The result contained in this paper is an application of a fixed point formula associated with Hamiltonian loop group action. We obtain a G-space which is a geometric dual of the Verlinde's fusion product. Also we obtain a proof of Verlinde formula. The calculation of weights at the fixed point set is based on fundamental properties of root systems of affine Lie algebras.
This paper was written in early 1997, the current version has a computer time-stamp 4/4/97.
We prove that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We then describe the topology of the regular and singular fibres, in particular calculating their middle Betti numbers. For the example of sl(2,C) we compute the Fukaya--Seidel category of Lagrangian vanishing cycles.
We prove that every orbit of the adjoint representation of any connected reductive algebraic group $G$ is a rational algebraic variety. For complex simply connected semisimple $G$, this implies rationality of affine Hamiltonian $G$-varieties (which we classify).
We present a Macaulay2 package that produces compactifications of adjoint orbits and of the fibres of symplectic Lefschetz fibrations on them. We use Macaulay2 functions to calculate the corresponding Hodge diamonds, which then reveal topological information about the Lefschetz fibrations.
This work is concerned with Mishchenko-Fomenko subalgebras and their restrictions to the adjoint orbits in a finite-dimensional complex semisimple Lie algebra. In this setting, it is known that each Mishchenko-Fomenko subalgebra restricts to a completely integrable system on every orbit in general position. We improve upon this result, showing that each Mishchenko-Fomenko subalgebra yields a completely integrable system on all regular orbits (i.e. orbits of maximal dimension). Our approach incorporates the theory of regular $\mathfrak{sl}_2$-triples and associated Slodowy slices, as developed …
A recent theorem of [GGSM1] showed that adjoint orbits of semisimple Lie algebras have the structure of symplectic Lefschetz fibrations. We investigate the behaviour of their fibrewise compactifications. Expressing adjoint orbits and fibres as affine varieties in their Lie algebra, we compactify them to projective varieties via homogenisation of the defining ideals. We find that their Hodge diamonds vary wildly according to the choice of homogenisation, and that extensions of the potential to the compactification must acquire degenerate singularities.
We study the hyperkaehler geometry of a regular semisimple adjoint orbit of SL(k,C) via the algebraic geometry of the corresponding reducible spectral curve.
A short note to show that the elements of the (open) cone underlying a nilpotent orbit on a period domain are pairwise congruent under the symmetry group of the period domain.
We studied an enhanced adjoint action of the general linear group on a product of its Lie algebra and a vector space consisting of several copies of defining representations and its duals. We determined regular semisimple orbits (i.e., closed orbits of maximal dimension) and the structure of enhanced null cone, including its irreducible components and their dimensions.
We explicitly construct a set of quadratic equations defining the highest weight vector orbit for adjoint representations of Chevalley groups of types D_l, E_6, E_7, and E_8. The combinatorics of these equations is related to the combinatorics of embeddings of the root system of type A_3. We believe that the constructed equations provide a prominent framework for calculations with exceptional groups in adjoint representations, which is particularly interesting for groups of type E_8.
This expository article is an introduction to the adjoint orbits of complex semisimple groups, primarily in the algebro-geometric and Lie-theoretic contexts, and with a pronounced emphasis on the properties of semisimple and nilpotent orbits. It is intended to build a foundation for more specialized settings in which adjoint orbits feature prominently (ex. hyperkähler geometry, Landau-Ginzburg models, and the theory of symplectic singularities). Also included are a few arguments and observations that, to the author's knowledge, have not yet appeared in the research literature.
[GGSM2] showed that height functions give adjoint orbits of semisimple Lie algebras the structure of symplectic Lefschetz fibrations (superpotential of the LG model in the language of mirror symmetry). We describe how to extend the superpotential to compactifications. Our results explore the geometry of the adjoint orbit from 2 points of view: algebraic geometry and Lie theory.
We try to write (generic) elements of classical simple Lie algebras as sums of as few elements of the orbit C of long root vectors as possible. If the Lie algebra in question is sl_n or sp_n, then C consists of all rank 1 matrices in the Lie algebra, and an element of rank k is the sum of k elements of C. If, on the other hand, the Lie algebra is o_n, then C consists of all rank 2 matrices with square zero, and the situation is much more complicated. We explicitly describe, in this case as well, the closure of the set of all sums of k elements of C, also known as the (k-1)st secant variety of …
Algebraic hyperbolicity of adjoint linear systems on spherical varieties
Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Department of Mathematical Sciences, KAIST（韩国科学技术院数学科学系）；Center for Complex Geometry, Institute for Basic Science (IBS)（基础科学研究院复杂几何中心）
We give a brief introduction to the Berline-Vergne localization formula for the finite-dimensional setting and indicate how the Duistermaat-Heckman formula is derived from it. We consider applications of the localization formula when it is specialized to a maximal dimensional co-adjoint orbit. In particular, the case when the co-adjoint orbit is a quotient $G/T$ of a connected Lie group $G$ modulo a maximal torus $T$ is analyzed in detail. We describe also a generalization of the localization formula to non-compact group actions.
