Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups

Ignatev, M. V.

doi:10.1134/s0001434609070074

Cited by 11 publications

(23 citation statements)

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“…Let D be an orthogonal subset of Φ + , ξ a set of non-zero scalars from k and Ω = Ω D,ξ the orbit associated with D. Then dim Ω does not depend on ξ and is less or equal to l(σ) − s(σ). Remark 1.3. i) This Theorem proves Conjecture 1.4 from [10]. Note that in many cases (e.g., for elementary orbits) dim Ω is equal to l(σ) − s(σ), see Subsection 2.4.…”

Section: 3supporting

confidence: 57%

“…It's easy to see that dim Ω = |S(β)| [15, Section 4]. It's straightforward to check that l(σ) − s(σ) coincides with |S(β)| (see [10,Section 4] for the case of classical groups).…”

Section: 4mentioning

confidence: 99%

“…In order to generalize the results of A. N. Panov, we introduced the concept of orbits associated with orthogonal subsets of root systems. In the paper [10], we studied these orbits for the case of classical root systems. For orthogonal subsets of special kind of the root systems of types B n and D n , we also obtained a formula involving the corresponding irreducible characters, see [9,Theorem 3.8]).…”

Section: Introductionmentioning

confidence: 99%

“…The main goal of this paper is to generalize the results of [10] to the general case of an arbitrary root system, not only the classical one. The structure of the paper ia as follows.…”

Section: Introductionmentioning

confidence: 99%

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Orthogonal subsets of root systems and the orbit method

Ignatev¹

2011

St. Petersburg Math. J.

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Abstract. Let k be the algebraic closure of a finite field, G a Chevalley group over k, U the maximal unipotent subgroup of G. To each orthogonal subset D of the root system of G and each set ξ of |D| nonzero scalars in k one can assign the coadjoint orbit of U . It is proved that the dimension of such an orbit does not depend on ξ. An upper bound for this dimension is also given in terms of the Weyl group. §0. Introduction 0.1. In studying irreducible complex representations of finite unipotent groups, the main tool is the orbit method. It was created by Kirillov for nilpotent Lie groups over R, see [12] and [13], and then adapted by Kazhdan [11] to the case of finite groups (see also [14] and the paper [3], where the theory of -adic sheaves for unipotent groups was explained). Here we consider the groups U (q) and U , the maximal unipotent subgroups of Chevalley groups over a finite field F q and its algebraic closure, respectively.The orbit method establishes a bijection between the set of equivalence classes of irreducible representations of U (q) and the set of orbits of the coadjoint representation of U (q). Note that many questions about representations can be interpreted in terms of orbits, and that the problem of giving a complete description of orbits remains unsolved and seems to be very difficult. On the other hand, much information about some special types of orbits, representations, and characters is known.In particular, a description is known for the regular orbits (i.e., those of maximal dimension) of the group UT n of all unipotent triangular matrices of size n × n; see [13]. The subregular orbits (i.e., orbits of second maximal dimension) and the corresponding characters 1 were described in [7] and [8]. As a generalization, Panov considered orbits of the group UT n associated with involutions in the symmetric group. In [16], he obtained a formula for the dimension of such an orbit.It is well known that the group UT n corresponds to the root system of type A n−1 . In order to generalize the results of Panov, we introduced the concept of orbits associated with orthogonal subsets of root systems. In the paper [10], we studied these orbits for the case of classical root systems. (For some special orthogonal subsets of the root systems of types B n and D n , we also obtained a formula involving the corresponding irreducible characters; see [9, Theorem 3.8].)Our main goal in this paper is to extend the results of [10] to the general case of an arbitrary root system, not only the classical one. The paper is organized as follows. In the remaining part of this section, we give the necessary definitions and formulate 2010 Mathematics Subject Classification. Primary 17B22. Key words and phrases. Orthogonal subsets of root systems, coadjoint orbits. 1 The description of the irreducible character corresponding to a given orbit is itself a nontrivial problem; see, e.g., the papers of André and Neto [1] and [2] for the description of the so-called supercharacters. In this paper, we concentrate on orbits, not...

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Section: 3supporting

confidence: 57%

“…It's easy to see that dim Ω = |S(β)| [15, Section 4]. It's straightforward to check that l(σ) − s(σ) coincides with |S(β)| (see [10,Section 4] for the case of classical groups).…”

Section: 4mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…The main goal of this paper is to generalize the results of [10] to the general case of an arbitrary root system, not only the classical one. The structure of the paper ia as follows.…”

Section: Introductionmentioning

confidence: 99%

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Orthogonal subsets of root systems and the orbit method

Ignatev¹

2011

St. Petersburg Math. J.

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“…(Obviously, f D = f D,ξ 1 , where ξ 1 (α) = 1 for all α ∈ D.) Given an orthogonal subset D ⊆ Φ + , we say that the orbits Ω D = Ω f D and Θ D,ξ = Θ f D,ξ are associated with D. Note that U -orbits associated with orthogonal subsets and their generalizations were studied, in particular, in [Pa], [Ig1], [Ig2], [IV]. Now, let W be the Weyl group of Φ.…”

Section: Acknowledgementsmentioning

confidence: 99%

On Involutions in the Weyl Group and B-Orbit Closures in the Orthogonal Case

Ignatyev

2019

Representations and Nilpotent Orbits of Lie Algebraic Systems

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We study coadjoint B-orbits on n * , where B is a Borel subgroup of a complex orthogonal group G, and n is the Lie algebra of the unipotent radical of B. To each basis involution w in the Weyl group W of G one can assign the associated B-orbit Ω w . We prove that, given basis involutions σ, τ in W , if the orbit Ω σ is contained in the closure of the orbit Ω τ then σ is less than or equal to τ with respect to the Bruhat order on W . For a basis involution w, we also compute the dimension of Ω w and present a conjectural description of the closure of Ω w .

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Combinatorics of B-orbits and Bruhat–Chevalley order on involutions

Ignatyev

2012

Transformation Groups

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Orthogonal subsets of classical root systems and coadjoint orbits of unipotent groups

Cited by 11 publications

References 3 publications

Orthogonal subsets of root systems and the orbit method

Orthogonal subsets of root systems and the orbit method

On Involutions in the Weyl Group and B-Orbit Closures in the Orthogonal Case

Combinatorics of B-orbits and Bruhat–Chevalley order on involutions

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