- A nilpotent orbit is an orbit of the adjoint action of G on g which contains the zero element of g in its closure. For the special linear group 2 G = SLn,C, whose Lie algebra 9 is all n x n matrices with trace zero, an adjoint orbit consists of all matrices with a given Jordan canonical form; such an orbit is nilpotent if the Jordan form has only zeros on the diagonal. primitive ideal, denoted J, and called the Joseph ideal, associated to Omin . It is an important problem to be able to determine “many” com- pletely prime ideals see, for example, [ 16, 171. 1.3. For g2 the nilpotent orbits are of dimension 0, 6, 8, 10, 12. We denote by 0, the unique nilpotent.
This is it the book Nilpotent Orbits, Primitive Ideals, And Characteristic Classes: A Geometric Perspective In Ring Theory Progress In Mathematics, By Walte. W. Borho, J.-L. Brylinski, R. MacPherson: Nilpotent Orbits, Primitive Ideals, and Characteristic Classes—A Geometric Perspective in Ring Theory Progress in Math. DOI: 10.1090/S0273-0979-1992-00271-9 Corpus ID: 11873593. Nilpotent orbits, normality and Hamiltonian group actions @inproceedingsBrylinski1992NilpotentON, title=Nilpotent orbits, normality and Hamiltonian group actions, author=Ranee Brylinski and Bertram Kostant, year=1992 . In chapter 1, we study the Springer correspondence for g when G is of type B, C or D p = 2. The correspondence is a bijective map from W to 2t. In particular, we classify nilpotent G-orbits in g type B, D over finite fields of characteristic 2. In chapter 2, we study the Springer correspondence for g when G is of type B, C or D p = 2.
This project is in pure mathematics, at the interface between the fields of representation theory, Lie theory, and algebraic geometry. For 50 years, the nilpotent orbits have become classical objects of study for their geometric properties and their role in representation theory. These two aspects are the main directions of the project. Sep 13, 2002 · Let X be an F-rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of X is F-split. This property has several consequences. When F is complete with respect to a discrete valuation with either finite or. algebraically closed ﬁeld of characteristic 2. We construct a Springer corre-spondence for the Lie algebra of G. In particular, for orthogonal Lie algebras in characteristic 2, the structure of component groups of nilpotent centralizers is determined and the number of nilpotent orbits over ﬁnite ﬁelds is obtained. 1. Introduction.
Apr 01, 1988 · There is a unique completely prime primitive ideal, denoted J0 and called the Joseph ideal, associated to Omin . It is an important problem to be able to determine "many" completely prime ideals see, for example, [16, 17]. 1.3. For g2 the nilpotent orbits are of dimension 0, 6, 8, 10, 12. Download Representations and Nilpotent Orbits of Lie Algebraic Systems Progress in Mathematics or any other file from Books category. HTTP download also available at fast speeds. Primitive ideals Invariant theory Geometry of Lie group actions Quantum affine algebras Yangians. characteristics; moreover, their centralizers and conjugacy classes have been worked out rather uniformly. However, unipotent elements other than 1 di er considerably in characteristics 0 and p > 0: they have in nite order in G in the former case, but are characterized in the latter case as the elements of G having order equal to some power of p.
3. Nilpotent Orbits There is a natural partial order on the set of nilpotent orbits. For any two nilpotent orbits Oand O0, de ne O O0if O O0. Using this partial order we can identify certain orbits of interest. 1 There is the zero orbit consisting of a single element 0. This is the smallest orbit in the partial ordering. Through the 1990s, a circle of ideas emerged relating three very different kinds of objects associated to a complex semisimple Lie algebra: nilpotent orbits, representations of a Weyl group, and primitive ideals in an enveloping algebra. This volume, a celebration of Anthony Joseph’s fundamental influence on classical and quantized representation theory, explores a wide array of current topics in Lie theory by experts in the area. The chapters are based on the 2017 conferences titled Algebraic Modes of Representations. Orbits of Nilpotent Matrices D. Z. Djokovi Department of Pure Mathematics University of Waterloo Waterloo, Ontario, Canada N2L 3G1 and J. Malzan Department of Mathematics University of Toronto Toronto, Ontario, Canada M5S IA1 Submitted by Hans Schneider ABSTRACT The action under conjugation of invertible lower triangular n X n matrices over an infinite field on lower triangular nilpotent.
Get this from a library! Nilpotent orbits, primitive ideals, and characteristic classes: a geometric perspective in ring theory. [W Borho; J -L Brylinski; Robert MacPherson] -- 1. The Subject Matter. Consider a complex semisimple Lie group G with Lie algebra g and Weyl group W. In this book, we present a geometric perspective on the following circle of ideas: polynomials. The nilpotent onec Nˆg is one of the bres of the adjoint quotient, namely N:= ˜ 10. Since it is de ned by the anishingv of some homogeneous polynomials, Nis stable under scalar multiplication this is the sense in which it is a cone. By Theorem 1.4, Nis the union of nitely many adjoint orbits, which are called the nilpotent orbits of g. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.
