On the Orbits of a Borel Subgroup in Abelian Ideals

Panyushev, Dmitri I.

doi:10.1007/s00031-016-9391-8

Cited by 12 publications

(34 citation statements)

References 21 publications

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“…In particular, every abelian ideal

fraktura

frakturb

contains finitely many

B

‐orbits. In , the same result has been proved avoiding the use of the sphericity of

G a

. In Section , we prove, along the same lines, the finiteness theorem quoted in (ii), in the more general context of finite‐order automorphisms of

G

(see Theorem ).…”

Section: Introductionsupporting

confidence: 60%

“…Thanks to previous lemmas, we can now reproduce the same argument given in [, Theorem 2.2] to prove Theorem .…”

Section: B0‐orbits In B0‐stable Subalgebras Contained In G1mentioning

confidence: 90%

“…As in , the key step to prove Theorem is the following combinatorial lemma, which generalizes [, Lemma 1.2] to the graded setting. Lemma Let

α, β \in Ψ (a)

be orthogonal weights and let

γ \in {normalΦ}_{0}

.…”

Section: B0‐orbits In B0‐stable Subalgebras Contained In G1mentioning

confidence: 99%

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Spherical nilpotent orbits and abelian subalgebras in isotropy representations

Gandini

Frajria

Papi

2017

J. London Math. Soc.

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Abstract. Let G be a simply connected semisimple algebraic group with Lie algebra g, let G 0 ⊂ G be the symmetric subgroup defined by an algebraic involution σ and let g 1 ⊂ g be the isotropy representation of G 0 . Given an abelian subalgebra a of g contained in g 1 and stable under the action of some Borel subgroup B 0 ⊂ G 0 , we classify the B 0 -orbits in a and we characterize the sphericity of G 0 a. Our main tool is the combinatorics of σ-minuscule elements in the affine Weyl group of g and that of strongly orthogonal roots in Hermitian symmetric spaces.

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“…In particular, every abelian ideal

fraktura

frakturb

contains finitely many

B

‐orbits. In , the same result has been proved avoiding the use of the sphericity of

G a

. In Section , we prove, along the same lines, the finiteness theorem quoted in (ii), in the more general context of finite‐order automorphisms of

G

(see Theorem ).…”

Section: Introductionsupporting

confidence: 60%

“…Thanks to previous lemmas, we can now reproduce the same argument given in [, Theorem 2.2] to prove Theorem .…”

Section: B0‐orbits In B0‐stable Subalgebras Contained In G1mentioning

confidence: 90%

See 1 more Smart Citation

Spherical nilpotent orbits and abelian subalgebras in isotropy representations

Gandini

Frajria

Papi

2017

J. London Math. Soc.

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“…It is known that the conjecture is true for sl n (cf. [5]). The proof is straightforward and we provide it in short here since we use it in what follows.…”

Section: Link Patterns and ℓ(σ) For The Weyl Groupmentioning

confidence: 99%

“…

Let B be a Borel subgroup of a semisimple algebraic group G and let m be an abelian nilradical in b = Lie(B). Using subsets of strongly orthogonal roots in the subset of positive roots corresponding to m, D. Panyushev [5] gives in particular classification of B−orbits in m and m * and states general conjectures on the closure and dimensions of the B−orbits in both m and m * in terms of involutions of the Weyl group. Using Pyasetskii correspondence between B−orbits in m and m * he shows the equivalence of these two conjectures.

…”

mentioning

confidence: 99%

B–Orbits in Abelian Nilradicals of Types B, C and D: Towards a Conjecture of Panyushev

Barnea

Melnikov

2016

Springer Proceedings in Mathematics &Amp; Statistics

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Abstract. Let B be a Borel subgroup of a semisimple algebraic group G and let m be an abelian nilradical in b = Lie(B). Using subsets of strongly orthogonal roots in the subset of positive roots corresponding to m, D. Panyushev [5] gives in particular classification of B−orbits in m and m * and states general conjectures on the closure and dimensions of the B−orbits in both m and m * in terms of involutions of the Weyl group. Using Pyasetskii correspondence between B−orbits in m and m * he shows the equivalence of these two conjectures. In this Note we prove his conjecture in types B n , C n and D n for adjoint case.1. Abelian nilradicals and Panyushev's conjecture 1.1. Minimal nilradicals. Let G be a semisimple linear algebraic group over C and let g be its Lie algebra. Let B be its Borel subgroup and b = Lie(B). Let g = n ⊕ h ⊕ n − be its corresponding triangular decomposition, where b = n ⊕ h. B acts adjointly on n. For x ∈ n let B.x denote its orbit.Since the description of B−orbits in n immediately reduces to simple Lie algebras in what follows we assume that g is simple.Let R be the root system of g and W its Weyl group. For α ∈ R let s α be the corresponding reflection in W .Let R + (resp. R − ) denote the subset of positive (resp. negative) roots. For α ∈ R let X α denote the standard root vector in g so that n =+ be a set of simple roots. Let θ be the maximal root in R + . Recall that any standard parabolic subgroup P of G is of the form P = L ⋉ M where L is a standard Levy subgroup and M is the unipotent radical of P. If R L is the root system of l = Lie(L) then ∆ L = ∆ ∩ R L . Let W P denote Weyl group of l. Let w be the longest element of W P .P is maximal if and only if ∆ L = ∆ \ {α i }. We will write

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B-Orbits of Square Zero in Nilradical of the Symplectic Algebra

Barnea

Melnikov

2016

Transformation Groups

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Let SP n (C) be the symplectic group and sp n (C) its Lie algebra. Let B be a Borel subgroup of SP n (C), b = Lie(B) and n its nilradical. Let X be a subvariety of elements of square 0 in n. B acts adjointly on X . In this paper we describe topology of orbits X /B in terms of symmetric link patterns.Further we apply this description to the computations of the closures of orbital varieties of nilpotency order 2 and to their intersections. In particular we show that all the intersections of codimension 1 are irreducible. N. Barnea: partially supported by Israel Scientific Foundation grant 797/14.

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On the Orbits of a Borel Subgroup in Abelian Ideals

Cited by 12 publications

References 21 publications

Spherical nilpotent orbits and abelian subalgebras in isotropy representations

Spherical nilpotent orbits and abelian subalgebras in isotropy representations

B–Orbits in Abelian Nilradicals of Types B, C and D: Towards a Conjecture of Panyushev

B-Orbits of Square Zero in Nilradical of the Symplectic Algebra

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