PARABOLIC SUBALGEBRAS, PARABOLIC BUILDINGS AND PARABOLIC PROJECTION

DAVID M. J. CALDERBANK AND PASSAWAN NOPPAKAEW

Abstract. Reductive (or semisimple) algebraic groups, Lie groups and Lie algebras have a rich geometry determined by their parabolic subgroups and subalgebras, which carry the structure of a building in the sense of J. Tits. We present herein an elementary approach to the geometry of parabolic subalgebras, over an arbitrary field of characteristic zero, which does not rely upon the structure theory of semisimple Lie algebras. Indeed we derive such structure theory, from root systems to the Bruhat decomposition, from the properties of parabolic subalgebras. As well as constructing the Tits building of a reductive Lie algebra, we establish a “parabolic projection” process which sends parabolic subalgebras of a reductive Lie algebra to parabolic subalgebras of a Levi subquotient. We indicate how these ideas may be used to study geometric configurations and their moduli.

Parabolic subgroups and their Lie algebras are fundamental in Lie theory and the theory of algebraic groups [2, 16, 17, 19, 21, 25, 29, 35, 38]. They play a key role in combinatorial, differential and integrable geometry through Tits buildings and parabolic invariant theory [1, 3, 5, 8, 31, 33, 36]. Traditional approaches define parabolic subalgebras of semisimple or reductive Lie algebras as those containing a Borel (maximal solvable) subalgebra in some field extension, and typically develop the theory of parabolic subalgebras using the root system associated to a Cartan subalgebra of such a Borel subalgebra. Such approaches are far from elementary, and provide limited insight when the field is not algebraically closed.

The present paper is motivated by a programme to study geometric configurations in projective spaces and other generalized flag manifolds (i.e., adjoint orbits of parabolic subalgebras) using a process called parabolic projection. This process relies upon two observations: first, if 𝔭, 𝔮 are parabolic subalgebras of a (reductive or semisimple) Lie algebra 𝔤, then so is 𝔭 ∩𝔮 +𝔫𝔦𝔩(𝔮) (where 𝔫𝔦𝔩(𝔮) is the nilpotent radical of 𝔮); secondly, if 𝔯 is a Lie subalgebra of 𝔮, and 𝔮 is parabolic subalgebra of 𝔤, then 𝔯 is a parabolic subalgebra of 𝔤 if and only if it is the inverse image of a parabolic subalgebra of the reductive Levi quotient of 𝔮_{0} := 𝔮/ 𝔫𝔦𝔩(𝔮)—thus two senses in which 𝔯 might be called a parabolic subalgebra of 𝔮 coincide. These observations point towards a theory of parabolic subalgebras of arbitrary Lie algebras, which may be developed using straightforward methods of linear algebra over an arbitrary field of characteristic zero, without relying on the structure theory of semisimple Lie algebras. Indeed, much of the latter can be derived as a consequence.

This development combines two approaches to parabolic subalgebras. In the first approach, inspired by work of V. Morozov [22, 23, 27] and other work emphasising the role of nilpotent elements in Lie algebra theory (see also [2, 21, 28, 33]), the Borel subalgebras in the traditional definition of parabolic subalgebras are replaced by the normalizers of maximal nil subalgebras (i.e., subalgebras contained in the nilpotent cone). In the second approach, initiated by A. Grothendieck [15] and developed by F. Burstall [4] and others (see [5, 6, 7, 10]), a subalgebra of a semisimple Lie algebra is parabolic if its Killing perp is a nilpotent subalgebra. The latter definition appears to rely upon the Killing form of a semisimple Lie algebra, but is easily adapted to the reductive case.

The set of parabolic subalgebras of a reductive Lie algebra 𝔤 has a rich structure: it may be decomposed into flag varieties under the action of the adjoint group G of 𝔤, but is also equipped with an incidence geometry relating these varieties. Let us illustrate this in the case that G = PGL(V) is the projective general linear group of a vector space V of dimension n + 1. Among the parabolic subalgebras of 𝔤, the maximal (proper) parabolic subalgebras play a distinguished role. The corresponding flag varieties are the grassmannians Gr_{k}(V) (1 ⩽ k ⩽ n) of k-dimensional subspaces of V. Two such subspaces are incident if one contains the other, and more general parabolic subalgebras (and their flag varieties) can be described using sets of mutually incident subspaces (called “flags”). In particular, the minimal parabolic subalgebras of 𝔤 are the infinitesimal stabilizers of complete flags, which are nested chains of subspaces of V, one of each dimension.

Date: September 2017.

The general picture is similar. Among adjoint orbits of parabolic subalgebras (generalized flag varieties), the maximal ones are distinguished. For algebraically closed fields, these may be identified with the nodes of the Dynkin diagram, and with the nodes of the Coxeter diagram of the (restricted) root system in general. Other parabolic subalgebras may be described by the maximal (proper) parabolic subalgebras containing them (which are mutually incident), so that their adjoint orbits correspond to subsets of nodes of the Coxeter or Dynkin diagram. This situation is abstracted by J. Tits’ theory of buildings [36], which deserves (in our opinion) a more central place in Lie theory and representation theory than it currently enjoys.

Our purpose in this paper is to present a self-contained treatment of parabolic subalgebras and their associated Lie theory, sufficient to give a novel proof that the set of minimal parabolic subalgebras of a Lie algebra are the chambers of a strongly transitive Tits building. In the algebraically closed case, this establishes the conjugacy theorems for Cartan subalgebras and Borel subalgebras, together with the Bruhat decomposition, using no algebraic geometry and very little structure theory. It also provides a natural framework in which to develop the basic properties of parabolic projection. In subsequent work, we shall apply parabolic projection to the construction of geometric configurations and discrete integrable geometries.

Although the results herein are essentially algebraic in nature, we wish to emphasise the geometry behind them. We thus begin in Section 1 by introducing incidence systems, and two examples which we shall use throughout the paper to illustrate the theory. Example 1A is the incidence geometry of vector (or projective) subspaces of a vector (or projective) space over an arbitrary field, while Example 1B is the incidence geometry of isotropic subspaces of a real inner product space of indefinite signature.

Section 2 provides a self-contained treatment of the Lie algebra theory we need, modulo some basic facts about Jordan decompositions and invariant bilinear forms which we summarize in Appendix A. In order to work over an arbitrary field of characteristic zero, we develop the nilpotent–reductive dichotomy for Lie algebras rather than the solvable–semisimple dichotomy. We thus ignore Lie’s theorem, Weyl’s theorem, the Levi–Malcev decomposition, and even most of the representation theory of 𝔰𝔩_{2}. Instead, we emphasise the role played by filtrations and Engel’s theorem, nilpotency ideals and the nilpotent cone, trace forms and Cartan’s criterion (which, in its most primitive form, concerns nilpotency rather than solvability). In Theorem 2.20 we thus establish, in a novel way, the basic result that (in characteristic zero) a Lie algebra is reductive if and only if it admits a nondegenerate trace form. Using this, Proposition 2.22 extends one of Cartan’s criteria from 𝔤𝔩(V) to reductive Lie algebras, a result which we have not been able to find in the literature.

We define parabolic subalgebras in Section 3 as those subalgebras containing the normalizer of a maximal nil subalgebra, but immediately obtain, in Theorem 3.4 several equivalent definitions using “admissible” trace forms and nilpotency ideals, some of which are new, cf. [2, 4]. We then consider pairs of parabolic subalgebras, their “oppositeness”, and their incidence properties. Intersections of opposite minimal parabolic subalgebras define minimal Levi subalgebras, also known as anisotropic kernels, which govern the structure theory of root systems for semisimple Lie algebras in characteristic zero—see Theorem 3.28. The theory of such “restricted” root systems is well known in the context of algebraic groups [35] or when the underlying field is the real numbers [16, 19, 25, 38]; here, though, we are forced to discard concepts such as Cartan decompositions which are particular to the real case.

The core results of the paper appear in Section 4, where we show that the parabolic subalgebras of a reductive Lie algebra 𝔤 form the simplices of a Tits’ building. This theory has a formidable reputation, but has become more approachable in recent years as the key concepts have become better understood. More recent approaches emphasise chamber systems [13, 30, 37, 39] rather than simplicial complexes [14, 36]. In particular, these approaches make explicit the labelling corresponding to the nodes of the Coxeter–Dynkin diagram in the parabolic case. Unfortunately, the modern definition of buildings incorporates an abstraction of the Bruhat decomposition, which is a nontrivial result in representation theory. In order to address these issues, we adopt a concise hybrid approach to buildings, which combines the original viewpoint (using “apartments”) with more recent approaches using chamber systems. Following [1, 30, 39], we develop sufficient theory to derive an abstract Bruhat decomposition (Theorem 4.21) for strongly transitive buildings. We use this to obtain (in Theorem 4.22) the Bruhat decomposition from conjugacy results for minimal parabolic subalgebras and their Levi subalgebras, which we also prove. This result is well-known in the context of algebraic groups, using Tits’ (B, N)-pairs, but our methods are almost entirely different.

In Section 5, we turn finally to parabolic projection, which, for a given fixed parabolic subalgebra 𝔮 ⩽ 𝔤, projects arbitrary parabolic subalgebras of 𝔤 onto parabolic subalgebras of the reductive Levi quotient 𝔮_{0} of 𝔮. This idea was originally developed by A. Macpherson and the first author [20] using root systems and standard parabolic subgroups. The analysis here is based instead on Proposition 3.16 which gives an explicit formula for the projection. The main result is Theorem 5.11, in which parabolic projection is shown to be a morphism of chamber systems when restricted to weakly opposite subalgebras. This is the basis for constructions of geometric configurations (see [24]) that we shall pursue elsewhere.

Acknowledgements. The first author thanks Vladimir Souček, Paul Gauduchon, Robert Marsh, Tony Dooley, Daniel Clarke and Amine Chakhchoukh for helpful discussions, and the Eduard Čech Institute, grant number GA CR P201/12/G028, for financial support. We are also immensely grateful to Fran Burstall and Alastair King for their comments and ideas.

1. Elements of incidence geometry

Incidence geometry is conveniently described using graph theory, where graphs (herein) are undirected with no loops or multiple edges. We use |Γ| and E_{Γ} to denote the vertex and edge sets of a graph Γ; an edge is determined by its two endpoints, so E_{Γ} may be viewed as a collection of two element subsets of |Γ|, or equivalently, a symmetric irreflexive relation on |Γ|. For v, w ∈ |Γ|, we write v—w for the reflexive closure of this relation (which is the structure preserved by graph morphisms). By a “subgraph”, we always mean a subset of vertices with the induced relation.

Definition 1.1 ([31, 32, 39]). An incidence system over a (usually finite) set I is an Imultipartite graph Γ, i.e., a graph equipped with a type function t = t_{Γ} : |Γ| → I such that ∀ v, w ∈ |Γ|, v—w and t(v) = t(w) imply v = w. An incidence morphism Γ_{1} → Γ_{2} of incidence systems over I is a type-preserving graph morphism.

A flag or clique in Γ of type J ∈ P(I) (i.e., J ⊆ I) is a set of mutually incident elements, one of each type j ∈ J (a J-flag). We denote the set of J-flags by FΓ(J); together with the obvious “face maps” FΓ(J_{2}) → FΓ(J_{1}) for J_{1} ⊆ J_{2}, these form an (abstract, I-labelled) simplicial complex FΓ (a presheaf or functor P(I)^{op} → Set) called the flag complex—FΓ is also an incidence system over P(I): two flags are incident iff their union is a flag. A full flag is an I-flag σ ∈ FΓ(I), i.e., σ contains one element of each type j ∈ I.

Example 1A. The proper nonempty subsets B of an n + 1 element set S form an incidence system Γ^{S} over I_{n} := {1, 2, … n}, where t(B) is the number of elements of B, and B_{1}—B_{2} iff B_{1} ⊆ B_{2} or B_{2} ⊆ B_{1}. We “linearize” this example as follows.

Let V be a vector space of dimension n + 1 over a field 𝔽. The proper nontrivial subspaces W ⩽ V are the elements of an incidence system Γ^{V} over I_{n}, where t(W) = dim W and W_{1}—W_{2} iff W_{1} ⩽ W_{2} or W_{2} ⩽ W_{1}. For J ⊆ I_{n}, a J-flag is a family of subspaces W_{j} : j ∈ J of V with dim W_{j} = j and W_{j} ⩽ W_{k} for j ⩽ k. Thus a full flag is a nested sequence 0 ⩽ W_{1} ⩽ W_{2} ⩽ ⋯ ⩽ W_{n} ⩽ V with dim W_{j} = j.

Example 1B. Let U be a vector space of dimension 2n + k over ℝ equipped with a quadratic form Q_{U} of signature (n + k, n), where k ⩾ 1. The nontrivial isotropic subspaces of U (on which Q_{U} is identically zero) have 1 ⩽ dim U ⩽ n, and are the elements (typed by dimension) of an incidence system Γ^{U,Q_{U}} over I_{n}, where the incidence relation is again given by containment. We shall provide a discrete model for Γ^{U,Q_{U}} in Example 4B.

2. Elementary Lie theory

2.1. Lie algebra notions and notations. Recall that a Lie algebra 𝔤 over a field 𝔽 is an 𝔽-vector space equipped with a skew-symmetric bilinear operation [·, ·]: 𝔤 × 𝔤 → 𝔤 satisfying the Jacobi identity [x, [y, z]] = [[x, y], z]+[y, [x, z]]. The commutator bracket (α, β) ↦ [α, β] = α∘β −β ∘α makes End_{𝔽}(V) into a Lie algebra, denoted 𝔤𝔩(V). A representation of Lie algebra 𝔤 on a vector space V is a Lie algebra homomorphism ρ: 𝔤 → 𝔤𝔩(V) (i.e., a linear map with ρ([x, y]) = [ρ(x), ρ(y)]). We write ρ(x, v) or x · v as a shorthand for the action ρ(x)(v) of x ∈ 𝔤 on v ∈ V. For subspaces 𝔭 ⊆ 𝔤 and U ⊆ V, we define ρ(𝔭, U) = 𝔭 · U to be the span of {ρ(x)(u) | x ∈ 𝔭, u ∈ U}, and introduce shorthands ρ(x, U) = x · U and ρ(𝔭, u) = 𝔭 · u when 𝔭 = span{x} or U = span{u}.

The action of subspaces 𝔭 ⊆ 𝔤 on subspaces U ⊆ V has an upper adjoint in each variable:

𝔭 · U ⊆ W iff 𝔭 ⊆ 𝔠_{𝔤}(U, W) := {x ∈ 𝔤 | x · U ⊆ W }
iff U ⊆ 𝔠_{V} (𝔭, W) := {v ∈ V | 𝔭 · v ⊆ W }.

In particular 𝔠_{𝔤}(U, U) = 𝔰𝔱𝔞𝔟_{𝔤}(U) is the stabilizer of U and 𝔠_{V} (𝔭, 0) = ker ρ(𝔭) is the (joint) kernel of the 𝔭 action; U is 𝔭-invariant in V iff 𝔭 · U ⊆ U iff 𝔭 ⊆ 𝔰𝔱𝔞𝔟_{𝔤}(U).

The adjoint representation ad : 𝔤 → 𝔤𝔩(𝔤) is defined by ad(x)(y) = [x, y] for x, y ∈ 𝔤 (which is a representation by the Jacobi identity). Thus ad(𝔭, 𝔮) is the bracket [𝔭, 𝔮] of subspaces 𝔭, 𝔮 ⊆ 𝔤 and we set [x, 𝔮] := [span{x}, 𝔮]. The upper adjoint specializes to give:

[𝔭, 𝔮] ⊆ 𝔯 iff 𝔭 ⊆ 𝔠_{𝔤}(𝔮, 𝔯) := {x ∈ 𝔤 | [x, 𝔮] ⊆ 𝔯}.

