Wonderful subgroups of reductive groups and spherical systems

Bravi, Paolo; Pezzini, Guido

doi:10.1016/j.jalgebra.2014.03.018

Cited by 22 publications

(30 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, the set ∆ corresponds to a minimal surjective equivariant morphism with connected fibers of type L in the sense of [5,Proposition 2.3.5]. In particular, the minimal quotients of higher defect have been studied in [7,Section 5.3]. Let us recall their description.…”

Section: Parabolic Inductions and Trivial Factorsmentioning

confidence: 99%

“…, W d in Lie Q u . One has to consider the set S ∆ , whose general definition involves the notion of external negative color (see [5,Section 2.3.5] and [7,Section 5.2]). Without going into technical details, in our cases it holds…”

Section: Parabolic Inductions and Trivial Factorsmentioning

confidence: 99%

“…The centralizer of e is K e = L e Q u where L e ∼ = SO(2n − 1). 7. so(2p + 2q + 1)/so(2p + 1) + so(2q).…”

Section: Collapsed Tails Of Type Dmentioning

confidence: 99%

See 2 more Smart Citations

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

Bravi¹,

Chirivì²,

Gandini³

2017

Kyoto J. Math.

View full text Add to dashboard Cite

show abstract

Section: Parabolic Inductions and Trivial Factorsmentioning

confidence: 99%

Section: Parabolic Inductions and Trivial Factorsmentioning

confidence: 99%

See 1 more Smart Citation

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

Bravi¹,

Chirivì²,

Gandini³

2017

Kyoto J. Math.

View full text Add to dashboard Cite

show abstract

“…(2) Then one classifies spherical subgroups H of G. This classification is now complete in characteristic 0 by work of Luna [Lun], Losev [Los], Cupit-Foutou [CuFou], and Bravi and Pezzini [BrPe1], [BrPe2], [BrPe3]. But it is still open in positive characteristic.…”

Section: Introductionmentioning

confidence: 99%

Spherical Spaces

Wedhorn¹

2018

Ann. inst. Fourier

View full text Add to dashboard Cite

The notion of a spherical space over an arbitrary base scheme is introduced as a generalization of a spherical variety over an algebraically closed field. It is studied how the sphericity condition behaves in families. In particular it is shown that sphericity of subgroup schemes is an open and closed condition over arbitrary base schemes generalizing a result by Knop and Röhrle. Moreover spherical embeddings are classified over arbitrary fields generalizing and simplifying results by Huruguen.

show abstract

“…Our main results are the definition of a combinatorial object called the homogeneous spherical datum of G/H (see Definition 17.3), generalizing the one given in [Lu01] in the finite-dimensional setting, and the proof that this datum satisfies the same combinatorial properties as in the finite-dimensional case (see Theorem 17.5). We recall that these objects classify finite-dimensional spherical homogeneous spaces (see the papers [Lu01,Lo09,Bra09,BP14,BP16], and also [Cu09] for a different approach).…”

Section: Introductionmentioning

confidence: 99%

Spherical subgroups of Kac–Moody groups and transitive actions on spherical varieties

Pezzini

2017

Advances in Mathematics

View full text Add to dashboard Cite

We define and study spherical subgroups of finite type of a Kac-Moody group. In analogy with the standard theory of spherical varieties, we introduce a combinatorial object associated with such a subgroup, its homogeneous spherical datum, and we prove that it satisfies the same axioms as in the finite-dimensional case. Our main tool is a study of varieties that are spherical under the action of a connected reductive group L, and come equipped with a transitive action of a group containing L as a Levi subgroup.1 is a nilpotent Lie subalgebra of p, and we denote by U α = Exp( n>0 p nα ) the corresponding unipotent subgroup of P .We come to our crucial assumption: unless otherwise stated, in the whole paper H denotes a subgroup of P such that P/H is a spherical L-variety, i.e. it has a dense B-orbit. Invariants of spherical varietiesWe begin by recalling some notions and notations of the standard theory of spherical varieties. With any B-variety X one associates the latticewhose rank is by definition the rank of X, and the vector spaceAny discrete valuation v of C(X) can be restricted to the subset C(X) (B) . If X has an open B-orbit then this yields a well-defined element ρ(v) of N B (X), by identifying Ξ B (X) with the multiplicative group C(X) (B) modulo the constant functions. If X is normal then this applies in particular to the valuation associated with any B-stable prime divisor D of X, and in this case we denote simply by ρ(D) (or ρ X (D)) the corresponding element of N B (X).These definitions apply in particular if X is a spherical L-variety. In this case the set D(X) B \ D(X) L is called the set of colors (or L-colors) of X, and is denoted by ∆ L (X). If K is the stabilizer in L of a point in the open L-orbit of X, then X is called an embedding of its open orbit L/K. Intersecting with L/K gives a bijection between ∆ L (X) and D(L/K) B . Preliminaries on the geometry of P/HThe case where H is not contained in any proper parabolic subgroup of P will have a prominent role in the paper. We discuss in this section some consequences of this assumption.4.1. Definition. Let G be a connected reductive algebraic group, and K a subgroup. Then K is very reductive if it is not contained in any parabolic subgroup of G.Recall that a very reductive subgroup of a reductive group is reductive.The following first lemma is known, and will be useful in later proofs. We recall that, given G a reductive group and V a G-module, if V is a spherical G-variety then it is also called a spherical module. Lemma.Assume that H is not contained in any parabolic subgroup of P . Then H u ⊆ P u , and up to conjugating H in P we can assume that H has a Levi subgroup K contained in L. Under this assumption we have that (1) K is a spherical very reductive subgroup of L, and the homogeneous space P/H has a unique closed L-orbit isomorphic to L/K; 4

show abstract

Wonderful subgroups of reductive groups and spherical systems

Cited by 22 publications

References 19 publications

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

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

Spherical Spaces

Spherical subgroups of Kac–Moody groups and transitive actions on spherical varieties

Contact Info

Product

Resources

About