[Paper Review] Parametrization, structure and Bruhat order of certain spherical quotients

Pierre–Emmanuel Chaput, Lucas Fresse+1 more|arXiv (Cornell University)|Jan 13, 2020

Advanced Algebra and Geometry23 references4 citations

TL;DR

This paper studies the structure and Bruhat order of $Z_G(e)$-orbits on the flag variety $G/B$, where $G$ is a reductive group and $e$ is a nilpotent element of height 2. It establishes a parametrization of orbits via $W^P \times (\mathcal{B}_L / M)$, proves each orbit is an algebraic affine bundle over an $M$-orbit, and shows in type $A$ that orbits admit a natural cell decomposition. The key contribution is a combinatorial description of the strong Bruhat order on these orbits using an abstract order on quotients of Coxeter systems.

ABSTRACT

Let $G$ be a reductive algebraic group and let $Z$ be the stabilizer of a nilpotent element $e$ of the Lie algebra of $G$. We consider the action of $Z$ on the flag variety of $G$, and we focus on the case where this action has a finite number of orbits (i.e., $Z$ is a spherical subgroup). This holds for instance if $e$ has height $2$. In this case we give a parametrization of the $Z$-orbits and we show that each $Z$-orbit has a structure of algebraic affine bundle. In particular, in type $A$, we deduce that each orbit has a natural cell decomposition. In the aim to study the (strong) Bruhat order of the orbits, we define an abstract partial order on certain quotients associated to a Coxeter system. In type $A$, we show that the Bruhat order of the $Z$-orbits can be described in this way.

Motivation & Objective

• To understand the parametrization and geometric structure of $Z_G(e)$-orbits on the flag variety $G/B$ when $Z_G(e)$ is spherical.
• To describe the inclusion relations between orbit closures, particularly in terms of the (strong) Bruhat order.
• To establish a combinatorial model for the Bruhat order on $Z_G(e)$-orbits using quotients of Coxeter systems.
• To show that in type $A$, each $Z_G(e)$-orbit admits a natural cell decomposition.
• To clarify the relationship between $Z_G(e)$-orbits and $N_G(e)$-orbits, showing they coincide under spherical conditions.

Proposed method

• Parametrize $Z_G(e)$-orbits using the product $W^P \times (\mathcal{B}_L / M)$, where $W^P$ is the parabolic quotient of the Weyl group and $\mathcal{B}_L$ is the flag variety of the Levi subgroup $L$.
• Prove that each $Z_G(e)$-orbit is an algebraic affine bundle over an $M$-orbit in $\mathcal{B}_L$, using the Levi decomposition $Z_G(e) = L_Z \ltimes U_Z$.
• Define an abstract partial order on certain quotients of a Coxeter system to model the strong Bruhat order on $Z_G(e)$-orbits.
• Use the tableau criterion and properties of involution-induced orders to compare elements in the Weyl group and relate them to orbit inclusions.
• Leverage results from [BR12] on covering relations in the poset $\mathcal{D}_n$ to characterize the Bruhat order via diagram switches in directed graphs associated to permutations.
• Show that for height 2 nilpotent elements, $U_Z = U$, so $Z_G(e)$ arises via parabolic induction from a symmetric subgroup of $L$, simplifying the structure.

Experimental results

Research questions

• RQ1 How can the $Z_G(e)$-orbits on the flag variety be parametrized when $Z_G(e)$ is spherical?
• RQ2 What is the geometric structure of each $Z_G(e)$-orbit, particularly in terms of fibrations or cell decompositions?
• RQ3 How can the strong Bruhat order on $Z_G(e)$-orbits be described combinatorially?
• RQ4 Under what conditions do $Z_G(e)$-orbits and $N_G(e)$-orbits coincide?
• RQ5 Can the Bruhat order on $Z_G(e)$-orbits be modeled via an abstract order on quotients of Coxeter systems?

Key findings

• The $Z_G(e)$-orbits on $G/B$ are parametrized by $W^P \times (\mathcal{B}_L / M)$, where $W^P$ is the set of minimal length representatives of $W/W_P$ and $\mathcal{B}_L / M$ is the flag variety of $L$ modulo the spherical subgroup $M$.
• Each $Z_G(e)$-orbit admits a structure of algebraic affine bundle over an $M$-orbit in $\mathcal{B}_L$, implying a fibration with affine space fibers.
• In type $A$, each $Z_G(e)$-orbit has a natural cell decomposition, arising from the affine bundle structure and the geometry of $\mathcal{B}_L / M$.
• The strong Bruhat order on $Z_G(e)$-orbits is described via an abstract partial order on quotients of a Coxeter system, generalizing the Bruhat order on the Weyl group.
• For nilpotent elements of height 2, the unipotent radical $U_Z$ coincides with the full unipotent radical $U$ of the parabolic $P$, so $Z_G(e) = L_Z \ltimes U$ with $L_Z$ a symmetric subgroup of $L$.
• The $Z_G(e)$-orbit and $N_G(e)$-orbit sets coincide when $Z_G(e)$ is spherical, though $N_G(e)$ may be spherical even when $Z_G(e)$ is not.