In particular 𝔫_{𝔤}(𝔮) := 𝔠_{𝔤}(𝔮, 𝔮) is the normalizer of 𝔮, 𝔠_{𝔤}(𝔮) := 𝔠_{𝔤}(𝔮, 0) is the centralizer of 𝔮, and 𝔷(𝔤) := 𝔠_{𝔤}(𝔤) is the centre of 𝔤. Thus 𝔭 ⊆ 𝔤 is a subalgebra (𝔭 ⩽ 𝔤) iff [𝔭, 𝔭] ⊆ 𝔭 iff 𝔭 ⊆ 𝔫_{𝔤}(𝔭), and an ideal (𝔭 ⊴ 𝔤) iff [𝔤, 𝔭] ⊆ 𝔭 iff 𝔫_{𝔤}(𝔭) = 𝔤. We note a useful lemma.

Lemma 2.1. If 𝔟 ⊆ 𝔞 ⊆ 𝔥 and 𝔤 = 𝔥 ⊕ 𝔪, where 𝔥 ⩽ 𝔤 and [𝔥, 𝔪] ⊆ 𝔪, then 𝔠_{𝔤}(𝔞, 𝔟) = 𝔠_{𝔥}(𝔞, 𝔟) ⊕ (𝔠_{𝔤}(𝔞) ∩ 𝔪).

A Lie algebra 𝔤 is reductive if it has a faithful semisimple representation (see Appendix A.1). This holds in particular if 𝔤 is nonabelian with irreducible adjoint representation (i.e., 𝔤 has no proper nontrivial ideals); then 𝔤 is said to be simple. More generally, the adjoint representation of 𝔤 is faithful and semisimple if and only if 𝔤 is semisimple, i.e., a direct sum of simple ideals. Thus any semisimple Lie algebra 𝔤 is reductive with [𝔤, 𝔤] = 𝔤.

2.2. Filtered and graded Lie algebras.

Definition 2.2. A ℤ-graded vector space is a vector space V equipped with a ℤ-grading, i.e., a direct sum decomposition V = ⨁_{k∈ℤ} V_{k}. A filtration of a vector space V is a family V^{(k)} : k ∈ ℤ of subspaces of V such that i ⩽ j ⇒ V^{(i)} ⊆ V^{(j)}.

Let V^{+} := ⋃_{k∈ℤ} V^{(k)} and V_{−} := ⋂_{k∈ℤ} V^{(k)}. If V^{+} = V and V_{−} = 0, we say V is a filtered vector space and refer to gr(V) := ⨁_{k∈ℤ} V^{(k)}/V^{(k−1)} as the associated graded vector space.

Definition 2.3 (See e.g. [8]). A ℤ-graded Lie algebra is a Lie algebra 𝔤 equipped with a ℤ-grading 𝔤 = ⨁_{k∈ℤ} 𝔤_{k} such that ∀ i, j ∈ ℤ, [𝔤_{i}, 𝔤_{j}] ⊆ 𝔤_{i+j}.

A filtration of a Lie algebra 𝔤 is a filtration 𝔣^{(k)} : k ∈ ℤ of the underlying vector space such that ∀ i, j ∈ ℤ, [𝔣^{(i)}, 𝔣^{(j)}] ⊆ 𝔣^{(i+j)}. If 𝔤_{k} := 𝔣^{(k)}/𝔣^{(k−1)}, this induces a Lie algebra structure on gr_{𝔣}(𝔤) := ⨁_{k∈ℤ} 𝔤_{k}. If 𝔣^{+} = 𝔤 and 𝔣_{−} = 0, we say 𝔤 is a filtered Lie algebra with associated graded Lie algebra gr_{𝔣}(𝔤).

Proposition 2.4. Let 𝔣^{(−1)} := 𝔫 ⩽ 𝔤, 𝔣^{(0)} := 𝔭 ⩽ 𝔫_{𝔤}(𝔫), and for j > 0 inductively define 𝔣^{(−j−1)} = [𝔫, 𝔣^{(−j)}] and 𝔣^{(j)} = 𝔠_{𝔤}(𝔫, 𝔣^{(j−1)}). Then for all i, j ∈ ℤ, [𝔣^{(i)}, 𝔣^{(j)}] ⊆ 𝔣^{(i+j)} and if 𝔣^{(−1)} ⊆ 𝔣^{(0)} (i.e., 𝔫 ⊴ 𝔭) then 𝔣^{(k)} : k ∈ ℤ is a filtration of 𝔤.

Proof. By construction [𝔣^{(−1)}, 𝔣^{(j)}] ⊆ 𝔣^{(j−1)} for all j ∈ ℤ, and 𝔣^{(j−1)} ⊆ 𝔣^{(j)} for j ≠ 0. Next, by Jacobi, for any i > 0, [𝔣^{(−i−1)}, 𝔣^{(j)}] = [[𝔣^{(−1)}, 𝔣^{(−i)}], 𝔣^{(j)}] ⊆ [𝔣^{(−1)}, [𝔣^{(−i)}, 𝔣^{(j)}]] + [𝔣^{(−i)}, 𝔣^{(j−1)}] for all j ∈ ℤ, so induction on i shows that [𝔣^{(−i)}, 𝔣^{(j)}] ⊆ 𝔣^{(−i+j)} for i > 0 and j ∈ ℤ.

We now show [𝔣^{(i)}, 𝔣^{(j)}] ⊆ 𝔣^{(i+j)} for i, j ⩾ 0 by induction on i+j: [𝔣^{(0)}, 𝔣^{(0)}] ⊆ 𝔣^{(0)} since 𝔣^{(0)} = 𝔭 is a subalgebra, and for i + j > 0, Jacobi implies [[𝔣^{(i)}, 𝔣^{(j)}], 𝔫] ⊆ [𝔣^{(i)}, 𝔣^{(j−1)}] + [𝔣^{(i−1)}, 𝔣^{(j)}] ⊆ 𝔣^{(i+j−1)} (inductively), i.e., [𝔣^{(i)}, 𝔣^{(j)}] ⊆ 𝔠_{𝔤}(𝔫, 𝔣^{(i+j−1)}) = 𝔣^{(i+j)}. □

We call this the filtration of 𝔤 induced by 𝔫 ⊴ 𝔭, or by 𝔫 if 𝔭 = 𝔫_{𝔤}(𝔫). Its negative part is the lower central series of 𝔫, and if 𝔣^{(−k)} = 0 for sufficiently large k, we say 𝔫 is nilpotent.

Definition 2.5. Let 𝔤, 𝔣^{(k)} : k ∈ ℤ be a filtered Lie algebra. A filtration of a representation ρ: 𝔤 → 𝔤𝔩(V) of 𝔤 is a filtration V^{(k)} : k ∈ ℤ of V such that ∀ i, j ∈ ℤ, 𝔣^{(i)} · V^{(j)} := ρ(𝔣^{(i)}, V^{(j)}) ⊆ V^{(i+j)}. This induces a representation \overline{ρ} of gr_{𝔣}(𝔤) on gr(V) := ⨁_{k∈ℤ} V_{k}, where V_{k} = V^{(k)}/V^{(k−1)}, such that ∀ i, j ∈ ℤ, \overline{ρ}(𝔤_{i}, V_{j}) ⊆ V_{i+j}. If V is filtered, we call V (or ρ) a filtered representation of 𝔤 with associated graded representation gr(V) (or \overline{ρ}).

2.3. Engel’s theorem and nilpotence.

Theorem 2.6 (Engel). Let ρ: 𝔤 → 𝔤𝔩(V) be a finite dimensional representation of a Lie algebra 𝔤 and 𝔫 ⊴ 𝔤. Then the following are equivalent:

(1) ρ(x) is nilpotent for all x ∈ 𝔫;

(2) 𝔫 acts trivially on any irreducible subquotient of ρ;

(3) V is a filtered representation for the filtration of 𝔤 induced by 𝔫.

Proof. The key is the following well-known lemma due to Engel; we omit the proof.

Lemma 2.7. Let 𝔲 ⩽ 𝔤𝔩(W) be a Lie subalgebra with σ nilpotent for all σ ∈ 𝔲; then if W is nonzero, ∃w ∈ W nonzero such that for all σ ∈ 𝔲, σ(w) = 0.

(1)⇒(2). Let ρ′: 𝔤 → 𝔤𝔩(W) be an irreducible subquotient of ρ and let W′ = {w ∈ W | ρ′(𝔫)(w) = 0}. Since 𝔫 ⊴ 𝔤, W′ is 𝔤-invariant. Since 𝔫 acts by nilpotent endomorphisms, Lemma 2.7 implies W′ is nonzero; hence W′ = W.

(2)⇒(3). This is an easy induction on dim V: if V = 0, we are done; otherwise V has a nontrivial irreducible 𝔤-invariant subspace U ⩽ V, which is in the kernel of ρ(𝔫) by (2). By induction, V/U by is filtered by some V^{(j)}/U : j ∈ ℤ. If k denotes the largest integer with V^{(k)} = U, we may redefine V^{(j)} = 0 for j < k to make V into a filtered representation.

(3)⇒(1). Given such a filtration V^{(j)} : j ∈ ℤ, we may assume V^{(0)} = V and V^{(j)} = 0 for j < −k. Now any x ∈ 𝔫 satisfies ρ(x)^{k+1} = 0. □

If any of these conditions hold, 𝔫 is called a nilpotency ideal [2] for ρ; it follows from (3) that ρ(𝔫) is a nilpotent Lie algebra. By (2), 𝔤 has a largest nilpotency ideal 𝔫𝔦𝔩_{ρ}(𝔤) for ρ, namely the intersection of the kernels of the simple subquotients of ρ. Thus ker ρ ⩽ 𝔫𝔦𝔩_{ρ}(𝔤) and equality holds if ρ is semisimple. Hence 𝔫𝔦𝔩_{ρ}(𝔤) = 0 if ρ is faithful and semisimple.

Henceforth, we assume 𝔤 is finite dimensional. An ideal 𝔫 ⊴ 𝔤 is a nilpotency ideal for the adjoint representation ad of 𝔤 if and only if it is nilpotent, and so 𝔫𝔦𝔩_{ad}(𝔤) is the largest nilpotent ideal of 𝔤, often called the nilradical.

Definition 2.8. The nilpotent radical 𝔫𝔦𝔩(𝔤) ⊴ 𝔤 is the intersection of its largest nilpotency ideals, or equivalently, the intersection of the kernels of the simple representations of 𝔤.

Since 𝔫𝔦𝔩(𝔤) ⩽ 𝔫𝔦𝔩_{ad}(𝔤), it is a nilpotent ideal, and it is the intersection of the kernels of finitely many simple representations of 𝔤, hence the kernel of a semisimple representation of 𝔤. Thus 𝔤/ 𝔫𝔦𝔩(𝔤) is reductive, and 𝔤 is reductive if and only if 𝔫𝔦𝔩(𝔤) = 0. The Lie algebra 𝔤 is a filtered Lie algebra via the canonical filtration 𝔤^{(k)} : k ∈ ℤ induced by 𝔫𝔦𝔩(𝔤) ⊴ 𝔤, and with respect to the canonical filtration, any (finite dimensional) representation of 𝔤 is a filtered representation by Engel’s theorem.

Definition 2.9. The nilpotent cone of a Lie algebra 𝔤 is N (𝔤) = {x ∈ 𝔤 | ρ(x) is nilpotent for any representation ρ of 𝔤}. A nil subalgebra 𝔫 ⩽ 𝔤 is a Lie subalgebra with 𝔫 ⊆ N (𝔤).

Remarks 2.10. If f: 𝔥 → 𝔤 is a Lie algebra homomorphism, then any representation ρ of 𝔤 induces a representation ρ ∘ f of h, so f(N (𝔥)) ⊆ N (𝔤). Clearly x ∈ N (𝔤) if and only if x is nilpotent in any semisimple representation of 𝔤. Thus N (𝔤) is the inverse image of the nilpotent cone in 𝔤/ 𝔫𝔦𝔩(𝔤), which is reductive. A nil subalgebra 𝔫 ⩽ 𝔤 is a nilpotency ideal for the adjoint representation of 𝔫_{𝔤}(𝔫) on 𝔤, hence nilpotent.

Example 2A. Let V be a vector space with dim_{𝔽} V = n+1 as in Example 1A. Then 𝔤𝔩(V) is a Lie algebra whose nilpotent cone N (𝔤𝔩(V)) consists of the nilpotent endomorphisms of V. A subalgebra 𝔫 of 𝔤𝔩(V) is therefore a nil subalgebra iff there is a filtration of V such that 𝔫 acts trivially on the associated graded representation. With respect to a basis adapted to such a filtration, the elements of 𝔫 are strictly upper triangular. In particular, 𝔤𝔩(V) is reductive, and its derived algebra is the subalgebra 𝔰𝔩(V) = [𝔤𝔩(V), 𝔤𝔩(V)] of traceless endomorphisms. Thus a nil subalgebra of 𝔤𝔩(V) is a nilpotent subalgebra of 𝔰𝔩(V).

Example 2B. Let U, Q_{U} be as in Example 1B, and let B_{U} be the associated symmetric bilinear form of signature (n+k, n) on U. Then 𝔰𝔬(U, Q_{U}) = {A ∈ 𝔤𝔩(U) | B_{U}(Au_{1}, u_{2})+B_{U}(u_{1}, Au_{2})} is a Lie subalgebra of 𝔤𝔩(U), and N (𝔰𝔬(U, Q_{U})) again consists of the elements of 𝔰𝔬(U, Q_{U}) which are nilpotent endomorphisms of U. If 𝔫 is a nonzero nil subalgebra of 𝔰𝔬(U, Q_{U}) then 𝔫 · U is nontrivial, hence so is its intersection with 𝔠_{U}(𝔫, 0) = ⋂_{A∈𝔫} ker A (because 𝔫 acts trivially on any irreducible summand of 𝔫 · U). The intersection (𝔫 · U) ∩ 𝔠_{U}(𝔫, 0) is isotropic, hence contains a 1-dimensional isotropic subspace π_{1}. Applying the same argument inductively to π_{1}^{⊥}/π_{1}, we obtain a filtration 0 ⩽ π_{1} ⩽ π_{2} ⩽ ⋯ ⩽ π_{n} ⩽ π_{n}^{⊥} ⩽ ⋯ ⩽ π_{2}^{⊥} ⩽ π_{1}^{⊥} ⩽ U, with dim π_{j} = j, preserved by 𝔫, and 𝔫 acts trivially on the associated graded representation.

2.4. Invariant forms and trace-forms. We summarize basic properties of invariant symmetric bilinear forms in Appendix A.3.

Definition 2.11. An invariant (symmetric bilinear) form on a filtered Lie algebra 𝔤 is compatible with the filtration iff 𝔣^{(j−1)} ⊆ (𝔣^{(−j)})^{⊥} for all j ∈ ℤ. The restriction of a compatible invariant form to 𝔣^{(j)} × 𝔣^{(−j)} descends to a pairing 𝔤_{j} × 𝔤_{−j} → 𝔽 and hence induces a invariant form on gr_{𝔣}(𝔤), called the associated graded invariant form.

Proposition 2.12. If 𝔣^{(j−1)} = (𝔣^{(−j)})^{⊥} for all j ∈ ℤ, the associated graded invariant form on gr_{𝔣}(𝔤) is nondegenerate, i.e., 𝔤_{j}^{⊥} = ⨁_{k≠−j} 𝔤_{k}.

Proof. x is in gr_{𝔣}(𝔤)^{⊥} if and only if its homogeneous components are. Now for x ∈ 𝔤_{j}, we have x ∈ gr_{𝔣}(𝔤)^{⊥} if and only if x is orthogonal to 𝔤_{−j}, i.e., any lift x̃ to 𝔣^{(j)} is in (𝔣^{(−j)})^{⊥}. On the other hand x = 0 if and only if the lift is in 𝔣^{(j−1)}. □

Definition 2.13. The trace form on 𝔤 associated to a representation ρ: 𝔤 → 𝔤𝔩(V), is the invariant form (x, y) ↦ tr(ρ(x)ρ(y)).

