On the set of orbits for a Borel subgroup

Knop, Friedrich

doi:10.1007/bf02566009

Cited by 81 publications

(119 citation statements)

References 14 publications

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“…For a conjugacy class O = O x we shall denote by V the set of B-orbits into which O is parted. It is well-known ( [3], [25] in characteristic 0, [15], [17] in positive characteristic) that X is a spherical homogeneous G-space if and only if the set of B-orbits in X is finite.…”

Section: Preliminariesmentioning

confidence: 99%

Spherical conjugacy classes and the Bruhat decomposition

Carnovale¹

2009

Ann. inst. Fourier

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Section: Preliminariesmentioning

confidence: 99%

Spherical conjugacy classes and the Bruhat decomposition

Carnovale¹

2009

Ann. inst. Fourier

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“…Let B be a Borel subgroup of H . Recall that the complexity of X (with respect to the H -action on X ) is defined as [Brion 1995;Knop 1995;Luna and Vust 1983;Panyushev 1994;Vinberg 1986]). …”

Section: Introductionmentioning

confidence: 99%

“…A G-variety X is called spherical if a Borel subgroup of G acts on X with a dense orbit, that is, κ G (X ) = 0. We recall some standard facts concerning spherical varieties [Brion 1995;Knop 1995;Panyushev 1994].…”

Section: Introductionmentioning

confidence: 99%

Spherical nilpotent orbits in positive characteristic

Fowler¹,

Röhrle²

2008

Pacific J. Math.

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Let G be a connected reductive linear algebraic group defined over an algebraically closed field of characteristic p. Assume that p is good for G. In this note we classify all the spherical nilpotent G-orbits in the Lie algebra of G. The classification is the same as in the characteristic zero case obtained by D.I. Panyushev in 1994, [32]: for e a nilpotent element in the Lie algebra of G, the G-orbit G • e is spherical if and only if the height of e is at most 3. Contents 1. Introduction 2. Preliminaries 2.1. Notation 2.2. Complexity 2.3. Spherical Varieties 2.4. Homogeneous Spaces 2.5. Kempf-Rousseau Theory 2.6. Associated Cocharacters 2.7. Fibre Bundles 2.8. Borel Subgroups of Levi Subgroups Acting on Unipotent Radicals 3. The Classification of the Spherical Nilpotent Orbits 3.1. Height Two Nilpotent Orbits 3.2. Even Gradings 3.3. Nilpotent Orbits of Height at Least Four 3.4. Nilpotent Orbits of Height Three 3.5. Height Three Nilpotent Elements of so 2n+1 (k) 3.6. Height Three Nilpotent Elements of so 2n (k) 3.7. Height Three Nilpotent Elements of the Exceptional Lie Algebras 3.8. The Classification 4. Applications and Complements 4.1. Spherical Distinguished Nilpotent Elements 4.2. Orthogonal Simple Roots and Spherical Nilpotent Orbits 4.3. Spherical Orbits and ad-Nilpotent Ideals 4.4. A Geometric Characterization of Spherical Orbits 4.5. Bad Primes and Spherical Nilpotent Orbits References

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“…In [Kno95], F. Knop introduced an action of a monoid (constructed from the Weyl group of G) on H(B). This action is called "weak order" and studied by M. Brion in [Bri01].…”

Section: Introductionmentioning

confidence: 99%

“…This action is called "weak order" and studied by M. Brion in [Bri01]. But the most spectacular combinatoric structure of the set H(B) was discovered by Knop in [Kno95]: he defined an action of the Weyl group W of G on H(B). Actually, the results of Knop are stated in a more general context.…”

Section: Introductionmentioning

confidence: 99%