K-orbits on the flag variety and strongly regular nilpotent matrices (original) (raw)
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The Gelfand–Zeitlin integrable system and K-orbits on the flag variety
Progress in Mathematics, 2014
In this expository paper, we provide an overview of the Gelfand-Zeiltin integrable system on the Lie algebra of n × n complex matrices gl(n, C) introduced by Kostant and Wallach in 2006. We discuss results concerning the geometry of the set of strongly regular elements, which consists of the points where Gelfand-Zeitlin flow is Lagrangian. We use the theory of K n = GL(n−1, C)×GL(1, C)-orbits on the flag variety B n of GL(n, C) to describe the strongly regular elements in the nilfiber of the moment map of the system. We give an overview of the general theory of orbits of a symmetric subgroup of a reductive algebraic group acting on its flag variety, and illustrate how the general theory can be applied to understand the specific example of K n and GL(n, C).
On the Nilpotency of Certain Subalgebras of Kac-Moody Lie Algebras
Journal of Lie Theory, 2004
Let g = n − ⊕ h ⊕ n + be an indecomposable Kac-Moody Lie algebra associated with the generalized Cartan matrix A = (a ij) and W be its Weyl group. For w ∈ W , we study the nilpotency index of the subalgebra S w = n + ∩ w(n −) and find that it is bounded by a constant k =k(A) which depends only on A but not on w for all A = (a ij) finite, affine of type other than E or F and indefinite type with |a ij | ≥ 2. In each case we find the best possible bound k. In the case when A = (a ij) is hyperbolic of rank two we show that the nilpotency index is either 1 or 2.
On the existence of smooth orbital varieties in simple Lie algebras
Journal of the London Mathematical Society, 2019
The orbital varieties are the irreducible components of the intersection between a nilpotent orbit and a Borel subalgebra of the Lie algebra of a reductive group. There is a geometric correspondence between orbital varieties and irreducible components of Springer fibers. In type A, a construction due to Richardson implies that every nilpotent orbit contains at least one smooth orbital variety and every Springer fiber contains at least one smooth component. In this paper, we show that this property is also true for the other classical cases. Our proof uses the interpretation of Springer fibers as varieties of isotropic flags and van Leeuwen's parametrization of their components in terms of domino tableaux. In the exceptional cases, smooth orbital varieties do not arise in every nilpotent orbit, as already noted by Spaltenstein. We however give a (non-exhaustive) list of nilpotent orbits which have this property. Our treatment of exceptional cases relies on an induction procedure for orbital varieties, similar to the induction procedure for nilpotent orbits.
On the variety of Lagrangian subalgebras, II
Annales Scientifiques de l’École Normale Supérieure, 2006
VERSION FRANÇ AISE: Motivé par le théorème de Drinfeld sur les espaces de Poisson homogènes, nousétudions la variété L des sous-algèbres de Lie Lagrangiennes de g ⊕ g pour g, une algèbre de Lie complexe semisimple. Soit G le groupe adjointe de g. Nous montrons que les adhérences des (G × G)-orbites dans L sont les variétés sphériques et lisses. Aussi, nous classifions les composantes irréductibles de L et nous montrons qu'elles sont lisses. Nous employons quelques méthodes de M. Yakimov pour donner une nouvelle description et une nouvelle preuve de la classification de Karolinsky des orbites diagonales de G dans L, quel, comme cas spécial, donne la classification de Belavin-Drinfeld des r-matrices quasitriangulaires de g. En outre, L possède une structure de Poisson canonique, et nous calculons son rang a chaque point and nous décrivons sa décomposition en feuilles symplectiques en termes des intersections des orbites des deux sous-groupes de G × G.
B-Orbits of Square Zero in Nilradical of the Symplectic Algebra
Transformation Groups, 2016
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.
On commuting varieties of nilradicals of Borel subalgebras of reductive Lie algebras
Eprint Arxiv 1209 1289, 2012
Let GGG be a connected reductive algebraic group defined over an algebraically closed field mathbbmk\mathbbm kmathbbmk of characteristic zero. We consider the commuting variety mathcalC(mathfraku)\mathcal C(\mathfrak u)mathcalC(mathfraku) of the nilradical mathfraku\mathfrak umathfraku of the Lie algebra mathfrakb\mathfrak bmathfrakb of a Borel subgroup BBB of GGG. In case BBB acts on mathfraku\mathfrak umathfraku with only a finite number of orbits, we verify that mathcalC(mathfraku)\mathcal C(\mathfrak u)mathcalC(mathfraku) is equidimensional and that the irreducible components are in correspondence with the {\em distinguished} BBB-orbits in mathfraku\mathfrak umathfraku. We observe that in general mathcalC(mathfraku)\mathcal C(\mathfrak u)mathcalC(mathfraku) is not equidimensional, and determine the irreducible components of mathcalC(mathfraku)\mathcal C(\mathfrak u)mathcalC(mathfraku) in the minimal cases where there are infinitely many BBB-orbits in mathfraku\mathfrak umathfraku.
A LiE Subroutine for Computing Prehomogeneous Spaces Associated with Real Nilpotent Orbits
Lecture Notes in Computer Science, 2005
We develop a LiE subroutine to compute the irreducible components of certain prehomogeneous spaces that are associated with complex nilpotent orbits. An understanding of these spaces is necessary for solving more general problems in the theory of nilpotent orbits and the representation theory of Lie groups. The output is a set of L A T E X statements that can be compiled in a L A T E X environment in order to produce tables. Although the algorithm is used to solve the problem in the case of exceptional complex reductive Lie groups [2], it does describe these prehomogeneous spaces for the classical cases also. Complete tables for the exceptional groups can be found at
Progress in Mathematics, 2018
Let G be any connected and simply connected complex semisimple Lie group, equipped with a standard holomorphic multiplicative Poisson structure. We show that the Hamiltonian flows of all the Fomin-Zelevinsky twisted generalized minors on every double Bruhat cell of G are complete in the sense that all the integral curves of their Hamiltonian vector fields are defined on C. It follows that the Kogan-Zelevinsky integrable systems on G have complete Hamiltonian flows, generalizing the result of Gekhtman and Yakimov for the case of SL(n, C). We in fact construct a class of integrable systems with complete Hamiltonian flows associated to generalized Bruhat cells which are defined using arbitrary sequences of elements in the Weyl group of G, and we obtain the results for double Bruhat cells through the so-called open Fomin-Zelevinsky embeddings of (reduced) double Bruhat cells in generalized Bruhat cells. The Fomin-Zelevinsky embeddings are proved to be Poisson, and they provide global coordinates on double Bruhat cells, called Bott-Samelson coordinates, in which all the Fomin-Zelevinsky minors become polynomials and the Poisson structure can be computed explicitly.