Nilpotent orbits are finite in number. The Zariski closure of a nilpotent orbit is a union of nilpotent orbits. Jacobson–Morozov theorem: over a field of characteristic zero, any nilpotent element e can be included into an sl 2-triple e,h,f and all such triples are conjugate by Z G e, the centralizer of e in G. We consider the closed subscheme of the nilpotent variety of Mr over K defined by the conditions char_AT=T^r, A^e=0. We prove that when the characteristic of K is zero this scheme is reduced. Jun 08, 1999 · Recently, V.Ginzburg introduced and studied in depth the notion of a principal nilpotent pair in a semisimple Lie algebra \\g. Our aim is to contribute to the general theory of nilpotent pairs. Roughly speaking, a nilpotent pair e_1,e_2 consists of two commuting elements in \\g that can independently be contracted to the origin. A principal nilpotent pair is a double counterpart of a. - A nilpotent orbit is an orbit of the adjoint action of G on g which contains the zero element of g in its closure. For the special linear group 2 G = SLn,C, whose Lie algebra 9 is all n x n matrices with trace zero, an adjoint orbit consists of all matrices with a given Jordan canonical form; Read more.
Recall also that under our assumptions there exists a canonical bijection from the set of nilpotent orbits of G K in g K to the set of nilpotent G F -orbits in g F, see [AHR1, §4.1] for. A quantization of conjugacy classes of matrices. Adv. Math. 196 2005, no. 1, 124–146. For a general $\mathfrakg,$ this amounts to the induction of zero-dimensional orbits and to the correspondence between semisimple and regular orbits. In particular, every Richardson nilpotent orbit can be obtained as a degeneration in the same way. being nilpotent is a polynomial condition. Use the Zarinski topology. Nilpotent orbits form a stratification of N: every nilpotent matrix is in exactly one conjugacy class stratum, and the closure of a stratum is a union of strata. dimg Partial Ordering of Nilpotent orbits: OA OB OA⊆OB Analytically: rankAk rankB≤ k for all k>0.
Sorry, our data provider has not provided any external links therefore we are unable to provide a link to the full text. CiteSeerX - Document Details Isaac Councill, Lee Giles, Pradeep Teregowda: Abstract. Let G be an adjoint algebraic group of type B, C, orD over an algebraically closed field of characteristic 2. We construct a Springer correspondence for the Lie algebra of G. In particular, for orthogonal Lie algebras in characteristic 2, the structure of component groups of nilpotent centralizers is. of characteristic 2, F q denotes a finite field of characteristic 2, and F q denotes an algebraic closure of F q. In ref. 1, Hesselink determines the nilpotent orbits in classical Lie algebras under the adjoint action of classical groups over k. In ref. 2, Spaltenstein gives a parametrization of these nilpotent orbits by pairs of partitions. We. with equivalence classes of nilpotent orbits for strong involutions of G, -y corresponding to the same inner class of holomorphic involutions. The notion of equivalence classes of nilpotent orbits for strong involutions of G, -ywill be explained in detail in section 3.1..
The character map for a nilpotent orbit of G follows directly from its Characteristic Dynkin:1957um , and we use Characteristics to label nilpotent orbits. The nilpotent orbits of a group can be organised into a Hasse diagram Kraft:1982fk , which displays their partial ordering, in terms of dimensions and moduli space inclusion relations. Oct 19, 2017 · Through the 1990s, a circle of ideas emerged relating three very different kinds of objects associated to a complex semisimple Lie algebra: nilpotent orbits, representations of a Weyl group, and primitive ideals in an enveloping algebra.
In particular, as Jyrki notes, they contain no non-zero nilpotent elements. Thus the set of nilpotents in the quaternions is an ideal, the zero ideal. Regards, $\endgroup$ – Matt E Apr 16 '12 at 4:35. CiteSeerX - Document Details Isaac Councill, Lee Giles, Pradeep Teregowda: Throughout this paper, k denotes an algebraically closed field of characteristic 2, Fq denotes a finite field of characteristic 2 and ¯ Fq denotes an algebraic closure of Fq. In , Hesselink determines the nilpotent orbits in classical Lie algebras under the.
Beyond the subregular case, there are indeed pairs of nilpotent orbits of dimensions differing again by 2, but almost nothing can be said about the geometry of the associated Springer fibers. There have been a couple of dissertations over the years by Lorist, Fung approaching this topic and getting partial results, but so far progress has. SEMINAR: SEMISIMPLE AND NILPOTENT ORBITS IN ALGEBRAIC GROUPS 3 8. Dynkin-Kostant classification II • Explain the strategy to the classiﬁcation of nilpotent orbits • Explain the relation between the set of G ad conjugacy classes of standard triples in g and nilpotent orbits. Jan 29, 2008 · Throughout this paper, k denotes an algebraically closed field of characteristic 2, F q denotes a finite field of characteristic 2, and F ̄ q denotes an algebraic closure of F q. In ref. 1, Hesselink determines the nilpotent orbits in classical Lie algebras under the adjoint action of classical groups over k.In ref. 2, Spaltenstein gives a parametrization of these nilpotent orbits by pairs of. In good prime characteristic, all of this agrees with the classical DynkinKostant theory in characteristic 0 as developed in [2, 3]. We use throughout the language of nilpotent orbits. The orbits are partially ordered in a natural way by the closure ordering, which for partitions in type A ‘ is the usual partial ordering for partitions of. This paper is about nilpotent orbits of reductive groups over local non-Archimedean fields. In this paper we will try to identify for which groups there are only finitely many nilpotent orbits, for which groups the nilpotent orbits are separable and for which groups Howe's conjecture holds. For general reductive groups we get some partial results.
A principal nilpotent pair is a double counterpart of a regular nilpotent element. Consequently, the theory of nilpotent pairs should stand out as double counterpart of the theory of nilpotent orbits. We show that any nilpotent pair has a characteristic h_1,h_2, which is unique within to conjugacy.
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