Proposition 2.14. Let ρ: 𝔤 → 𝔤𝔩(V) be a filtered representation of a filtered Lie algebra. Then the trace form of ρ is compatible with the filtration 𝔣^{(j)} : j ∈ ℤ of 𝔤, and the associated graded invariant form on gr_{𝔣}(𝔤) is the trace form of the induced representation \overline{ρ}.

Proof. If x ∈ 𝔣^{(j−1)} and y ∈ 𝔣^{(−j)} then ρ(x)ρ(y) maps V^{(k)} to V^{(k−1)} for all k ∈ ℤ, hence is nilpotent. Thus ρ(x)ρ(y) is trace-free, and hence x and y are orthogonal.

To compute the associated graded form on x ∈ 𝔤_{j} and y ∈ 𝔤_{−j}, choose lifts x̃ ∈ 𝔣^{(j)}, ỹ ∈ 𝔣^{(−j)} and a splitting V ≅ gr(V) of the filtration of V. Then the trace of ρ(x̃)ρ(ỹ) may be computed by restricting and projecting onto V_{k}, for each k ∈ ℤ, computing the trace, and summing over k, which yields the trace of \overline{ρ}(x)\overline{ρ}(y). □

Proposition 2.15. Let ⟨·, ·⟩ be the trace form of ρ: 𝔤 → 𝔤𝔩(V).

(1) 𝔫𝔦𝔩_{ρ}(𝔤) ⩽ 𝔤^{⊥} and the induced invariant form on 𝔤/ 𝔫𝔦𝔩_{ρ}(𝔤) is a trace form.

(2) If ⟨·, ·⟩ is nondegenerate, then 𝔫𝔦𝔩_{ρ}(𝔤) = 0.

(3) For a Lie homomorphism f: 𝔥 → 𝔤, ⟨·, ·⟩ pulls back to a trace form associated to ρ ∘ f.

(4) If 𝔥 ⩽ 𝔤 is a subalgebra and ρ(𝔥) is nilpotent, then 𝔥 ⩽ 𝔫_{𝔤}(𝔥)^{⊥}.

Proof. (1) By Engel’s theorem, V is a filtered representation for the filtration of 𝔤 induced by 𝔫𝔦𝔩_{ρ}(𝔤); hence the trace-form is compatible and 𝔫𝔦𝔩_{ρ}(𝔤) ⩽ 𝔤^{⊥}.

(2) is immediate from (1) and (3) is obvious.

(4) Pull back (i.e., restrict) ⟨·, ·⟩ to 𝔫_{𝔤}(𝔥); since 𝔥 is a nilpotency ideal for the restriction of ρ to 𝔫_{𝔤}(𝔥), 𝔥 ⊆ 𝔫𝔦𝔩_{ρ}(𝔫_{𝔤}(𝔥)) ⊆ 𝔫_{𝔤}(𝔥)^{⊥} ∩ 𝔫_{𝔤}(𝔥) ⊆ 𝔫_{𝔤}(𝔥)^{⊥}. □

Corollary 2.16. Let ⟨·, ·⟩ be a trace form on 𝔤.

(1) 𝔫𝔦𝔩(𝔤) ⩽ 𝔤^{⊥} and the induced invariant form on 𝔤/ 𝔫𝔦𝔩(𝔤) is a trace form.

(2) If ⟨·, ·⟩ is nondegenerate, then 𝔤 is reductive.

(3) For a Lie homomorphism f: 𝔥 → 𝔤, ⟨·, ·⟩ pulls back to a trace form on 𝔥.

(4) If 𝔥 ⩽ 𝔤 is a nil subalgebra, then 𝔥 ⩽ 𝔫_{𝔤}(𝔥)^{⊥}.

2.5. Cartan criteria for nilpotency and reductive Lie algebras. In this subsection, we assume the underlying field 𝔽 has characteristic zero (hence is perfect). Then we have the following straightforward characterization of the nilpotent cone N (𝔤).

Proposition 2.17. Let 𝔤 be a Lie algebra. Then x ∈ N (𝔤) if and only if x ∈ [𝔤, 𝔤] and ad(x) is nilpotent. In particular 𝔫𝔦𝔩(𝔤) = 𝔫𝔦𝔩_{ad}(𝔤) ∩ [𝔤, 𝔤].

Proof. If x ∈ N (𝔤) then ad x is nilpotent and x is trivial in any 1-dimensional representation. Conversely, if x ∈ [𝔤, 𝔤] with ad(x) nilpotent, then ρ(x) is nilpotent in any semisimple representation ρ: 𝔤 → 𝔤𝔩(V) by Proposition A.3. Hence x ∈ N (𝔤). □

Lemma 2.18. Suppose 𝔤𝔩(V) = 𝔥 ⊕ 𝔪 with 𝔥 ⩽ 𝔤, 𝔪 ⊆ 𝔥^{⊥} and [𝔥, 𝔪] ⊆ 𝔪. For subspaces 𝔟 ⊆ 𝔞 ⊆ 𝔥, let 𝔲 = 𝔠_{𝔥}(𝔞, 𝔟). Then any element of 𝔲 ∩ 𝔲^{⊥} ⊆ 𝔤𝔩(V) is nilpotent.

Proof (see [2, 17]). By Lemma 2.1, 𝔠_{𝔤𝔩(V)}(𝔞, 𝔟) = 𝔲 ⊕ (𝔠_{𝔤𝔩(V)}(𝔞) ∩ 𝔪), and 𝔪 ⊆ 𝔥^{⊥} so 𝔲^{⊥} ∩ 𝔥 ⊆ 𝔠_{𝔤𝔩(V)}(𝔞, 𝔟)^{⊥}. Hence it suffices to prove the result for 𝔪 = 0.

Let x = x_{s} + x_{n} be the Jordan decomposition of x ∈ 𝔲 ⩽ 𝔤𝔩(V), let 𝔽^{c} be a splitting field for x_{s}, let S ⊆ 𝔽^{c} be the set of eigenvalues x_{s}^{c}, and let f: 𝔽^{c} → ℚ be a ℚ-linear form on 𝔽^{c}. Define y ∈ 𝔤𝔩(V^{c}) to be scalar multiplication by f(λ) on the λ-eigenspace of x_{s} for all λ ∈ S. Then by Lemma A.1, ad y is a polynomial with no constant term in ad x_{s}^{c}, hence in ad x^{c}, so that y ∈ 𝔲^{c}. If also x ∈ 𝔲^{⊥}, then 0 = tr(x^{c}y) = ∑_{λ∈S} m(λ)λf(λ), where m(λ) ∈ ℤ^{+} is the multiplicity of λ. Applying f, we obtain f(λ) = 0 for all ℚ-linear forms f and all λ ∈ S. Hence x_{s} = 0 and x is nilpotent. □

Proposition 2.19 (Cartan’s criterion). Suppose ρ: 𝔤 → 𝔤𝔩(V) is a representation. Then, with respect to the induced trace form, 𝔤^{⊥} ∩ [𝔤, 𝔤] ⩽ 𝔫𝔦𝔩_{ρ}(𝔤).

Proof. Take 𝔟 = ρ(𝔤^{⊥}), 𝔞 = ρ(𝔤) and 𝔪 = 0 in Lemma 2.18, so that 𝔲 = 𝔠_{𝔤𝔩(V)}(ρ(𝔤), ρ(𝔤^{⊥})) (and hence ρ(𝔤^{⊥}) ⊴ 𝔲). Since 𝔲 ⩽ 𝔠_{𝔤𝔩(V)}(ρ(𝔤), ρ(𝔤)^{⊥}), ρ([𝔤, 𝔤]) = [ρ(𝔤), ρ(𝔤)] ⩽ 𝔲^{⊥}. Hence ρ(𝔤^{⊥} ∩ [𝔤, 𝔤]) ⩽ 𝔲 ∩ 𝔲^{⊥}, and so 𝔤^{⊥} ∩ [𝔤, 𝔤] ⊴ 𝔤 is a nilpotency ideal for ρ. □

Theorem 2.20. A finite dimensional Lie algebra 𝔤 over a field of characteristic zero is reductive if and only if it admits a nondegenerate trace form. Then 𝔤 = 𝔷(𝔤) ⊕ [𝔤, 𝔤], the sum is orthogonal, and [𝔤, 𝔤] ≅ ad(𝔤) = 𝔠_{𝔡𝔢𝔯(𝔤)}(𝔷(𝔤)) ≅ 𝔡𝔢𝔯([𝔤, 𝔤]) is semisimple.

Proof. If 𝔤 is reductive, it has a faithful semisimple representation ρ, and 𝔤^{⊥} ∩ [𝔤, 𝔤] = 0 with respect to the induced trace form by Proposition 2.19, so 𝔤^{⊥} ⊆ [𝔤, 𝔤]^{⊥} = 𝔠_{𝔤}(𝔤, 𝔤^{⊥}) = 𝔷(𝔤) by Proposition A.4 (1). Now [𝔤, 𝔤]^{⊥} = 𝔷(𝔤) ⊆ 𝔫𝔦𝔩_{ad}(𝔤) has trivial intersection with [𝔤, 𝔤] by Proposition 2.17, so the trace form is nondegenerate on [𝔤, 𝔤]. Since 𝔤^{⊥} ⩽ 𝔷(𝔤), we may add a representation of 𝔤/[𝔤, 𝔤] to ρ to make the trace form nondegenerate. Conversely, the existence of a nondegenerate trace form implies 𝔤 is reductive by Corollary 2.16.

Since 𝔫𝔦𝔩_{ad}(𝔤)∩[𝔤, 𝔤] = 0, Dieudonné’s famous observation [12] (see Proposition A.5) shows that [𝔤, 𝔤] is semisimple. Thus [𝔤, 𝔤] ≅ ad(𝔤) is a nondegenerate ideal in 𝔡𝔢𝔯(𝔤) ⩽ 𝔤𝔩(𝔤), and hence 𝔥 := ad(𝔤)^{⊥} ∩ 𝔡𝔢𝔯(𝔤) is a complementary ideal in 𝔡𝔢𝔯(𝔤). Now for any D ∈ 𝔥 and x ∈ 𝔤, 0 = [D, ad(x)] = ad(D(x)), so D(x) ∈ 𝔷(𝔤) and hence D vanishes on [𝔤, 𝔤]; thus ad(𝔤) = 𝔠_{𝔡𝔢𝔯(𝔤)}(𝔷(𝔤)) ≅ 𝔡𝔢𝔯([𝔤, 𝔤]). □

Definition 2.21. An admissible form on a Lie algebra 𝔤 is a trace form with 𝔤^{⊥} = 𝔫𝔦𝔩(𝔤).

Admissible forms always exist: just pull back a nondegenerate trace form on the reductive quotient 𝔤/ 𝔫𝔦𝔩(𝔤). They can be used to construct linear subspaces of N (𝔤).

Proposition 2.22. Let 𝔟 ⊆ 𝔞 ⊆ 𝔤 be subspaces of a Lie algebra 𝔤 which contain 𝔫𝔦𝔩(𝔤) and let 𝔲 = 𝔠_{𝔤}(𝔞, 𝔟). Then for any admissible form on 𝔤, 𝔲 ∩ 𝔲^{⊥} ⊆ N (𝔤).

Proof. Any admissible form is induced by a semisimple representation ρ: 𝔤 → 𝔤𝔩(V). Now 𝔤𝔩(V) = ρ(𝔤) ⊕ ρ(𝔤)^{⊥} and ρ(𝔲) = 𝔠_{ρ(𝔤)}(ρ(𝔞), ρ(𝔟)), so for any x ∈ 𝔲 ∩ 𝔲^{⊥}, ρ(x) is nilpotent by Lemma 2.18, and so ad(ρ(x)) is nilpotent on ρ(𝔤). Since 𝔠_{𝔤}(𝔤, 𝔤^{⊥}) ⊆ 𝔲, 𝔲^{⊥} ⊆ [𝔤, 𝔤] by Proposition A.4 (1) and so ρ(x) ⊆ [ρ(𝔤), ρ(𝔤)]. Thus ρ(x) ∈ N (ρ(𝔤)) by Proposition 2.17, and so x ∈ N (𝔤). □

3. Parabolic subalgebras

3.1. General definition. Henceforth, the characteristic of underlying field 𝔽 will be zero.

Definition 3.1. A subalgebra 𝔭 of a Lie algebra 𝔤 is parabolic in 𝔤 if it contains the normalizer 𝔫_{𝔤}(𝔪) of a maximal nil subalgebra 𝔪 ⊆ N (𝔤) of 𝔤.

Any Lie algebra 𝔤 is a parabolic subalgebra of itself, and more generally, if 𝔭 ⩽ 𝔮 ⩽ 𝔤 are subalgebras such that 𝔭 is parabolic in 𝔤 then 𝔮 is parabolic in 𝔤. At the other extreme, the minimal parabolic subalgebras of 𝔤 are the normalizers of its maximal nil subalgebras.

Proposition 3.2. Let 𝔭 be a subalgebra of 𝔤 such that 𝔫𝔦𝔩(𝔭) contains 𝔫𝔦𝔩(𝔤). Then 𝔭 is parabolic in 𝔤 if and only if 𝔭/ 𝔫𝔦𝔩(𝔤) is parabolic in 𝔤/ 𝔫𝔦𝔩(𝔤).

Proof. Since 𝔫𝔦𝔩(𝔤) is an ideal in 𝔤 and N (𝔤/ 𝔫𝔦𝔩(𝔤)) = N (𝔤)/ 𝔫𝔦𝔩(𝔤), 𝔪 is a maximal nil subalgebra of 𝔤 if and only if 𝔪 contains 𝔫𝔦𝔩(𝔤) and 𝔪/ 𝔫𝔦𝔩(𝔤) is a maximal nil subalgebra of 𝔤/ 𝔫𝔦𝔩(𝔤); now 𝔫_{𝔤}(𝔪)/ 𝔫𝔦𝔩(𝔤) = 𝔫_{𝔤/ 𝔫𝔦𝔩(𝔤)}(𝔪/ 𝔫𝔦𝔩(𝔤)) and the result follows. □

For 𝔮 ⩽ 𝔤, define 𝔫𝔦𝔩_{𝔤}(𝔮) := 𝔫𝔦𝔩_{ad_{𝔤}}(𝔮) ∩ [𝔤, 𝔤], the largest ideal of 𝔮 contained in N (𝔤).

Proposition 3.3. Suppose 𝔫𝔦𝔩(𝔤) ⩽ 𝔪 ⩽ 𝔤 and let 𝔮 = 𝔫_{𝔤}(𝔪). Then for any admissible form on 𝔤, (1) 𝔪 = 𝔫𝔦𝔩_{𝔤}(𝔮) ⟹ 𝔪 = 𝔮 ∩ 𝔮^{⊥}, and (2) 𝔪 = 𝔮 ∩ 𝔮^{⊥} ⟹ 𝔪 = 𝔮^{⊥}.

Proof (see [2, 23, 27]). Since 𝔤^{⊥} ⩽ 𝔪 ⊴ 𝔮 ⩽ 𝔤, [𝔮, 𝔮^{⊥}] ⩽ 𝔮^{⊥} and 𝔮 ∩ 𝔮^{⊥} is an ideal in 𝔮, which is contained in N (𝔤) by Proposition 2.22. If 𝔪 ⊆ N (𝔤) then 𝔪 ⊆ 𝔮∩𝔮^{⊥} by Corollary 2.16 (4), and (1) follows. Now 𝔮 acts on 𝔮^{⊥}/𝔪 with 𝔪 as a nilpotency ideal. However, the 𝔪-action has kernel (𝔮 ∩ 𝔮^{⊥})/𝔪 = 0. Hence 𝔮^{⊥} = 𝔪 by Engel’s theorem. □

This result leads to the following equivalences, cf. [2, 4, 7, 15].

Theorem 3.4. For a subalgebra 𝔭 ⩽ 𝔤 with 𝔫𝔦𝔩(𝔤) ⩽ 𝔫𝔦𝔩(𝔭), the following are equivalent.

