On divisible weighted Dynkin diagrams and reachable elements

Let D(e) denote the weighted Dynkin diagram of a nilpotent element $e$ in complex simple Lie algebra \g. We say that D(e) is divisible if D(e)/2 is again a weighted Dynkin diagram. (That is, a necessary condition for divisibility is that $e$ is even.) The corresponding pair of nilpotent orbits is said to be friendly. In this note, we classify the friendly pairs and describe some of their properties. We also observe that any subalgebra sl(3) in \g determines a friendly pair. Such pairs are called A2-pairs. It turns out that the centraliser of the lower orbit in an A2-pair has some remarkable properties. Let $Gx$ be such an orbit and $h$ a characteristic of $x$. Then $h$ determines the Z-grading of the centraliser $z=z(x)$. We prove that $z$ is generated by the Levi subalgebra $z(0)$ and two elements in $z(1)$. In particular, (1) the nilpotent radical of $z$ is generated by $z(1)$ and (2) $x\in [z,z]$. The nilpotent elements having the last property are called reachable.Comment: 17 pages; v2 minor corrrections; final version, to appear in Transformation Groups (2010

On the structure of Borel stable abelian subalgebras in infinitesimal symmetric spaces

Let g=g_0+g_1 be a Z_2-graded Lie algebra. We study the posets of abelian subalgebras of g_1 which are stable w.r.t. a Borel subalgebra of g_0. In particular, we find out a natural parametrization of maximal elements and dimension formulas for them. We recover as special cases several results of Kostant, Panyushev, Suter.Comment: Latex file, 35 pages, minor corrections, some examples added. To appear in Selecta Mathematic

Resolution of null fiber and conormal bundles on the Lagrangian Grassmannian

We study the null fiber of a moment map related to dual pairs. We construct an equivariant resolution of singularities of the null fiber, and get conormal bundles of closed $K_C$-orbits in the Lagrangian Grassmannian as the categorical quotient. The conormal bundles thus obtained turn out to be a resolution of singularities of the closure of nilpotent $K_C$-orbits, which is a "quotient" of the resolution of the null fiber.Comment: 17 pages; completely revised and add reference

Affine spherical homogeneous spaces with good quotient by a maximal unipotent subgroup

For an affine spherical homogeneous space G/H of a connected semisimple algebraic group G, we consider the factorization morphism by the action on G/H of a maximal unipotent subgroup of G. We prove that this morphism is equidimensional if and only if the weight semigroup of G/H satisfies some simple condition.Comment: v2: title and abstract changed; v3: 16 pages, minor correction