonal roots γ 1 , . . . , γ r+s . Moreover, if among these roots r are long and s are short, then r + 2s does not exceed the width of the unipotent radical of the ith maximal parabolic subgroup in G. A version of this result was first announced in a paper by the author in Soviet Mathematics: Doklady in 1988. From a technical viewpoint, this amounts to the determination of Borel orbits of a Levi factor of a parabolic subgroup with Abelian unipotent radical and generalizes some results of Richardson, Röhrle, and Steinberg. These results are instrumental in the description of overgroups of a split maximal torus and in the recent papers by the author and V. Nesterov on the geometry of tori.

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Section: Introductionmentioning

confidence: 89%

Weight elements of Chevalley groups

Vavilov

2008

St. Petersburg Math. J.

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“…The description of the intersection of some conjugacy class of a reductive group with its Bruhat cells has been studied for instance, in [4,[8][9][10][17][18][19][21][22][23][24][25]. A basic question here is under what conditions is such an intersection not empty.…”

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confidence: 99%

“…For a given pair k ≤ l, condition (i) is equivalent to the inequalities rank(g − αE n ) ≥ min{k, n − l} for every α ∈ K. This can be proved using the description of the intersection of conjugacy classes in GL(V ) with Bruhat cells ([9]). Corollary 1.5.…”

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confidence: 99%

Big elements in irreducible linear groups

Gordeev

Rehmann

2014

Arch. Math.

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Let V be a linear space over a field K of dimension n > 1, and let G ≤ GL(V ) be an irreducible linear group. In this paper we prove that the group G contains an element g such that rank(g − αEn) ≥ n 2 for every α ∈ K, where En is the identity operator on V . This estimate is sharp for any n = 2 m . The existence of such an element implies that the conjugacy class of G in GL(V ) intersects the big Bruhat cell Bẇ0B of GL(V ) non-trivially (here B is a fixed Borel subgroup of G). The latter fact is equivalent to the existence of a complete flag F such that the flags g(F), F are in general position for some g ∈ G.Introduction. Let B ≤ GL(V ) be a fixed Borel subgroup of GL(V ), let T denote a maximal split torus of B, and let N denote the normalizer of T in GL(V ). Then GL(V ) has a split BN -pair and the Bruhat decomposition GL(V ) = w∈W BẇB, where W = N/T is the Weyl group of GL(V ), T = B ∩ N is a maximal group of semisimple matrices, andẇ is any fixed representative of w ∈ W in the group N (cf.[3], 2.5). We say that an element g ∈ GL(V ) is big if the conjugacy class C g of g in GL(V ) intersects the big Bruhat cell Bẇ 0 B non-trivially (here w 0 ∈ W is the longest element in the Weyl group). A necessary and sufficient condition for an element g ∈ GL(V ) to be big is given by the set of inequalities

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“…The sets W C have been studied by several authors (see, e.g., [10], [11] by Ellers and Gordeev and [4] by Carnovale) and are not easy to determine even for the case of G = SL(n, k) (see [11]). On the other hand, for a conjugacy class C in G, let m C be the unique element in W such that C ∩ (Bm C B) is dense in C (see Lemma 2.4).…”

mentioning

confidence: 99%

“…where l 2 (w) is the number of distinct 2-cycles in the cycle decomposition of w, and r(C) = min{rank(g − cI) : c ∈ k} for any g ∈ C. Theorem 4.2 is proved in Section 4.4 using a criterion of Ellers and Gordeev [11,Theorem 3.20]. The same criterion of Ellers and Gordeev also leads to a simple condition for a conjugacy class O in W to lie entirely in W C for a given conjugacy class C in SL(n + 1, k).…”

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confidence: 99%

On intersections of conjugacy classes and bruhat cells

Chan

2010

Transformation Groups

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For a connected semisimple algebraic group G over an algebraically closed field k and a fixed pair (B, B − ) of opposite Borel subgroups of G, we determine when the intersection of a conjugacy class C in G and a double coset BwB − is nonempty, where w is in the Weyl group W of G. The question comes from Poisson geometry, and our answer is in terms of the Bruhat order on W and an involution mC ∈ W associated to C. We prove that the element mC is the unique maximal length element in its conjugacy class in W , and we classify all such elements in W . For G = SL(n + 1, k), we describe mC explicitly for every conjugacy class C, and when w ∈ W ∼ = S n+1 is an involution, we give an explicit answer to when C ∩ (BwB) is nonempty.

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Intersection of conjugacy classes with Bruhat cells in Chevalley groups

Abstract: Let G = G(K) where G is a simple and simply-connected algebraic group that is defined and quasi-split over a field K. We investigate properties of intersections of Bruhat cells BẇB of G with conjugacy classes C of G, in particular, we consider the question, when is BẇB ∩ C = ∅.

Cited by 23 publications

References 8 publications

Weight elements of Chevalley groups

Weight elements of Chevalley groups

Big elements in irreducible linear groups

On intersections of conjugacy classes and bruhat cells

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