(1) There is an admissible form on 𝔤 such that 𝔭^{⊥} is a nilpotent subalgebra of [𝔤, 𝔤].

(2) There is an admissible form on 𝔤 such that 𝔭^{⊥} is a nil subalgebra of 𝔤.

(3) 𝔭 is parabolic in 𝔤.

(4) For any admissible form on 𝔤, 𝔭^{⊥} ⊆ 𝔭 and 𝔫_{𝔤}(𝔭) = 𝔭.

(5) 𝔭 = 𝔫_{𝔤}(𝔫𝔦𝔩_{𝔤}(𝔭)).

(6) For any admissible form on 𝔤, 𝔭^{⊥} = 𝔫𝔦𝔩_{𝔤}(𝔭) = 𝔫𝔦𝔩(𝔭).

(7) dim 𝔤 − dim 𝔭 = dim 𝔫𝔦𝔩(𝔭) − dim 𝔫𝔦𝔩(𝔤).

Proof. By Corollary 2.16, 𝔫𝔦𝔩(𝔭) ⩽ 𝔫𝔦𝔩_{𝔤}(𝔭) ⩽ 𝔭^{⊥} for any admissible form on 𝔤; also, since 𝔤^{⊥} ⩽ 𝔭 ⩽ 𝔤, 𝔭 ⊆ 𝔫_{𝔤}(𝔭) = 𝔫_{𝔤}(𝔭^{⊥}) = [𝔭, 𝔭^{⊥}]^{⊥} by Proposition A.4 (1). In particular (6) and (7) are equivalent, since dim 𝔭^{⊥} = dim 𝔤 − dim 𝔭 + dim 𝔤^{⊥} and 𝔤^{⊥} = 𝔫𝔦𝔩(𝔤).

(1)⇔(2) (cf. [4]). Since 𝔭^{⊥} ⩽ 𝔤 and 𝔭 ⩽ 𝔫_{𝔤}(𝔭^{⊥}), Proposition 2.4 yields subspaces 𝔭^{(j)} of 𝔤 with 𝔭^{(−1)} = 𝔭^{⊥}, 𝔭^{(0)} = 𝔭 and [𝔭^{⊥}, 𝔭^{(j)}] ⩽ 𝔭^{(j−1)} for all j ∈ ℤ. Now 𝔠_{𝔤}(𝔭^{⊥}, (𝔭^{(−j)})^{⊥}) = [𝔭^{⊥}, 𝔭^{(−j)}]^{⊥}, so induction on j shows that 𝔭^{(j−1)} = (𝔭^{(−j)})^{⊥} for j > 0, and hence ⋃_{k⩾0} 𝔭^{(k)} = (⋂_{k<0} 𝔭^{(k)})^{⊥} = {0}^{⊥} = 𝔤, since 𝔭^{⊥} is nilpotent. Hence 𝔭^{⊥} is an ad_{𝔤}-nilpotent subalgebra of [𝔤, 𝔤], i.e., a nil subalgebra of 𝔤 by Proposition 2.17. The converse is immediate.

(2)⇔(3). 𝔭^{⊥} is a nil subalgebra of 𝔤 iff 𝔭^{⊥} ⊆ 𝔪 iff 𝔪^{⊥} ⊆ 𝔭 for a maximal nil subalgebra 𝔪, and by Proposition 3.3, 𝔪^{⊥} = 𝔫_{𝔤}(𝔪).

(1–3)⇒(4). Corollary 2.16 (4) implies 𝔭^{⊥} ⊆ 𝔫_{𝔤}(𝔭^{⊥})^{⊥}, hence 𝔭^{⊥} ⊆ 𝔫_{𝔤}(𝔭^{⊥}) ⊆ 𝔭^{⊥⊥} = 𝔭.

(4)⇒(5). Since 𝔭^{⊥} ⊆ 𝔭 = 𝔫_{𝔤}(𝔭^{⊥}), 𝔭^{⊥} = 𝔭 ∩ 𝔭^{⊥} ⊴ 𝔭 and is nil by Proposition 2.22, so 𝔭^{⊥} ⩽ 𝔫𝔦𝔩_{𝔤}(𝔭); hence equality holds and 𝔭 = 𝔫_{𝔤}(𝔫𝔦𝔩_{𝔤}(𝔭)).

(5)⇒(6) By Proposition 3.3, 𝔫𝔦𝔩_{𝔤}(𝔭) = 𝔭^{⊥}, and hence 𝔭 = 𝔫_{𝔤}(𝔭^{⊥}) = [𝔭, 𝔭^{⊥}]^{⊥}, so 𝔭^{⊥} ⊆ [𝔭, 𝔭^{⊥}] + 𝔤^{⊥} ⩽ 𝔫𝔦𝔩(𝔭) since [𝔭, 𝔫𝔦𝔩_{𝔤}(𝔭)] ⊆ 𝔫𝔦𝔩(𝔭).

(6)⇒(1). Immediate, since 𝔫𝔦𝔩(𝔭) ⊆ [𝔭, 𝔭]. □

Proposition 3.5. Suppose 𝔭 ⩽ 𝔮 ⩽ 𝔤 are subalgebras with 𝔮 parabolic in 𝔤. Then 𝔭 is parabolic in 𝔮 if and only if it is parabolic in 𝔤.

Proof. Fix an admissible form on 𝔤; its restriction to 𝔮 is admissible, since 𝔮^{⊥} = 𝔫𝔦𝔩(𝔮). Since 𝔭 ⊆ 𝔮 ⊆ 𝔤, we have 𝔤^{⊥} ⊆ 𝔮^{⊥} ⊆ 𝔭^{⊥}, i.e., 𝔫𝔦𝔩(𝔤) ⊆ 𝔫𝔦𝔩(𝔮) ⊆ 𝔭^{⊥} ∩ 𝔮.

If 𝔭 is parabolic in 𝔮, 𝔮^{⊥} = 𝔫𝔦𝔩(𝔮) ⊆ 𝔭 and 𝔫𝔦𝔩(𝔭) = 𝔭^{⊥} ∩ 𝔮. It follows that 𝔫𝔦𝔩(𝔤) ⊆ 𝔫𝔦𝔩(𝔭) and 𝔭^{⊥} ⊆ 𝔮^{⊥⊥} = 𝔮, hence 𝔫𝔦𝔩(𝔭) = 𝔭^{⊥}, as required. Conversely if 𝔭 is parabolic in 𝔤, then 𝔭^{⊥} = 𝔫𝔦𝔩(𝔭) contains 𝔫𝔦𝔩(𝔮), and since 𝔭^{⊥} ⊆ 𝔭 ⊆ 𝔮, 𝔭 is parabolic in 𝔮. □

Corollary 3.6. Let 𝔭 ⩽ 𝔮 ⩽ 𝔤 with 𝔤 reductive, and 𝔮 parabolic in 𝔤. Then 𝔭 is parabolic in 𝔤 if and only if 𝔫𝔦𝔩(𝔭) contains 𝔫𝔦𝔩(𝔮) and 𝔭/ 𝔫𝔦𝔩(𝔮) is parabolic in 𝔮/ 𝔫𝔦𝔩(𝔮).

We refer to 𝔮_{0} = 𝔮/ 𝔫𝔦𝔩(𝔮) as the (reductive) Levi quotient of 𝔮.

3.2. Grading elements and splittings.

Definition 3.7. A grading element for a Lie algebra 𝔭 is an element χ ∈ gr(𝔭) (the associated graded algebra of the canonical filtration) with [χ, x] = jx for all j ∈ ℤ and x ∈ 𝔭_{j}. A (reductive) Levi subalgebra is a Lie subalgebra of 𝔭 complementary to 𝔫𝔦𝔩(𝔭).

A grading element χ exists if and only if the derivation D of gr(𝔭) defined by Dx = jx (for all j ∈ ℤ and x ∈ 𝔭_{j}) is inner; χ then belongs to the centre of 𝔭_{0}, and is unique modulo the centre of gr(𝔭). If 𝔭 is reductive, i.e., 𝔫𝔦𝔩(𝔭) = 0, then 0 ∈ 𝔭 = 𝔭_{0} is a grading element.

Proposition 3.8 (see [5, 7]). If 𝔭 has a grading element χ ∈ 𝔭_{0}, then the following sets are in canonical exp(𝔫𝔦𝔩(𝔭))-equivariant bijection:

• lifts of χ to 𝔭;

• splittings 𝔭 ≅ gr(𝔭) of the canonical filtration of 𝔭;

• Levi subalgebras of 𝔭. Furthermore, the action of exp(𝔫𝔦𝔩(𝔭)) is free and transitive.

Proof. A lift of χ to 𝔭 determines a splitting gr(𝔭) ≅ 𝔭 via its eigenspace decomposition. For any such splitting, the image of 𝔭_{0} is a Levi subalgebra of 𝔭, and any Levi subalgebra of 𝔭 contains a unique lift of χ. Since 𝔭_{0} = 𝔭/ 𝔫𝔦𝔩(𝔭), lifts of χ form an affine space modelled on 𝔫𝔦𝔩(𝔭), which is nilpotent, so exp(𝔫𝔦𝔩(𝔭)) acts freely and transitively on the lifts. □

For a parabolic subalgebra 𝔭 ⩽ 𝔤, we let 𝔭^{(j)} be the filtration of 𝔤 induced by 𝔫𝔦𝔩(𝔭) ⊴ 𝔭 ⩽ 𝔤 (cf. Theorem 3.4), and denote the associated graded Lie algebra by gr_{𝔭}(𝔤).

Proposition 3.9. Let 𝔭 be parabolic in a reductive Lie algebra 𝔤. Then the associated graded algebra gr_{𝔭}(𝔤) is reductive, and its centre is contained in 𝔭_{0} = 𝔭/ 𝔫𝔦𝔩(𝔭).

Proof. Since 𝔤 is reductive, it admits a nondegenerate trace form, induced by a representation ρ: 𝔤 → 𝔤𝔩(V). Since 𝔭^{⊥} = 𝔫𝔦𝔩(𝔭), it acts nilpotently on V, and hence induces a filtration on V with V^{(j)} = V for j ⩾ 0 (say) and V^{(j−1)} = ρ(𝔭^{⊥}, V^{(j)}) for j ⩽ 0. This makes V into a filtered representation of 𝔤 (with the filtration by 𝔭^{(j)} : j ∈ ℤ).

We have already observed that 𝔭^{(j−1)} = (𝔭^{(−j)})^{⊥} for j > 0, hence writing k = 1 − j and taking perps, we have (𝔭^{(−k)})^{⊥} = 𝔭^{(k−1)} for k ⩽ 0. The induced trace form on gr_{𝔭}(𝔤) is therefore nondegenerate, and hence gr_{𝔭}(𝔤) is reductive.

For any x ∈ 𝔷(gr_{𝔭}(𝔤)), its homogeneous components are also in the centre. Now for a homogeneous central element x ∈ 𝔭_{j} := 𝔭^{(j)}/𝔭^{(j−1)}, any lift centralizes 𝔫𝔦𝔩(𝔭), so it must lie in 𝔭, i.e., j ⩽ 0. However, since 𝔭^{(k)} = [𝔫𝔦𝔩(𝔭), 𝔭^{(k+1)}] for k < 0, we must also have j ⩾ 0, i.e., the centre of gr_{𝔭}(𝔤) is contained in 𝔭_{0}. □

Corollary 3.10. A parabolic subalgebra 𝔭 of a reductive Lie algebra 𝔤 has a unique grading element χ ∈ 𝔷(𝔭_{0}) ∩ [gr_{𝔭}(𝔤), gr_{𝔭}(𝔤)].

Indeed, the derivation D of gr_{𝔭}(𝔤) defined by Dx = jx for x ∈ 𝔤_{j} vanishes on the centre of gr_{𝔭}(𝔤) and preserves its semisimple complement. It follows that D is an inner derivation, i.e., D = ad χ for χ ∈ 𝔷(𝔭_{0}) which is determined uniquely by requiring it is in the complement [gr_{𝔭}(𝔤), gr_{𝔭}(𝔤)] to centre of gr_{𝔭}(𝔤).

3.3. Pairs of parabolic subalgebras.

Definition 3.11 (see [10]). Parabolic subalgebras 𝔭, 𝔮 of a reductive Lie algebra 𝔤 are said to be costandard if 𝔭 ∩ 𝔮 is parabolic in 𝔤.

Proposition 3.12. For parabolics 𝔭, 𝔮 ⩽ 𝔤, with 𝔤 reductive, the following are equivalent:

(1) 𝔭 and 𝔮 are costandard;

(2) 𝔭 ∩ 𝔮 contains a minimal parabolic subalgebra of 𝔤;

(3) for some (hence any) admissible form on 𝔤, 𝔭^{⊥} ⩽ 𝔮 (or equivalently, 𝔮^{⊥} ⩽ 𝔭).

Proof. The first two conditions are manifestly equivalent. Now for any admissible form on 𝔤, (𝔭 ∩ 𝔮)^{⊥} = 𝔭^{⊥} + 𝔮^{⊥}. Thus (1) and (2) imply (3). For the converse, it suffices to show that (3) implies 𝔭^{⊥} + 𝔮^{⊥} is nilpotent. For this, let 𝔯^{(−i)} = ∑_{j⩾0,k⩾0,i=j+k} 𝔭^{(−i)} ∩ 𝔮^{(−j)}. Then 𝔯^{(−1)} = 𝔭^{⊥} + 𝔮^{⊥}, and [𝔯^{(−1)}, 𝔯^{(−i)}] ⊆ 𝔯^{(−i−1)}, so 𝔯^{(−1)} is nilpotent. □

Proposition 3.13. If 𝔭^{1}, … 𝔭^{k} ⩽ 𝔤 are parabolic and pairwise costandard in a reductive Lie algebra 𝔤, then 𝔭^{1} ∩ ⋯ ∩ 𝔭^{k} is parabolic in 𝔤.

Proof. Induction on k, with k = 1 being trivial: if 𝔭^{1} ∩ ⋯ ∩ 𝔭^{k−1} is parabolic, and 𝔭^{k} is costandard with each 𝔭^{i}, then (𝔭^{k})^{⊥} ⊆ 𝔭^{i} for all i = 1, … k − 1, hence (𝔭^{k})^{⊥} ⊆ 𝔭^{1} ∩ ⋯ ∩ 𝔭^{k−1}, which proves the result by Proposition 3.12. □

Definition 3.14. Parabolic subalgebras 𝔭, 𝔭̂ of a reductive Lie algebra 𝔤 are said to be opposite if (for some, hence any, admissible form) 𝔭 +𝔭̂^{⊥} = 𝔤 (i.e., 𝔭^{⊥} ∩𝔭̂ = 0) and 𝔭^{⊥} +𝔭̂ = 𝔤 (i.e., 𝔭 ∩ 𝔭̂^{⊥} = 0); in other words, 𝔤 = 𝔭 ⊕ 𝔭̂^{⊥} (i.e., 𝔭 ∩ 𝔭̂ is a Levi subalgebra of 𝔭̂) which means equivalently that 𝔤 = 𝔭^{⊥} ⊕ 𝔭̂ (i.e., 𝔭 ∩ 𝔭̂ is a Levi subalgebra of 𝔭).

