Long Abelian ideals

We study Abelian ideals of a Borel subalgebra consisting of long roots. It is shown that methods of Cellini and Papi can be extended to this situation. A uniform expression for the number of long Abelian ideals is given. We also show that there is a one-to-one correspondence between the long Abelian ideals and B-stable commutative subalgebras in the little adjoint representation of the Langlands dual Lie algebra.     

On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Let be a reductive Lie algebra over an algebraically closed field of characteristic zero and an arbitrary -grading. We consider the variety , which is called the commuting variety associated with the -grading. Earlier it was proved by the author that is irreducible, if the -grading is of maximal rank. Now we show that is irreducible for and (E₆,F₄). In the case of symmetric pairs of rank one, we show that the number of irreducible components of is equal to that of nonzero non--regular nilpotent G ₀-orbits in . We also discuss a general problem of the irreducibility of commuting varieties.     

Weight Multiplicity Free Representations, g-Endomorphism Algebras, and Dynkin Polynomials

Throughout this paper, G is a connected semisimple algebraicgroup defined over an algebraically closed field k of characteristiczero, and g is its Lie algebra.     

Isotropy Representations, Eigenvalues of a Casimir Element, and Commutative Lie Subalgebras

Let h be a reductive subalgebra of a semisimple Lie algebrag and C_h U(h) be the Casimir element determined by the restrictionof the Killing form on g to h. The paper studies eigenvaluesof C_h on the isotropy representation mg/h. Some general estimatesconnecting the eigenvalues and the Dynkin indices of m are given.If h is a symmetric subalgebra, it is shown that describingthe maximal eigenvalue of C_h on exterior powers of m is connectedwith possible dimensions of commutative Lie subalgebras in m,thereby extending a result of Kostant. In this situation, aformula is also given for the maximal eigenvalue of C_h on m.More generally, a similar picture arises if h = g, where isan automorphism of finite order m and m is replaced by the eigenspaceof corresponding to a primitive mth root of unity.     

Multiplicities of weights of some representations and convex polytopes

Moscow Institute of Engineering, Electronics and Automation. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 28, No. 4, pp. 88–90, October–December, 1994.     

Rationality of singularities and the Gorenstein property for nilpotent orbits

Ordzhonikidze Aviation Institute, Moscow. Translated from Funktsional'yi Analiz i Ego Prilozheniya, Vol. 25, No. 3, pp. 76–78, July–September, 1991.     

On divisible weighted Dynkin diagrams and reachable elements

Let e be a nilpotent element of a complex simple Lie algebra $ \mathfrak{g} Let e be a nilpotent element of a complex simple Lie algebra \mathfrakg \mathfrak{g} . The weighted Dynkin diagram of e, D(e) \mathcal{D}(e) , is said to be divisible if D(e)

The work of Vladimir Morozov on Lie algebras

We survey the works of V. V. Morozov on Lie algebras concentrating on the following three results on subalgebras of a semisimple complex Lie algebra: the theorem on a nilpotent element, the triangulizability theorem for solvable subalgebras, and the regularity theorem for nonsemisimple maximal subalgebras.