Primitive wonderful varieties

Abstract

We complete the classification of wonderful varieties initiated by D. Luna. We review the results that reduce the problem to the family of primitive varieties, and report the references where some of them have already been studied. Finally, we analyze the rest case-by-case

Regular functions on spherical nilpotent orbits in complex symmetric pairs: Classical Hermitian cases

“…This case is equal to the previous one up to an external automorphism of k. We can take e = ϕ 5 ∧ϕ 4 ∧ϕ 3 +ϕ 4 ∧ϕ 3 +ϕ 5 ∧ϕ 2 −ϕ 2 . We have l = k h ∼ = gl(1) ⊕2 ⊕so (8) and k e = l e + n, where l e ∼ = g 2 .…”

“…Take θ such that t θ = t and take the following root vectors for the simple roots of k: We can take e = ϕ 8 . We have l = k h ∼ = gl (8) and k e = l e + n, where l e ∼ = sl(8). We can take e = ϕ 8 ∧ ϕ 7 ∧ ϕ 6 − ϕ 5 .…”

“…We can take e = ϕ 8 ∧ ϕ 7 ∧ ϕ 6 − ϕ 5 . We have l = k h ∼ = gl(4) ⊕ so (8) and k e = l e + n, where l e ∼ = sl(4) ⊕ so (7). We have l = k h ∼ = e 6 ⊕ gl(1) ⊕2 and k e = l e + n, where l e ∼ = gl(1) ⊕ e 6 .…”

The dual group of a spherical variety

“…A G-variety X is called spherical if B has an open orbit in G. Homogeneous spherical varieties have been classified by Luna and Bravi-Pezzini, [Lun01,BP16], in terms of a combinatorial structure called a homogeneous spherical datum. In addition to the based root datum of G, it consists of a quintuple (Ξ, Σ, D, c, M ) where Ξ is a subgroup of Λ (the weight lattice), Σ is a finite subset of Ξ (the spherical roots), D is a finite set (the colors), c is a map D → Ξ ∨ , and M is a subset of D × S. These objects are subject to a number of axioms (see e.g.…”

“…Instead of directly working with spherical varieties, we only study them through an intermediate combinatorial structure which we call a weak spherical datum. This structure is a weakening (whence the name) of the homogeneous spherical datum of [Lun01] which is used to classify homogeneous spherical varieties (by Bravi-Pezzini [BP16]). Additionally, one can associate a weak spherical datum to any G-variety (Proposition 5.4) which widens the scope of our theory to this generality.…”