Remarks 3.15. If 𝔭 ⩽ 𝔤 is a parabolic subalgebra, then the set of parabolic subalgebras 𝔭̂ opposite to 𝔭 is an exp(𝔫𝔦𝔩(𝔭))-torsor (a G-torsor for a group G is a simply transitive G-set); Proposition 3.8 yields canonical isomorphisms between the following exp(𝔫𝔦𝔩(𝔭))-torsors:

• lifts of the grading element χ ∈ 𝔭_{0} to 𝔭 = 𝔭^{(0)};

• splittings gr_{𝔭}(𝔤) ≅ 𝔤;

• parabolic subalgebras 𝔭̂ opposite to 𝔭 in 𝔤;

• Levi subalgebras of 𝔭. An (algebraic) Weyl structure for 𝔭 is an element of this (i.e., any of these) torsor(s). An opposite pair 𝔭, 𝔭̂ ⩽ 𝔤 gives a vector space direct sum decomposition of 𝔤 into subalgebras 𝔫𝔦𝔩(𝔭) ⊕ 𝔭 ∩ 𝔭̂ ⊕ 𝔫𝔦𝔩(𝔭̂), where 𝔭 ∩ 𝔭̂ is a Levi subalgebra of both 𝔭 and 𝔭̂, hence is orthogonal to both 𝔫𝔦𝔩(𝔭) and 𝔫𝔦𝔩(𝔭̂) with respect to any admissible form on 𝔤. Thus any admissible form on 𝔤 restricts to an admissible form on 𝔭 ∩ 𝔭̂ and a duality between 𝔫𝔦𝔩(𝔭) and 𝔫𝔦𝔩(𝔭̂).

Proposition 3.16. Let 𝔭 and 𝔮 be parabolic in 𝔤. Then 𝔯 := 𝔭 ∩ 𝔮 + 𝔫𝔦𝔩(𝔮) is parabolic in 𝔤 with 𝔫𝔦𝔩(𝔯) = 𝔫𝔦𝔩(𝔭) ∩ 𝔮 + 𝔫𝔦𝔩(𝔮).

Proof. It suffices to prove that 𝔯/ 𝔫𝔦𝔩(𝔮) is parabolic in 𝔮_{0} := 𝔮/ 𝔫𝔦𝔩(𝔮). Introduce an admissible form on 𝔤 and the induced admissible form on 𝔮_{0}. First note that (𝔭∩𝔮+𝔮^{⊥})^{⊥} = (𝔭∩𝔮)^{⊥}∩𝔮 = (𝔭^{⊥} + 𝔮^{⊥}) ∩ 𝔮. Since 𝔮^{⊥} ⩽ 𝔮, this gives 𝔯^{⊥} = 𝔭^{⊥} ∩ 𝔮 + 𝔮^{⊥}. Then (𝔯/ 𝔫𝔦𝔩(𝔮))^{⊥} = 𝔯^{⊥}/ 𝔫𝔦𝔩(𝔮) is contained in 𝔯/ 𝔫𝔦𝔩(𝔮) and in [𝔮_{0}, 𝔮_{0}], since 𝔭^{⊥} = [𝔭, 𝔭^{⊥}] + 𝔫𝔦𝔩(𝔤). Now 𝔯^{⊥}/ 𝔫𝔦𝔩(𝔮) is nilpotent since 𝔭^{⊥} = 𝔫𝔦𝔩(𝔭) is nilpotent. □

Proposition 3.17. Let 𝔭, 𝔮 be parabolic subalgebras of a reductive Lie algebra 𝔤. Then the following are canonically isomorphic exp(𝔫𝔦𝔩(𝔭) ∩ 𝔫𝔦𝔩(𝔮))-torsors:

• lifts ξ_{𝔭} and ξ_{𝔮} of the grading elements of 𝔭 and 𝔮 with [ξ_{𝔭}, ξ_{𝔮}] = 0 (thus ξ_{𝔭}, ξ_{𝔮} ∈ 𝔭 ∩ 𝔮);

• Levi subalgebras of 𝔭 in 𝔭 ∩ 𝔮;

• Levi subalgebras of 𝔮 in 𝔭 ∩ 𝔮.

Proof. Let 𝔯 = 𝔭 ∩ 𝔮 + 𝔫𝔦𝔩(𝔮). Then 𝔯 is a parabolic subalgebra of 𝔤 contained in 𝔮, so that 𝔯/ 𝔫𝔦𝔩(𝔮) is a parabolic subalgebra of 𝔮_{0} := 𝔮/ 𝔫𝔦𝔩(𝔮). The grading element χ_{𝔮} of 𝔮 lies in the centre of 𝔮/ 𝔫𝔦𝔩(𝔮) and hence in 𝔯/ 𝔫𝔦𝔩(𝔮). Consequently the lifts of χ_{𝔮} belong to an affine subspace of 𝔯 modelled on 𝔫𝔦𝔩(𝔮), which therefore meets 𝔭 ∩ 𝔮 in an affine subspace modelled on 𝔭 ∩ 𝔫𝔦𝔩(𝔮). Mutatis mutandis, there is an affine subspace of lifts of the grading element of 𝔭 to 𝔭 ∩ 𝔮, modelled on 𝔫𝔦𝔩(𝔭) ∩ 𝔫𝔦𝔩(𝔮).

Now if ξ_{𝔭} and ξ_{𝔮} are lifts of grading elements of 𝔭 and 𝔮 to 𝔭 ∩ 𝔮, then [ξ_{𝔭}, ξ_{𝔮}] belongs to 𝔫𝔦𝔩(𝔭)∩𝔫𝔦𝔩(𝔮), on which ad ξ_{𝔭} and ad ξ_{𝔮} are invertible. Hence the equation [ξ_{𝔭}, ξ_{𝔮}] = 0 uniquely determines either lift to 𝔭 ∩ 𝔮 from the other, and the compatible lifts form an affine space modelled on 𝔫𝔦𝔩(𝔭) ∩ 𝔫𝔦𝔩(𝔮), hence a torsor for exp(𝔫𝔦𝔩(𝔭) ∩ 𝔫𝔦𝔩(𝔮)). □

3.4. Minimal Levi subalgebras and maximal anisotropic subalgebras. We refer to Levi subalgebras of parabolic subalgebras of 𝔤 as Levi subalgebras of 𝔤. In particular a minimal Levi subalgebra of 𝔤 is a Levi subalgebra of a minimal parabolic subalgebra 𝔟 ⩽ 𝔤.

Corollary 3.18. If 𝔭, 𝔮 ⩽ 𝔤 are parabolic, 𝔭 ∩ 𝔮 contains a minimal Levi subalgebra of 𝔤.

Indeed, 𝔭 contains a minimal parabolic subalgebras 𝔟 and by Proposition 3.17 there is a minimal Levi subalgebra in 𝔟 ∩ 𝔮 ⩽ 𝔭 ∩ 𝔮.

Proposition 3.19. If 𝔥 is a Levi subalgebra of 𝔤 then N (𝔥) = N (𝔤)∩𝔥, and if 𝔥 is a minimal Levi subalgebra, every x ∈ 𝔥 is semisimple in (the adjoint representation of ) 𝔤.

Proof. Any admissible form on 𝔤 induces an admissible form on 𝔥 (see Remarks 3.15). Now by Corollary 2.16, any x ∈ N (𝔤) ∩ 𝔥 is also in 𝔠_{𝔤}(x)^{⊥} ∩ 𝔥 ⊆ 𝔷(𝔥)^{⊥} ∩ 𝔥 = [𝔥, 𝔥]. Since x is ad_{𝔥}-nilpotent, it is in N (𝔥), and the other inclusion is automatic. If 𝔥 is minimal, N (𝔥) = 0 (else 𝔥 would contain a proper parabolic subalgebra). Now 𝔥 = 𝔠_{𝔤}(𝔷(𝔥)) and hence ad_{𝔤}(𝔥) = 𝔠_{𝔡𝔢𝔯(𝔤)}(𝔷(𝔥)), which is closed under Jordan decomposition by Proposition A.2. □

Definition 3.20. A subalgebra 𝔨 of a reductive Lie algebra 𝔤 is called anisotropic, toral or ad-semisimple if every element of 𝔨 is semisimple in (the adjoint representation of) 𝔤.

Over an algebraically closed field, anisotropic subalgebras are abelian [17, 21].

Proposition 3.21. Let 𝔨 be an anisotropic subalgebra of 𝔤 containing a lift ξ_{𝔭} of the grading element of a parabolic 𝔭. Then ξ_{𝔭} ∈ 𝔷(𝔨), i.e., 𝔨 is in the Levi subalgebra 𝔠_{𝔤}(ξ_{𝔭}).

Proof. Since ad(ξ_{𝔭}) is semisimple (with integer eigenvalues), the invariant subspace 𝔨 ⩽ 𝔤 is a direct sum of eigenspaces for ad(ξ_{𝔭})|_{𝔨}. Now if [ξ_{𝔭}, x] = jx for some x ∈ 𝔨 and j ∈ ℤ, then ad(x)(ξ_{𝔭}) = −jx and ad(x)^{2}(ξ_{𝔭}) = 0. Now ad(x) is semisimple, so x = 0 or j = 0. □

Corollary 3.22. Minimal Levi subalgebras 𝔥 ⩽ 𝔤 are maximal anisotropic subalgebras.

The converse does not necessarily hold unless the underlying field is algebraically closed, in which case minimal Levi subalgebras are abelian and called Cartan subalgebras. A minimal parabolic subalgebra 𝔟 with abelian Levi factor is solvable (i.e., [𝔟, 𝔟] is nilpotent) and called a Borel subalgebra.

Let 𝔨 be a minimal Levi subalgebra of 𝔤, also called an anisotropic kernel. Since 𝔷(𝔨) is anisotropic and abelian, there is a splitting field for its action on 𝔤, i.e., a field extension 𝔽^{c} such that the adjoint action of 𝔷(𝔨) on 𝔤^{c} = 𝔤 ⊗_{𝔽} 𝔽^{c} is simultaneously diagonalizable. Let 𝔞 be the subspace of 𝔷(𝔨) whose elements have all eigenvalues in 𝔽.

Definition 3.23. For α ∈ 𝔞^{∗}, let 𝔤_{α} = {x ∈ 𝔤 | [h, x] = α(h)x for all h ∈ 𝔞}. If α ≠ 0 and 𝔤_{α} ≠ 0, we say that α ∈ 𝔞^{∗} is a (restricted) root of 𝔤, and call 𝔤_{α} the root space of α. The root lattice Λ_{r} of (𝔤, 𝔨) is the (free) ℤ-submodule of 𝔞^{∗} generated by the set Φ of roots. Elements of its dual Λ_{cw} = {ξ ∈ 𝔞 | α(ξ) ∈ ℤ for all α ∈ Φ} are called coweights.

Since 𝔨 is the centralizer of 𝔷(𝔨), 𝔤 has a root space decomposition

𝔤 = 𝔨 ⊕ ⨁_{α∈Φ} 𝔤_{α},

with 𝔤_{α}^{∗} ≅ 𝔤_{−α} (using any admissible form on 𝔤). The kernels of the roots have intersection 𝔷(𝔤) in 𝔞 = 𝔷(𝔤) ⊕ (𝔞 ∩ [𝔤, 𝔤]), hence span its annihilator 𝔷(𝔤)^{∘} ≅ (𝔞 ∩ [𝔤, 𝔤])^{∗}. We shall also need the following basic fact concerning the existence of coroots h_{α} : α ∈ Φ in 𝔞.

Proposition 3.24. For any α ∈ Φ, there is a unique h_{α} ∈ 𝔞 ∩ [𝔤_{α}, 𝔤_{−α}] with α(h_{α}) = 2; furthermore, for any nonzero x_{α} ∈ 𝔤_{α}, there exists y_{α} ∈ 𝔤_{−α} with h_{α} = [x_{α}, y_{α}].

Proof (see [2, 17]). For any x ∈ 𝔤_{α} and y ∈ 𝔤_{−α}, if [x, y] ∈ ker α ⊆ 𝔞 then x, y, [x, y] span a subalgebra 𝔰 ⩽ 𝔤 with [x, y] ∈ [𝔰, 𝔰] ∩ 𝔷(𝔰) ⊆ N (𝔰), so [x, y] ∈ [𝔤, 𝔤] ∩ 𝔨 is nilpotent and semisimple, hence zero. For x = x_{α} ≠ 0, [x_{α}, 𝔤_{−α}] ∩ 𝔞 = 𝔠_{𝔤}(x_{α})^{⊥} ∩ 𝔞 = (ker α)^{⊥} ∩ 𝔞 so there is a unique h_{α} ∈ 𝔞 with α(h_{α}) = 2 and [x_{α}, y_{α}] = h_{α} for some y_{α} ∈ 𝔤_{−α}. □

For any parabolic subalgebra 𝔭 containing 𝔨, the unique lift ξ_{𝔭} of its grading element to 𝔷(𝔨) acts on 𝔤 with integer eigenvalues, so ξ_{𝔭} ∈ Λ_{cw}. For any ξ ∈ Λ_{cw} and j ∈ ℤ, let Φ_{ξ}^{j} = {α ∈ Φ | α(ξ) = k} and Φ_{ξ}^{±} = {α ∈ Φ | ±α(ξ) ∈ ℤ^{+}}. Then:

• Φ_{ξ}^{±} and Φ_{ξ}^{0} are (relatively) additively closed in Φ, with Φ = Φ_{ξ}^{−} ⊔ Φ_{ξ}^{0} ⊔ Φ_{ξ}^{+}; • 𝔭_{ξ} := 𝔨 ⊕ ⨁_{α∈Φ\Φ_{ξ}^{+}} 𝔤_{α} is a parabolic subalgebra with nilpotent radical 𝔭_{ξ}^{⊥} = ⨁_{α∈Φ_{ξ}^{−}} 𝔤_{α};

• for any parabolic 𝔭 ⊇ 𝔨, {ξ ∈ Λ_{cw} | 𝔭_{ξ} = 𝔭} is additively closed and contains ξ_{𝔭}.

Remark 3.25. It may be illuminating to compare the above theory with standard approaches to the theory of real reductive Lie algebras [16, 19, 25, 38], in which the main ingredient is a Cartan decomposition of 𝔤, i.e., a symmetric decomposition 𝔤 = 𝔥 ⊕ 𝔪 (into the +1 and −1 eigenspaces of an involution) such that 𝔥 is a maximal compact subalgebra. Then 𝔞 is a maximal abelian subspace of 𝔪 and 𝔨 = (𝔨 ∩ 𝔥) ⊕ 𝔞 is the centralizer of 𝔞 in 𝔤. It follows that 𝔞 is the “split part” 𝔱 ∩ 𝔪 of a “maximally split” Cartan subalgebra 𝔱, where 𝔱 ∩ 𝔥 is a Cartan subalgebra of 𝔨 ∩ 𝔥. The (restricted) roots in 𝔞^{∗} are restrictions of roots in 𝔱^{∗}. We shall refer to 𝔞 ⩽ 𝔨 in general as a split Cartan subalgebra of 𝔤.

Example 3A. Let V be as in Examples 1A–2A. The parabolic subalgebras of 𝔤𝔩(V) are the stabilizers of flags in the incidence system Γ^{V} of proper nontrivial subspaces W ⩽ V. Elements W_{1}, W_{2} are incident if and only if their stabilizers (maximal proper parabolics) are costandard. A minimal parabolic subalgebra is the stabilizer of full flag, i.e., a Borel subalgebra, while a minimal Levi subalgebra 𝔨 is the stabilizer of a direct sum decomposition of V into one dimensional subspaces, i.e., a Cartan subalgebra of 𝔤𝔩(V). If 𝔽 = ℝ and V = ℝ^{n}, then 𝔰𝔬_{n} ⊕ℝ id is a maximal anisotropic subalgebra of 𝔤𝔩_{n}(ℝ), but not a minimal Levi subalgebra.

Example 3B. Let U, Q_{U} be as in Examples 1B and 2B. The parabolic subalgebras of 𝔰𝔬(U, Q_{U}) are stabilizers of flags in the incidence system Γ^{U,Q_{U}}, with incident flags corresponding to costandard parabolics. A minimal parabolic subalgebra is the stabilizer of a full flag π_{1} ⩽ π_{2} ⩽ ⋯ ⩽ π_{n} of isotropic subspaces, while a minimal Levi subalgebra 𝔨 is the stabilizer of an orthogonal decomposition U = (⨁ R_{𝔨}) ⊕ W_{𝔨} where Q_{U} is positive definite on W_{𝔨}, dim W_{𝔨} = k, and the n elements of R_{𝔨} have signature (1, 1). If W_{𝔨} ≅ ℝ^{k} and ⨁ R_{𝔨} ≅ ℝ^{n,n}, then 𝔨 ≅ 𝔰𝔬_{k} ⊕𝔞 ⊆ 𝔰𝔬_{n+k,n} where 𝔞 is a Cartan subalgebra of 𝔰𝔬_{n,n}. Thus 𝔰𝔬_{n+k} ⊕ 𝔰𝔬_{n} ⊆ 𝔰𝔬_{n+k,n} is maximal anisotropic, but not a minimal Levi subalgebra.

