On the coordinate ring of spherical conjugacy classes

Costantini, Mauro

doi:10.1007/s00209-008-0468-5

Cited by 16 publications

(23 citation statements)

References 41 publications

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“…Intersections of conjugacy classes with Bruhat cells, and other related problems were also addressed in recent papers by Nicoletta Cantarini, Giovanna Carnovale, and Mauro Costantini [53,54,60]. In particular, [54] completely describes the intersections C ∩ BwB for spherical conjugacy classes.…”

Section: Problem 5 Describe Intersections Of Conjugacy Classes Of Chmentioning

confidence: 98%

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Long root tori in Chevalley groups

Vavilov

Semenov

2013

St. Petersburg Math. J.

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In this paper we study some remarkable semisimple elements of an (extended) Chevalley group that are diagonalizable over the ground field -the 'weight elements'. In particular, we calculate the Bruhat decomposition of microweight elements. Results of the present paper are crucial for the description of overgroups of split maximal tori in Chevalley groups.The paper is devoted to a detailed study of the most important and, in general, the simplest semisimple elements in Chevalley groups G = G(Φ, K), namely, of the long root type elements gh α (ε)g −1 , where the root α is long, ε ∈ K * , and g ∈ G. We furnish detailed proofs of all previously announced results related to these elements. Letbe a long root torus. Let us fix a Borel subgroup B = B(Φ, K) and let U = U (Φ, K) be its unipotent radical. We prove a strong version of reduction to D 4 , asserting that there exists u ∈ U such that uQu −1 is contained in a Chevalley subgroup G(Δ, K) of type Δ ≤ Φ, where Δ is a twisted subsystem of D 4 . It turns out that all elements gh α (ε)g −1 , ε ∈ K * , apart from the identity element, and at most two further ones, lie in the same typical Bruhat cell Bw 0 B. In other words, there exist at most one element θ = 1 such that gh α (θ)g −1 ∈ BwB and gh α (θ −1 )g −1 ∈ Bw −1 B for some w = w 0 . Further, we reproduce -hitherto unpublished! -complete proofs of the results from the Thesis of the second author, on the number and depth of degenerations. In particular, we prove all results announced in [28,80], and, in fact, get sharper results, producing explicit lists of possible degenerations. Before, such lists were only available for the two simplest cases where Φ = A l and Φ = C l . These results are instrumental in the work of the first author and Vladimir Nesterov, devoted to the description of orbits of Chevalley groups on pairs of long root tori. download from IP 130.102.42.98. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 388 N. A. VAVILOV AND A. A. SEMENOV 1 As a scientometric curiosity, let us mention that the English translation, and thus also all usual databases, erroneously indicate A. V. Yakovlev as the author of [44]. Licensed to University of Queensland. Prepared on Sun Jun 14 08:50:17 EDT 2015 for download from IP 130.102.42.98. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use LONG ROOT TORI 389we correct a couple of petty inaccuracies 2 in the analysis of the cases where Φ = B l , F 4 . Therefore, we opted for inclusion of all details of calculations. Actually, one could write the proof in a more uniform way, skipping some of the case by case analysis, but a thorough understanding of the arising unipotent radicals is interesting in itself and is of vital importance for future applications, in particular, in the forthcoming papers by the first author and Vladimir Nesterov. Therefore, we decided to present all details for each case.From a technical viewpoint, Theorem 1 amounts to describing the Bor...

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Section: Problem 5 Describe Intersections Of Conjugacy Classes Of Chmentioning

confidence: 98%

“…In fact, devising such a conceptual proof would be an important step towards the solution of the following problem, whose systematic study was initiated by Erich Ellers and Nikolai Gordeev [58,59,60]. What is worse, we do not have any idea what such a proof would look like.…”

Section: Problemmentioning

confidence: 99%

Long root tori in Chevalley groups

Vavilov

Semenov

2013

St. Petersburg Math. J.

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“…In addition, the highest weights occurring with multiplicity 1 generate a finite index sub-lattice among those integral weights λ such that m γ λ = −λ and −w 0 λ = λ ( [5,12] We recall the alternative approach to strata in [29, §2]. The G-orbits of pairs of Borel subgroups in G are parametrized by the elements of W .…”

Section: Remark 54 For K = C and γ A Spherical Conjugacy Class M γ mentioning

confidence: 99%

Lusztig’s partition and sheets (with an Appendix by M. Bulois)

Carnovale

2015

Mathematical Research Letters

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We show that, for a connected reductive algebraic group G over an algebraically closed field of zero or good characteristic, the parts, called strata, in the partition of G recently introduced by Lusztig are unions of sheets of conjugacy classes. For G simple and adjoint we refine the parametrization of such sheets obtained in previous work with F. Esposito. We give a simple combinatorial description of strata containing spherical conjugacy classes, showing that Lusztig's correspondence induces a bijection between unions of spherical conjugacy classes and unions of classes of involutions in the Weyl group. Using ideas from the Appendix by M. Bulois, we show that the closure of a stratum is not necessarily a union of strata.

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“…The structure of R(O) along with its universal cover R(Õ) were given in [12]. ,2m 2 +1,...,2m l +1,0,...,0) .…”

Section: Dual Pair Correspondencementioning

confidence: 99%

Regular functions of symplectic spherical nilpotent orbits and their quantizations

Wong

2015

Represent. Theory

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Abstract. We study the ring of regular functions of classical spherical orbits R(O) for G = Sp(2n, C). In particular, treating G as a real Lie group with maximal compact subgroup K, we focus on a quantization model of O when O is the nilpotent orbit (2 2p 1 2q ). With this model, we verify a conjecture by McGovern and another conjecture by Achar and Sommers related to the character formula of such orbits. Assuming the results in a preprint of Barbasch, we will also verify the Achar-Sommers conjecture for a larger class of nilpotent orbits.

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On the coordinate ring of spherical conjugacy classes

Abstract: Let G be a simple algebraic group over an algebraically closed field k of characteristic zero and O be a spherical conjugacy class of G. We determine the decomposition of the coordinate ring k [O] of O into simple G-modules.

Cited by 16 publications

References 41 publications

Long root tori in Chevalley groups

Long root tori in Chevalley groups

Lusztig’s partition and sheets (with an Appendix by M. Bulois)

Regular functions of symplectic spherical nilpotent orbits and their quantizations

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