Dmitri I. Panyushev

Abelian ideals of a Borel subalgebra and long positive roots

Let b be a Borel subalgebra of a simple Lie algebra g. Let Ab denote the set of all Abelian ideals of b. It is easily seen that any a ∈ Ab is actually contained in the nilpotent radical of b. Therefore, a is determined by the corresponding set of roots. More precisely, let t be a Cartan subalgebra of g lying in b and let ∆ be the root system of the pair (g, t). Choose ∆, the system of positive roots, so that the roots of b are positive. Then a = ⊕γ∈Igγ, where I is a suitable subset of ∆ and gγ is the root space for γ ∈ ∆. It follows that there are finitely many Abelian ideals and that any question concerning Abelian ideals can be stated in terms of combinatorics of the root system. An amazing result of D. Peterson says that the cardinality of Ab is 2rk . His approach uses a one-to-one correspondence between the Abelian ideals and the so-called minuscule elements of the affine Weyl group Ŵ. An exposition of Peterson’s results is found in [5]. Peterson’s work appeared to be the point of departure for active recent investigations of Abelian ideals, ad-nilpotent ideals, and related problems of representation theory and combinatorics [1, 2, 3, 4, 5, 6, 7, 8]. We consider Ab as poset with respect to inclusion, the zero ideal being the unique minimal element of Ab. Our goal is to study this poset structure. It is easily seen that Ab is a ranked poset; the rank function attaches to an ideal its dimension. It was shown in [8] that there is a one-to-one correspondence between the maximal Abelian ideals and the long simple roots of g. (For each simple Lie algebra, the maximal Abelian ideals were determined in [10].) This correspondence possesses a number of nice properties, but the very existence of it was demonstrated in

On covariants of reductive algebraic groups

Recently, Lehrer and Springer have proved the surjectivity of some natural map associated with covariants of a finite reflection group in a complex vector space, see [LS, Theorem A]. The aim of this note is to show that a similar statement is valid in a greater generality; namely, for an arbitrary action of a reductive algebraic group G on an affine variety X and for a sufficiently good Gorbit (e.g. closed) in X. We also demonstrate some invariant-theoretic applications of it. The ground field k is algebraically closed and of characteristic zero. Let X be an affine variety, with coordinate ring k[X], which is acted upon by a reductive algebraic group G. For any (finite-dimensional) G-module M, the space k[X] 18 M is being identified with the space of all polynomial morphisms from X to M, denoted by P(X, M). Under this identification, f @ m cf E k[X], m E M) determines the mapping that takes x E X tof(x)m E M. The group G acts on k[X] by (g.f)(x) =f(g-‘.x). This yields a natural G-module structure on k[X] @ M. Furthermore, the subset of G-invariant elements, denoted by (k[X] 18 M)Gor P,(X, M), is nothing but the set of G-equivariant polynomial morphisms from X to M. Clearly, P,(X, M) IS a module over k[XIG, the module of covariunts (of type A4). For any x E X, there is the ‘evaluation’ map:

The exterior algebra and “spin” of an orthogonal\(\mathfrak{g}\)-module

AbstractA well known result of B. Kostant gives a description of theG-module structure for the exterior algebra of the Lie algebra

Normalizers of ad-nilpotent ideals

\mathfrak{g}

On homogeneous spaces of rank one

. We give a generalization of this result for the isotropy representations of symmetric spaces. If

Weight posets associated with gradings of simple Lie algebras, Weyl groups, and arrangements of hyperplanes

\mathfrak{g} = \mathfrak{g}_0 \oplus \mathfrak{g}_1