3.5. Minimal parabolic subalgebras and lowest weight representations. Let 𝔟 be a minimal parabolic subalgebra containing a minimal Levi subalgebra 𝔨. The corresponding Weyl structure ξ_{𝔟} induces a grading 𝔤 = ⨁_{j∈ℤ} 𝔟_{j} such that 𝔟 is the nonpositive part, with 𝔟_{0} = 𝔨 and 𝔟_{j} = ⨁_{α∈Φ^{j}} 𝔤_{α} for j ≠ 0 (where Φ^{j} = Φ_{ξ_{𝔟}}^{j}). Thus Φ^{0} = ∅ and Φ is the disjoint union of positive roots α ∈ Φ^{+} and negative roots α ∈ Φ^{−}. For λ, μ ∈ 𝔞^{∗}, we write λ ⩾ μ (λ > μ) if λ − μ is a (nonzero) sum of positive roots.

Definition 3.26. Let ρ: 𝔤 → 𝔤𝔩(V) be a representation and 𝔨 a minimal Levi subalgebra with split Cartan subalgebra 𝔞. For λ ∈ 𝔞^{∗}, the λ-weight space is the simultaneous eigenspace V_{λ} = {v ∈ V | ∀ h ∈ 𝔞, ρ(h)v = λ(h)v}, which is a representation of 𝔨. If V_{λ} ≠ 0, λ is called a weight of V. Let 𝔟 be a minimal parabolic containing 𝔨. A lowest weight vector with lowest weight λ is a vector v ∈ V_{λ} with ρ(𝔫𝔦𝔩(𝔟), v) = 0. If V is generated by a lowest weight vector, it is called a lowest weight representation.

Proposition 3.27. For 𝔨 ⊆ 𝔟 ⊆ 𝔤 as above, let V be a lowest weight representation with lowest weight λ. Then V is a direct sum of weight spaces with weights μ ⩾ λ, the lowest weight λ is unique, and V_{λ} = ρ(𝔟, v) = ρ(𝔨, v) for any lowest weight vector v.

Proof. Let v be a lowest weight vector and let W be the span of elements of the form ρ(y_{1}) ⋯ ρ(y_{k})v with each y_{j} either in 𝔨 or 𝔤_{α} with α ∈ Φ^{+}. A standard inductive argument using ρ(x)ρ(y) = ρ([x, y]) + ρ(y)ρ(x) shows (with x ∈ 𝔟_{−1}) that W is ρ(𝔤)-invariant, hence W = V, and (with x ∈ 𝔞) that W is a sum of weight spaces V_{μ} with μ ⩾ λ, where equality only holds if all y_{j} belong to 𝔨. □

Theorem 3.28. Let 𝔟 be a minimal parabolic subalgebra of 𝔤 containing a minimal Levi subalgebra 𝔨, so that Φ = Φ^{+} ⊔ Φ^{−} = ⨆_{j∈ℤ} Φ^{j} and 𝔤 = ⨁_{j∈ℤ} 𝔟_{j}.

(1) Any parabolic 𝔮 containing 𝔟 is the stabilizer of a 1-dimensional lowest weight space L_{𝔮} = V_{𝔮} (for 𝔟) in a lowest weight representation V_{𝔮} of 𝔤, and λ(h_{α}) = 0 if α(ξ_{𝔮}) = 0.

(2) For any α ∈ Φ^{1}, 𝔤_{α} is irreducible for 𝔨, and there is a maximal proper parabolic subalgebra 𝔮^{α} such that for all β ∈ Φ^{1}, β(ξ^{α}) = δ_{αβ}, where ξ^{α} = ξ_{𝔮_{α}}. If L_{𝔮^{α}} ⊆ V_{𝔮^{α}} as in (1), then the lowest weight is a negative multiple of λ^{α} ∈ 𝔞^{∗} with λ^{α}(h_{β}) = δ_{αβ}.

(3) Φ^{1} is a basis for Λ_{r}, and parabolic subalgebras containing 𝔟 are in bijection with subsets J of Φ^{1}, where the parabolic 𝔮_{J} ⩽ 𝔤 corresponding to J is ⋂_{α∈J} 𝔮^{α} and ξ_{𝔮_{J}} = ∑_{α∈J} ξ^{α}.

Proof (see [2, 17, 29]). (1) Let L_{𝔮} := ∧^{d} 𝔫𝔦𝔩(𝔮) ⩽ ∧^{d}𝔤 with d = dim 𝔫𝔦𝔩(𝔮) so dim L_{𝔮} = 1, and let V_{𝔮} ⊆ ∧^{d}𝔤 be the 𝔤-submodule generated by L_{𝔮}. Since L_{𝔮} ⊆ V_{𝔮} has stabilizer 𝔮, 𝔫𝔦𝔩(𝔟) ⩽ [𝔮, 𝔮] acts trivially and L_{𝔮} is a lowest weight space for 𝔟. If α(ξ_{𝔮}) = 0 then 𝔤_{±α} ⊆ 𝔮 and hence h_{α} ∈ [𝔮, 𝔮].

(2)–(3) Since 𝔟̂ = ∑_{j∈ℕ} 𝔟_{j} is a parabolic (opposite to 𝔟), 𝔟_{1} generates 𝔟_{+} = 𝔫𝔦𝔩(𝔟̂) = ∑_{j∈ℤ^{+}} 𝔟_{j} = ∑_{α∈Φ^{+}} 𝔤_{α}—hence Φ^{1} spans Λ_{r} and for any proper (parabolic) subalgebra 𝔮 ⩽ 𝔤 containing 𝔟, 𝔮 ∩ 𝔟_{1} is a proper 𝔨-invariant subspace of 𝔟_{1}.

If 𝔰 is a maximal proper 𝔨-invariant subspace of 𝔟_{1}, and 𝔫 is the (nilpotent) Lie subalgebra of 𝔟_{+} generated by 𝔰, then an inductive argument shows that [𝔟, 𝔫] ⊆ 𝔟 ⊕𝔫, so that 𝔮 := 𝔟 ⊕𝔫 is parabolic in 𝔤 with 𝔮 ∩ 𝔟_{1} = 𝔰. The positive eigenspaces of ξ_{𝔮} meet 𝔟_{1} in an irreducible complement to 𝔰, which must be a root space 𝔤_{α} with α ∈ Φ^{1}. Hence all such 𝔰 have the form 𝔰 = ⨁{𝔤_{β} | β ∈ Φ^{1}, β ≠ α} and the corresponding 𝔮 = 𝔮^{α} has β(ξ_{𝔮}) = δ_{αβ} for all β ∈ Φ^{1}. The rest of the theorem follows straightforwardly. □

Thus Φ^{1} is a basis of simple roots, i.e., a ℤ-basis for Λ_{r} with respect to which any α ∈ Φ ⊆ Λ_{r} either has all coefficients nonnegative, or all coefficients nonpositive. The corresponding fundamental coweights ξ^{α} : α ∈ Φ^{1} form a basis for 𝔞 ∩[𝔤, 𝔤] so that rank Λ_{r} = dim(𝔞 ∩[𝔤, 𝔤]). The weights λ^{α} ∈ (𝔞 ∩ [𝔤, 𝔤])^{∗} with λ^{α}(h_{β}) = δ_{αβ} are also uniquely determined, and called the fundamental weights.

Conversely, any basis of simple roots Ψ ⊆ Φ determines a unique element ξ of Λ_{cw} with α(ξ) = 1 for all α ∈ Ψ, and then 𝔭_{ξ} is a minimal parabolic subalgebra of 𝔤 with Ψ = Φ^{1}.

4. Global theory and parabolic buildings

We now have sufficient information to adopt a more global perspective on parabolic subalgebras of a reductive Lie algebra 𝔤. Let G be a connected algebraic group with Lie algebra 𝔤 (over a field 𝔽 of characteristic zero), so that the centre Z(G) has Lie algebra 𝔷(𝔤), and G/Z(G) ≅ G^{ss} ⩽ Aut(𝔤) is the identity component of the automorphism group, called the adjoint group of 𝔤; G^{ss} is generated by exp(ad x) for x ∈ N (𝔤) (these generate a connected normal subgroup of Aut(𝔤) whose Lie algebra meets every simple component of 𝔤 nontrivially). For a subspace 𝔰 ⊆ 𝔤 we let N_{G}(𝔰) ⩽ G be the stabilizer of 𝔰 and C_{G}(𝔰) the kernel of the action of N_{G}(𝔰) on 𝔰. These subgroups have Lie algebras 𝔫_{𝔤}(𝔰) and 𝔠_{𝔤}(𝔰) respectively.

4.1. Homogeneity and generalized flag varieties. Via its adjoint group, G acts on the set 𝒫^{𝔤} of parabolic subalgebras 𝔮 of 𝔤, and its orbits are called generalized flag varieties. Since any such 𝔮 is self-normalizing (𝔫_{𝔤}(𝔮) = 𝔮), its stabilizer N_{G}(𝔮) has Lie algebra 𝔮.

Proposition 4.1. Any generalized flag variety embeds into the projective space of a lowest weight representation of 𝔤.

Proof. Use Theorem 3.28 (1): for any g ∈ G, the infinitesimal stabilizer of g · L_{𝔮} = L_{g·𝔮} is g · 𝔮 and hence the adjoint orbit of 𝔮 is isomorphic to the G-orbit of L_{𝔮} ∈ P(V_{𝔮}). □

Proposition 4.2. For α ∈ Φ, let h_{α} = [x_{α}, y_{α}] ∈ 𝔞 with α(h_{α}) = 2, x_{α} ∈ 𝔤_{α} and y_{α} ∈ 𝔤_{−α}.

(1) The automorphism g = exp(ad(x_{α})) exp(ad(−y_{α})) exp(ad(x_{α})) of 𝔤 preserves 𝔞 ∩ [𝔤, 𝔤], sending h to g · h = h − α(h)h_{α};

(2) For all α, β ∈ Φ, β(h_{α}) ∈ ℤ and σ_{α}(β) := β − β(h_{α})α ∈ Φ.

Finally, if α_{1}, … α_{k} is a basis of simple roots, and Φ_{+} the corresponding set of positive roots, then for any i ∈ {1, … k}, σ_{α_{i}} permutes Φ_{+} \ span{α_{i}}.

Proof (see [2, 17]). (1) The automorphism g = exp(ad(x_{α})) exp(ad(−y_{α})) exp(ad(x_{α})) restricts to the identity on ker α and sends h_{α} to −h_{α}. It thus sends any h ∈ 𝔞 to h−α(h)h_{α} ∈ 𝔞 as required.

(2) For any α ∈ Φ, such a triple x_{α}, y_{α}, h_{α} exists and spans an 𝔰𝔩_{2} subalgebra of 𝔤. Since β(g · h) = β(h − α(h)h_{α}) = (β − β(h_{α})α)(h) = σ_{α}(β)(h), and g · [h, z] = [g · h, g · z], g restricts to an isomorphism 𝔤_{β} ≅ 𝔤_{σ_{α}^{−1}(β)} = 𝔤_{σ_{α}(β)}. Hence if β ∈ Φ, σ_{α}(β) ∈ Φ. The 𝔰𝔩_{2} relations ad h_{α} ∘ ad x_{α} = ad x_{α} ∘ (ad h_{α} + 2) and ad h_{α} ∘ ad y_{α} = ad y_{α} ∘ (ad h_{α} − 2) then show that the eigenvalue σ_{α}(β)(h_{α}) = −β(h_{α}) differs from β(h_{α}) by 2k ∈ 2ℤ, i.e., β(h_{α}) ∈ ℤ.

The last part is standard: any α = ∑_{j=1}^{k} n_{j}α_{j} ∈ Φ_{+} \ span{α_{i}} has n_{j} > 0 for some j ≠ i, as does σ_{α_{i}}(α) = α − α(h_{i})α_{i}; hence σ_{α_{i}}(α) is positive. □

This result shows that Φ is a root system in Λ_{r}, and we refer to σ_{α}, for α ∈ Φ, as a root reflection. The system need not be “reduced”: if α is a root, then there may be integer multiples of α other than ±α which are roots; however, σ_{mα} = σ_{α} for any m ∈ ℤ \ {0}. There are irreducible nonreduced systems denoted BC_{n} in addition to the Dynkin classification [17].

Proposition 4.3. N_{G}(𝔨) acts transitively on minimal parabolic subalgebras containing 𝔨.

Proof. To show any two minimal parabolic subalgebras 𝔟, 𝔠 containing 𝔨 are conjugate by an element of N_{G}(𝔨), let us say a root α is “shared” if the multiples α which are positive for 𝔟 are also positive for 𝔠; otherwise, they are negative for 𝔠 and we say α is “unshared”. We now use complete induction on the number of unshared roots.

If there are none then 𝔟 = 𝔠, otherwise there is a root space 𝔤_{α_{i}} in 𝔟_{1} ∩ 𝔠, so α_{i} (and its multiples) is unshared. By Proposition 4.2, there exists g ∈ N_{G}(𝔞) such that g · 𝔟 has positive roots σ_{α_{i}}(Φ_{+}), where Φ_{+} is the set of positive roots for 𝔟; now g · 𝔟 and 𝔠 have fewer unshared roots, hence are conjugate under N_{G}(𝔨). □

Note that the proof shows more: if we consider 𝔠 to be fixed, then the element of N_{G}(𝔞) needed to conjugate 𝔟 to 𝔠 is obtained by an iterative application of simple root reflections.

4.2. The global Weyl group and parabolic incidence system. Let ℬ^{𝔤} be the set of all minimal parabolic subalgebras of 𝔤, let 𝒲 ^{𝔤} be the set of all “BK-pairs” (𝔟, 𝔨) where 𝔟 is a minimal parabolic subalgebra of 𝔤 and 𝔨 is a Levi subalgebra of 𝔟, and let 𝒜^{𝔤} be the set of minimal Levi subalgebras in 𝔤. Given 𝔨 ∈ 𝒜^{𝔤}, let 𝔞 ⩽ 𝔷(𝔨) be the corresponding split Cartan subalgebra, so that 𝔨 = 𝔠_{𝔤}(𝔞) = 𝔫_{𝔤}(𝔞) is the Lie algebra of K := C_{G}(𝔞).

Theorem 4.4. G acts transitively on 𝒲 ^{𝔤}, hence also on ℬ^{𝔤} and 𝒜^{𝔤}. The stabilizer of (𝔟, 𝔨) ∈ 𝒲 ^{𝔤} is K = C_{G}(𝔞).

Proof. Any two minimal parabolic subalgebras 𝔟, 𝔠 ⩽ 𝔤 share a (minimal) Levi subalgebra 𝔨, hence are conjugate by an element of N_{G}(𝔨) ⩽ G. On the other hand, the set of such 𝔨 contained in a given 𝔟 is a torsor for exp(𝔫𝔦𝔩(𝔟)) ⩽ G. Now suppose that g ∈ G fixes (𝔟, 𝔨) and let λ_{1}, … λ_{k} ∈ (𝔞 ∩ [𝔤, 𝔤])^{∗} be the fundamental weights corresponding to 𝔟. For each j there is an irreducible lowest weight representation V_{j} whose lowest weight is a positive multiple of λ_{j}. Since the action of g on V_{j} sends weight spaces to weight spaces and lowest weight vectors to lowest weight vectors, the induced action on 𝔞 ∩ [𝔤, 𝔤] fixes λ_{j}. Since 𝔞 = 𝔷(𝔤) ⊕ (𝔞 ∩ [𝔤, 𝔤]) and the fundamental weights span (𝔞 ∩ [𝔤, 𝔤])^{∗}, g ∈ C_{G}(𝔞). □

Proposition 4.5. Let Q be an adjoint orbit of maximal (proper) parabolic subalgebras of 𝔤. Then for any minimal parabolic subalgebra 𝔟, there is a unique 𝔮 ∈ Q with 𝔟 ⩽ 𝔮.

