Generalization of the Bruhat Decomposition

Abstract

The problem of describing adjacency on the set of orbits of a Borel subgroup B of a reductive group G acting on a spherical variety (that is, a G-variety with a finite number of S-orbits) is considered. The adjacency relation on the set of B-orbits generalizes the classical Bruhat order on the Weyl group. For a special class of homogeneous spherical varieties G/H , where H is a product of a maximal torus and the commutator subgroup of a maximal unipotent subgroup of the group G , a satisfactory description of the set of 5-orbits with adjacency relation is obtained.

The components of the singular locus of a component of a Springer fiber over x^2 = 0

“…2 and this is the unique element of I n,k satisfying ω o ≤ σ for any σ ∈ I n,k . By this theorem and [13]…”

Finite Parabolic Conjugation on Varieties of Nilpotent Matrices

“…The concrete minimal degenerations are obtained easily from Proposition 5.1. Furthermore, each minimal degeneration is of codimension 1 (which is, as well, clear from the theory of spherical varieties, see [7]; a concrete proof is given in [22]).…”

Orbits of strongly solvable spherical subgroups on the flag variety

“…Since it is homogeneous under the action of a solvable group, B/H is smooth and affine (see e.g. [34,Lemma 2.12]). Notice also that B/H is irreducible since B is connected.…”

“…There is a natural bijection between the set of reduced pairs and the set of B-orbits in G/H. A very special example of a strongly spherical subgroup was treated by Timashev in [34], where the corresponding set of B-orbits is studied. More precisely, let U ′ be the derived subgroup of U , namely U ′ = Φ + ∆ U α , and consider the subgroup T U ′ ⊂ G: this is a spherical subgroup of G contained in B, and we have equalities Ψ = Ψ = ∆.…”

“…see[34]). Let U ′ = α∈Φ + ∆ U α be the derived subgroup of U , then H = T U ′ is a spherical subgroup of G which is contained in B.…”