Proof. Fix a Levi subalgebra 𝔨 ⩽ 𝔟. Since G acts transitively on minimal parabolic subalgebras, for any 𝔮′ ∈ Q, there exists g ∈ G with g · 𝔟 ⊆ 𝔮′, so 𝔟 ⊆ 𝔮 := g^{−1} · 𝔮′ ∈ Q. Suppose now that 𝔮, 𝔮′ = g · 𝔮 ∈ Q both contain 𝔟; then 𝔮 contains both 𝔟 and g^{−1} · 𝔟, which are therefore conjugate by an element of N_{G}(𝔮) (since N_{G}(𝔮) acts transitively on minimal parabolic subalgebras of 𝔮/𝔮^{⊥}). Thus we may assume 𝔮′ = g · 𝔮 with g ∈ N_{G}(𝔟), and fix a (minimal) Levi subalgebra 𝔨 of 𝔟. Now since 𝔨 and g^{−1} · 𝔨 are both Levi subalgebras of 𝔟, they are related by an element of exp(𝔟^{⊥}) ⊆ N_{G}(𝔟) ⊆ N_{G}(𝔮) so we may assume g ∈ N_{G}(𝔨) ∩ N_{G}(𝔟) = C_{G}(𝔞) by Theorem 4.4. Hence 𝔮′ = 𝔮. □

Thus costandard maximal parabolic subalgebras in the same adjoint orbit are equal.

We summarize the development so far with a double fibration of G-homogeneous spaces

\begin{aligned}& 𝒲 ^{𝔤} \\ 𝒜^{𝔤} & \underline{} & ℬ^{𝔤} & \overline{} \end{aligned}

in which the fibre over 𝔟 ∈ ℬ^{𝔤} is an exp(𝔫𝔦𝔩(𝔟))-torsor (the Weyl structures for ℬ^{𝔤}). If we choose a basepoint (𝔟, 𝔨) ∈ 𝒲 ^{𝔤}, then the orbit-stabilizer theorem provides isomorphisms 𝒲 ^{𝔤} ≅ G/K, ℬ^{𝔤} ≅ G/N_{G}(𝔟) and 𝒜^{𝔤} ≅ G/N_{G}(𝔨). The fibre of 𝒲 ^{𝔤} over 𝔨 ∈ 𝒜^{𝔤} is a torsor for the local Weyl group W_{𝔨}(𝔤) := N_{G}(𝔞)/C_{G}(𝔞) = N_{G}(𝔨)/K.

Definition 4.6. The global Weyl group W (𝔤) of 𝔤 is the automorphism group Aut_{G}(𝒲 ^{𝔤}) of the Weyl space 𝒲 ^{𝔤} of BK-pairs (𝔟, 𝔨), i.e., W (𝔤) is the set of bijections 𝒲 ^{𝔤} → 𝒲 ^{𝔤} commuting with the G-action.

We shall write the G-action on 𝒲 ^{𝔤} on the left, and the W (𝔤) action on the right.

Proposition 4.7. The Weyl space 𝒲 ^{𝔤} is a principal W (𝔤)-bundle over 𝒜^{𝔤}, i.e., the action of W (𝔤) on 𝒲 ^{𝔤} is fibre-preserving, and each fibre is a (W_{𝔨}(𝔤), W (𝔤))-bitorsor. In particular, any basepoint (𝔟, 𝔨) ∈ 𝒲 ^{𝔤} yields an isomorphism between W_{𝔨}(𝔤) and W (𝔤).

Proof. Note that W (𝔤) preserves the fibration of 𝒲 ^{𝔤} over 𝒜^{𝔤} and thus induces a right action of W (𝔤) on 𝒜^{𝔤} commuting with G. However, any 𝔨 ∈ 𝒜^{𝔤} is the Lie algebra of its stabilizer N_{G}(𝔨) in G, and for any w ∈ W (𝔤), N_{G}(𝔨w) = N_{G}(𝔨), so 𝔨w = 𝔨. Thus the induced action of W (𝔤) on 𝒜^{𝔤} is trivial, i.e., the action of W (𝔤) on 𝒲 ^{𝔤} is fibre-preserving. Since the G-action is transitive, the W (𝔤) action is free, and so any w ∈ W is uniquely determined by what it does to a base-point (𝔟, 𝔨) ∈ 𝒲 _{𝔨}^{𝔤}. If also (𝔠, 𝔨) ∈ 𝒲 _{𝔨}^{𝔤}, there is an automorphism sending g · (𝔟, 𝔨) to g · (𝔠, 𝔨) for all g ∈ G, because the effective quotient W_{𝔨}(𝔤) = N_{G}(𝔨)/K acts freely on 𝒲 _{𝔨}^{𝔤}. Hence W (𝔤) also acts transitively on fibres. □

Remark 4.8. Another description of W (𝔤) uses a natural groupoid structure on 𝒜^{𝔤}: the set of morphisms from 𝔨 to 𝔨′ in 𝒜^{𝔤} is N_{G}(𝔨, 𝔨′)/K, i.e., the set of elements of G which conjugate 𝔨 to 𝔨′, modulo the right action of K (or equivalently, the left action of K′). Since Hom_{𝒜} (𝔨, 𝔨) = W_{𝔤}(𝔨), we refer to 𝒜^{𝔤}, with this structure, as the Weyl groupoid of 𝔤.

Now 𝒲 ^{𝔤} determines a representation (or action) of the Weyl groupoid 𝒜^{𝔤}: any [g] ∈ Hom_{𝒜} (𝔨, 𝔨′) = N_{G}(𝔨, 𝔨′)/K induces a function 𝒲 _{𝔨}^{𝔤} → 𝒲 _{𝔨′}^{𝔤} sending (𝔟, 𝔨) to (g · 𝔟, g · 𝔨) = (g · 𝔟, 𝔨′); these determine a functor 𝒜^{𝔤} → Set. Furthermore, on each fibre 𝒲 _{𝔨}^{𝔤}, the action of W_{𝔤}(𝔨) is free and transitive. Thus we may equip 𝒲 ^{𝔤} with a trivial (2-connected) groupoid structure over 𝒜^{𝔤}, in which there is a unique morphism between any two points (𝔟, 𝔨) and (𝔠, 𝔨′), labelled by the unique element of Hom_{𝒜} (𝔨, 𝔨′) = N_{G}(𝔨, 𝔨′)/K whose representatives send 𝔟 to 𝔠. In other words, 𝒲 ^{𝔤} is a universal groupoid cover of 𝒜^{𝔤}, and W (𝔤) is its group of “deck transformations”.

In addition to its G-space structure, 𝒫^{𝔤} has an incidence relation: 𝔭, 𝔭′ ∈ 𝒫^{𝔤} are incident if they are costandard. This incidence structure of 𝒫^{𝔤} is determined by the maximal proper parabolic subalgebras 𝔭. Let I_{𝔤} be the set of adjoint orbits of such 𝔭.

Definition 4.9. The parabolic incidence system Γ^{𝔤} of 𝔤 is the set of all maximal proper parabolic subalgebras 𝔭 ⩽ 𝔤, equipped with the incidence relation

𝔭—𝔭′ iff 𝔭 and 𝔭′ are costandard, i.e., 𝔭 ∩ 𝔭′ is parabolic in 𝔤 and the type function t_{Γ^{𝔤}}: |Γ^{𝔤}| → I_{𝔤} sending 𝔭 to its adjoint orbit G · 𝔭.

Theorem 4.10. Let 𝔤 be a reductive Lie algebra. Then the map

Λ: ⨆_{J⊆I_{𝔤}} FΓ^{𝔤}(J) → 𝒫^{𝔤}; σ ↦ ⋂_{𝔭∈σ} 𝔭

induces an incidence isomorphism over P(I_{𝔤}), where parabolics 𝔭, 𝔮 ∈ 𝒫^{𝔤} are incident if they are costandard, and the fibres of the type function 𝒫^{𝔤} → P(I_{𝔤}) are adjoint orbits.

Proof. By Proposition 3.13, Λ is well-defined. For any 𝔮 ∈ 𝒫^{𝔤}, after choosing a minimal parabolic subalgebra 𝔟 contained on 𝔮, Theorem 3.28 and Proposition 4.5 show that the maximal parabolic subalgebras containing 𝔮 have intersection 𝔮 and form a J-flag in FΓ^{𝔤}(J) for some J ∈ P(I_{𝔤}). Now if 𝔮 = 𝔭^{1} ∩ ⋯ ∩ 𝔭^{k} and g ∈ G then g · 𝔮 = g · 𝔭^{1} ∩ ⋯ g · 𝔭^{k}, so the adjoint orbit of 𝔮 is in the image of FΓ^{𝔤}(J) and we may label it by J. Distinct adjoint orbits have distinct labels by Proposition 4.5 and Λ intertwines incidence of flags with incidence of parabolic subalgebras. Hence it is an incidence isomorphism. □

Henceforth we label parabolic adjoint orbits (i.e., generalized flag varieties) 𝒫^{𝔤}(J) in 𝔤 by subsets J of I_{𝔤} using this isomorphism.

Example 4A. Continuing Examples 1A–3A, we take G = GL(V), and the adjoint group is PGL(V). There is a bijection between minimal Levi subalgebras 𝔨 ⩽ 𝔤𝔩(V) and n + 1 element subsets S_{𝔨} ⊆ P(V) = Gr_{1}(V) with V = ⨁ S_{𝔨}: S_{𝔨} = {L ∈ P(V) | 𝔨 · L ⊆ L} and 𝔨 = ⋂_{L∈S_{𝔨}} 𝔰𝔱𝔞𝔟_{𝔤𝔩(V)}(L), which is the Cartan subalgebra of diagonal matrices with respect to any basis representing the elements of S_{𝔨}, hence 𝔞 = 𝔨. Thus N_{G}(𝔨) is the subgroup of GL(V) preserving the decomposition V = ⨁ S_{𝔨} and the local Weyl group W_{𝔨}(𝔤𝔩(V)) = N_{G}(𝔨)/C_{G}(𝔨) is canonically isomorphic to Sym(S_{𝔨}).

The minimal parabolic subalgebra 𝔟 stabilizing the full flag 0 = W_{0} ⩽ W_{1} ⩽ ⋯ ⩽ W_{n} ⩽ W_{n+1} = V contains 𝔨 if and only if there is a bijection j ↦ L_{j}: {1, … n + 1} → S_{𝔨} such that W_{j} = W_{j−1} ⊕ L_{j}, and hence W_{j} = L_{1} ⊕ ⋯ ⊕ L_{j}. Hence the fibre of 𝒲 ^{𝔤} over 𝔨 ∈ 𝒜^{𝔤} may be identified canonically with the set of bijections {1, … n+1} → S_{𝔨}, and the global Weyl group W (𝔤) with Sym_{n+1} = Sym({1, … n + 1}). Note that for each j ∈ {1, … n} the transposition σ_{j} = (j j + 1) sends 𝔟 to the stabilizer of the full flag W̃_{1} ⩽ ⋯ ⩽ W̃_{n} with W̃_{i} = W_{i} for i ≠ j and W̃_{j} = W_{j−1} ⊕ L_{j+1}. These transpositions generate Sym_{n+1}.

A maximal (proper) parabolic subalgebra contains 𝔨 if and only if the subspace it stabilizes is a sum of elements of S_{𝔨}. Hence the incidence system of such parabolic subalgebras is isomorphic to the incidence system of proper nontrivial subsets of S_{𝔨}, cf. Example 1A.

Example 4B. Continuing Examples 1B–3B, we take G to be the adjoint group SO_{0}(U, Q_{U}). Minimal Levi subalgebras 𝔨 now correspond to orthogonal direct sum decompositions U = (⨁ R_{𝔨}) ⊕ W_{𝔨} where Q_{U} is positive definite on W_{𝔨}, dim W_{𝔨} = k, and R_{𝔨} is a set of n signature (1, 1) subspaces of U. Again N_{G}(𝔨) = N_{G}(𝔞) is the subgroup of SO_{0}(U, Q_{U}) preserving this decomposition of U, and the local Weyl group W_{𝔨}(𝔰𝔬(U, Q_{U}))) = N_{G}(𝔨)/K is isomorphic to Sym(R_{𝔨})⋉ℤ_{2}^{S_{𝔨}}, where any representative of f ∈ ℤ_{2}^{S_{𝔨}} preserves any R ∈ R_{𝔨}, and if f(R) = 0, it preserves the two isotropic lines in R; otherwise, it swaps them.

The minimal parabolic subalgebra 𝔟 stabilizing the full flag π_{1} ⩽ ⋯ ⩽ π_{n} of isotropic subspaces of U contains 𝔨 if and only if for each j ∈ {1, … n} there is an isotropic line L_{j} ⩽ R ∈ R_{𝔨} such that π_{j} = π_{j−1} ⊕ L_{j} (where π_{0} = 0). The fibre of 𝒲 ^{𝔤} over 𝔨 ∈ 𝒜^{𝔤} is isomorphic to the space of such maps, which identifies the global Weyl group with Sym_{n} ⋉ℤ_{2}^{n}. The canonical generators are ρ_{j} = (j j + 1) for j ∈ {1, … n − 1}, while ρ_{n} ∈ ℤ_{2}^{n} with ρ_{n}(j) = δ_{jn}. Thus for any j ∈ {1, … n}, ρ_{j} sends 𝔟 to the stabilizer of π̃_{1} ⩽ ⋯ ⩽ π̃_{n} where π̃_{i} = π_{i} for i ≠ j, π̃_{j} = π_{j−1} ⊕ L_{j+1} for j ≠ n, and otherwise, if j = n, π̃_{n} = π_{n−1} ⊕ L̃_{n} where L̃_{n} is isotropic with L_{n} ⊕ L̃_{n} ∈ R_{𝔨}.

Let R be a set of n disjoint two element sets. A subset of ⋃ R is admissible if it contains at most one element from each two element set. The nonempty admissible subsets form an incidence system Γ^{R,±} over I_{n}, with the number of elements as the type, and incidence by containment. The incidence system of maximal parabolic subalgebras of 𝔰𝔬(U, Q_{U}) containing 𝔨 is isomorphic to Γ^{R,±}; this is the discrete model promised in Example 1B.

In general, the preceding analysis of parabolic subalgebras can be used to show that 𝒫^{𝔤} is a Tits building [1, 14, 36] with apartment complex 𝒜^{𝔤}. We find it more convenient to do this in the framework of chamber systems [3, 30, 39], initiated by Tits in [37].

4.3. Chamber systems and Coxeter groups.

Definition 4.11 ([30, 39]). A chamber system over I is a graph Δ with an edge labelling λ: E_{Δ} → I such that for each i ∈ I, the i-adjacency relation b—^{i}c (i.e., b = c or {b, c} is an edge with label i) is an equivalence relation. A chamber subsystem is a subgraph with the induced edge labelling, and a chamber morphism φ: Δ → Δ′ over I is morphism of edge labelled graphs, i.e., a map on vertices such that b—^{i}c implies φ(b)—^{i}φ(c).

The vertices of a chamber system are called chambers, and the equivalence class of a chamber c under i-adjacency is called its i-panel p_{i}(c). We shall often blur the distinction between Δ and its underlying set |Δ| of chambers. For an i-panel p and a chamber subsystem A, we say p ∈ A if A ∩ p is nonempty (i.e., for some chamber c ∈ A, p = p_{i}(c)).

Proposition 4.12. Let Γ be an incidence system over I. Then the set of full flags in Γ can be made into a chamber system Δ = CΓ via the relation b—^{i}c iff their subflags of type I \ {i} are equal: b ∩ t_{Γ}^{−1}(I \ {i}) = c ∩ t_{Γ}^{−1}(I \ {i}).

Definition 4.13. A firm i-panel is one containing at least two chambers; it is thin if it has exactly two, and thick if it has more than two. A firm, thin or thick chamber system is one whose i-panels (for all i ∈ I) are all firm, thin or thick respectively.

In a thin chamber system Δ, for each i ∈ I there is a free involution of |Δ|, which interchanges the two chambers in each i-panel. These involutions are not chamber automorphisms; they generate a right action on |Δ| of the free group over I with the property that b—^{i}c iff b = ci iff bi = c.

Definition 4.14. The structure group of a thin chamber system Δ is the subgroup of Sym(|Δ|) generated by the involutions b ↦ bi for i ∈ I (the effective quotient of the free group action generated by I). Its group elements are the vertices of a chamber system with w—^{i}w′ iff w′ = wi (the Cayley graph of the presentation) and we denote the structure group (viewed in this way) by W_{Δ}, or by I ↪ W_{Δ} (to indicate the generating set).

Proposition 4.15. Let Δ be a thin chamber system.

• A permutation of |Δ| is a chamber morphism if and only if it commutes with W_{Δ}.

• Δ is connected if and only if W_{Δ} acts transitively. In this case the automorphism group Aut(Δ) acts freely and the following are equivalent:

(1) for any adjacent b, c ∈ Δ, there is an automorphism of Δ interchanging b and c;

(2) Δ is homogeneous, i.e., Aut(Δ) acts transitively on (the chambers of ) Δ;

(3) W_{Δ} acts freely on Δ;

(4) there is a unique function δ_{Δ} : Δ × Δ → W_{Δ} with δ_{Δ}(b, c) = w iff c = bw.

Thus a connected homogeneous thin chamber system Δ is an (Aut(Δ), W_{Δ})-bitorsor, and, for fixed b ∈ Δ, the map δ_{Δ}(b, ·): Δ → W_{Δ} is an isomorphism of chamber systems.

4.4. Apartments and buildings. Henceforth, all chamber systems will be nonempty, firm and connected.

Definition 4.16. Let Δ be a chamber system and let I ↪ W be a group generated by involutions. An apartment in Δ is a (connected) homogeneous thin chamber subsystem A of Δ. A W -distance on Δ is a map δ_{Δ} : Δ × Δ → W such that for any b, c, c′ ∈ Δ, δ_{Δ}(b, c) = δ_{Δ}(c, b)^{−1} and c′—^{i}c in Δ implies δ_{Δ}(b, c′)—^{i}δ_{Δ}(b, c) in W. An apartment A is compatible with a W -distance δ_{Δ} if W_{A} ≅ W with δ_{A} = δ_{Δ}|_{A}. We say (Δ, δ_{Δ}) is a building (of type W) if for any b, c ∈ Δ there is a compatible apartment A containing b and c.

A thin building (Δ, δ_{Δ}) is just a homogeneous thin chamber system with its natural function δ_{Δ} : Δ × Δ → W = W_{Δ}. Strictly speaking, we should require that W is a Coxeter group (as it will be in our examples). In the literature, the notion of a building is usually defined either purely in terms of δ_{Δ} (with no mention of apartments) or in terms of apartments (often using the language of simplicial complexes rather than chamber systems). We now relate our hybrid approach to the latter.

The idea is that we can define δ_{Δ}(b, c) = δ_{A}(b, c) for an apartment A containing b and c, provided that the apartments we admit give the same answer, and yield a W -distance.

Lemma 4.17. Let Δ a chamber system and let φ: A_{1} → A_{2} be an isomorphism between apartments A_{1} and A_{2} fixing b ∈ A_{1} ∩ A_{2}. Then:

(1) φ fixes A_{1} ∩ A_{2} pointwise if and only if for any c ∈ A_{1} ∩ A_{2}, δ_{A_{1}}(b, c) = δ_{A_{2}}(b, c);

(2) for any c_{1} ∈ A_{1} and c_{2} ∈ A_{2} with c_{1}—^{i}c_{2}, i.e., p := p_{i}(c_{2}) = p_{i}(c_{2}), we have that

δ_{A_{1}}(b, c_{1})—^{i}δ_{A_{2}}(b, c_{2}) if and only if φ sends p ∩ A_{1} to p ∩ A_{2}.

Proof. (1) If c = bw ∈ A_{1} then φ(c) = bw in A_{2}, which is c iff δ_{A_{2}}(b, c) = w = δ_{A_{2}}(b, c).

(2) If w = δ_{A_{1}}(b, c_{1}), then φ sends p ∩ A_{1} = {c_{1}, c_{1}i} to {c′, c′i} ⊆ A_{2} for the unique c′ ∈ A_{2}

with δ_{A_{2}}(b, c′) = w. This is p ∩ A_{2} = {c_{2}, c_{2}i} if and only if δ_{A_{1}}(b, c_{1})—^{i}δ_{A_{2}}(b, c_{2}). □

In (2), we say that A_{1} and A_{2} share the i-panel p, that p ∈ A_{1} and p ∈ A_{2}, and that an isomorphism φ sending p ∩ A_{1} to p ∩ A_{2} preserves p.

Definition 4.18. An apartment complex 𝒜 in Δ is a set of apartments such that any two chambers in Δ belong to a common apartment in 𝒜 . We say 𝒜 is regular iff for any two intersecting apartments A_{1}, A_{2} ∈ 𝒜 , there is a chamber isomorphism A_{1} → A_{2} fixing A_{1} ∩A_{2} pointwise and preserving any i-panel they share.

Proposition 4.19. A chamber system Δ is a building with respect to some δ_{Δ} : Δ × Δ → W if and only if it admits a regular apartment complex 𝒜 (whose apartments have type W), in which case there is a unique δ_{Δ} such that the apartments in 𝒜 are compatible.

Proof. If (Δ, δ_{Δ}) is a building then the set 𝒜 of compatible apartments form an apartment complex, while given an apartment complex 𝒜 , there is at most one δ_{Δ} with δ_{Δ}|_{A} = δ_{A} for all A ∈ 𝒜 . The equivalence now follows easily from Lemma 4.17. □

The chamber isomorphisms A_{1} → A_{2} often arise as restrictions of automorphisms of Δ.

Definition 4.20. A group G of chamber automorphisms of a chamber system Δ preserving an apartment complex 𝒜 is said to be strongly transitive if it acts transitively on 𝒲 := {(a, A) | a ∈ A ∈ 𝒜 }. In this situation 𝒜 is regular if and only if

(R) for any chamber b ∈ Δ and any i-panel or chamber p, Stab_{G}(b) ∩ Stab_{G}(p) acts transitively on {A ∈ 𝒜 : b, p ∈ A}.

Indeed condition (R) implies regularity by Lemma 4.17. Conversely, if 𝒜 is regular, then for any A_{1}, A_{2} ∈ 𝒜 and any b ∈ A_{1} ∩ A_{2}, the unique isomorphism A_{1} → A_{2} fixing b is the restriction of some g ∈ G by strong transitivity, so condition (R) holds.

We can now summarize what we have established in the homogeneous case.

Theorem 4.21 (Abstract Bruhat Decomposition). Suppose Δ is a chamber system over I ↪ W with apartment complex 𝒜 and a strongly transitive group G of automorphisms satisfying condition (R). Then Δ is a building, where the apartments in 𝒜 are compatible with a W -distance δ_{Δ} which induces a bijection G\(Δ × Δ) → W.

Proof. Since 𝒜 is a regular apartment complex, it determines a W -distance δ_{Δ} : Δ × Δ → W which agrees with δ_{A} on any apartment A ∈ 𝒜 . Thus δ_{Δ} is surjective and G-invariant. Furthermore, for any w ∈ W, G acts transitively on pairs (b, c) with δ_{Δ}(b, c) = w (if (b′, c′) is another such pair, then after choosing apartments A, A′ ∈ 𝒜 containing b, c and b′, c′ respectively, the element of G sending (b, A) to (b′, A′) also sends c to c′). Hence the induced surjection G\(Δ × Δ) → W is also injective. □

4.5. Parabolic buildings and the Bruhat decomposition. We now apply Proposition 4.12 to the parabolic incidence system Γ^{𝔤}, to obtain a chamber system whose chambers are the full flags of Γ^{𝔤}, which we may identify with the minimal parabolic subalgebras of 𝔤, and hence denote ℬ^{𝔤}; two such subalgebras 𝔟, 𝔠 are then i-adjacent (for i ∈ I_{𝔤}) iff there is a (necessarily unique) parabolic subalgebra of type I_{𝔤} \ {i} containing both 𝔟 and 𝔠.

Theorem 4.22. The chamber system ℬ^{𝔤} of minimal parabolic subalgebras is a building over the global Weyl group W (𝔤), which is a Coxeter group, and there is a canonical bijection between G\(ℬ^{𝔤} × ℬ^{𝔤}) and W (𝔤).

Proof. For any minimal Levi subalgebra 𝔨 ∈ 𝒜^{𝔤}, the subgraph of ℬ^{𝔤} given by the minimal parabolic subalgebras containing 𝔨 is a thin chamber system of type W (𝔤) whose automorphism group is the local Weyl group N_{G}(𝔨)/C_{G}(𝔨) (whose restricted root system action implies W (𝔤) is a Coxeter group). Any two minimal parabolic subalgebras belong to such an apartment by Corollary 3.18, which also shows that the apartment complex of type W (𝔤) defined by 𝒜^{𝔤} satisfies (R). The result now follows from Theorem 4.21. □

We refer to ℬ^{𝔤} as the parabolic building of 𝔤. If we fix a minimal parabolic subalgebra 𝔟 with stabilizer B = N_{G}(𝔟) then ℬ^{𝔤} ≅ G/B (as a G-space) and G\(ℬ^{𝔤} ×ℬ^{𝔤}) ≅ G\(G/B×G/B) ≅ B\G/B, the set of double cosets of B in G.

Proposition 4.23. There is a unique and involutive map op_{𝔤}: I_{𝔤} → I_{𝔤} (i.e., op_{𝔤}^{2} = id_{I_{𝔤}}) such that for any J ⊆ I_{𝔤} and any parabolic subalgebra 𝔭 of type J, its opposite parabolic subalgebras have type op_{𝔤}(J).

Proof. For any parabolic subalgebra 𝔭, the parabolic subalgebras opposite to 𝔭 belong to a single adjoint orbit by Remarks 3.15. Furthermore, if 𝔭̂ is opposite to 𝔭 then g · 𝔭̂ is opposite to g · 𝔭, and if 𝔭′ is costandard with 𝔭 then by Corollary 5.5, there exists 𝔭̂′ opposite to 𝔭′ and costandard with 𝔭̂; it follows that 𝔭̂ ∩ 𝔭̂′ is opposite to 𝔭 ∩ 𝔭′. Hence the opposition involution of adjoint orbits is determined uniquely by the opposition relation on maximal parabolic subalgebras. □

Definition 4.24. The map op_{𝔤} is called the duality involution of 𝔤.

5. Parabolic projection and geometric configurations

5.1. Parabolic projection. Let 𝔮 be parabolic in 𝔤, let 𝔮_{0} = 𝔮/ 𝔫𝔦𝔩(𝔮) and let Q = N_{G}(𝔮) ⊆ G. By Corollary 3.6, the inverse image of any parabolic in 𝔮_{0} is a parabolic subalgebra of 𝔤 contained in 𝔮. By Proposition 3.16, this process has a right inverse.

Definition 5.1. Let 𝔮 be a parabolic subalgebra of a reductive Lie algebra 𝔤. Then parabolic projection Π_{𝔮}: 𝒫^{𝔤} → 𝒫^{𝔮_{0}} is defined by

Π_{𝔮}(𝔭) = (𝔭 ∩ 𝔮 + 𝔫𝔦𝔩(𝔮))/ 𝔫𝔦𝔩(𝔮).

The following property is immediate from the definition and the special case 𝔭 ⊆ 𝔭′.

Proposition 5.2. If 𝔭, 𝔭′ ⩽ 𝔤 are costandard parabolic subalgebras then Π_{𝔮}(𝔭 ∩ 𝔭′) ⊆ Π_{𝔮}(𝔭) ∩ Π_{𝔮}(𝔭′). Thus Π_{𝔮} preserves incidence.

By Theorem 4.10, any maximal (proper) parabolic subalgebra 𝔭 of 𝔮 is the intersection with 𝔮 of a maximal parabolic subalgebra of 𝔤 whose adjoint orbit in I_{𝔤} is uniquely determined by the adjoint orbit Q · 𝔭. There is thus a natural inclusion ι_{𝔮}: I_{𝔮_{0}} → I_{𝔤} whose image is the complement of the type J_{𝔮} ∈ P(I_{𝔤}) of 𝔮 in 𝔤. More generally, if 𝔭 ∈ 𝒫^{𝔮_{0}} has type J ∈ P(I_{𝔮_{0}}) then its inverse image in 𝔮 has type ι_{𝔮}(J) ∪ J_{𝔮} ∈ P(I_{𝔤}).

Parabolic projection has a more complicated behaviour in general, although on the set 𝒫_{𝔮-co}^{𝔤} of parabolic subalgebras of 𝔤 costandard with 𝔮, the behaviour is straightforward.

Proposition 5.3. The restriction of Π_{𝔮} to 𝒫_{𝔮-co}^{𝔤} sends parabolic subalgebras 𝔭 of type J ∈ P(I_{𝔤}) to parabolic subalgebras of type ι_{𝔮}^{−1}(J) ∈ P(I_{𝔮_{0}}).

Proof. If 𝔭 is costandard with 𝔮 then Π_{𝔮}(𝔭) = 𝔭 ∩ 𝔮/ 𝔫𝔦𝔩(𝔮). Since 𝔭 ∩ 𝔮 is in the adjoint orbit J ∪ J_{𝔮}, its projection to 𝔮_{0} is in ι_{𝔮}^{−1}(J \ J_{𝔮}) = ι_{𝔮}^{−1}(J). □

Parabolic subalgebras of 𝔤 are rarely costandard with 𝔮; generically, they are weakly opposite to 𝔮 in the following sense.

Definition 5.4 ([26]). Parabolic subalgebras 𝔭, 𝔮 ⩽ 𝔤 are said to be weakly opposite if 𝔭 + 𝔮 = 𝔤 (equivalently 𝔫𝔦𝔩(𝔭) ∩ 𝔫𝔦𝔩(𝔮) = 0).

The weakly opposite case is also amenable to analysis.

Lemma 5.5. Parabolic subalgebras 𝔭 and 𝔮 of 𝔤 are weakly opposite if and only if there is a parabolic subalgebra 𝔭̂ opposite to 𝔭 and costandard with 𝔮.

Proof. If 𝔭̂ is opposite to 𝔭 and costandard with 𝔮 then 𝔫𝔦𝔩(𝔮) ⩽ 𝔭̂ so 𝔫𝔦𝔩(𝔭) ∩ 𝔫𝔦𝔩(𝔮) = 0. Conversely, if 𝔫𝔦𝔩(𝔭) ∩ 𝔫𝔦𝔩(𝔮) = 0, we may take 𝔭̂ opposite to 𝔭 using a Weyl structure ξ_{𝔭} in 𝔭 ∩ 𝔮: 𝔫𝔦𝔩(𝔮) then has nonnegative eigenvalues for ξ_{𝔭}, hence lies in 𝔭̂. □

Lemma 5.6. Let 𝔭, 𝔭̂ be parabolic in 𝔤 with 𝔭̂ opposite to 𝔭 and costandard with 𝔮. Then Π_{𝔮}(𝔭̂) is opposite to Π_{𝔮}(𝔭) in 𝔮_{0}.

Proof. Since Π_{𝔮}(𝔭̂) = 𝔭̂ ∩ 𝔮/ 𝔫𝔦𝔩(𝔮), 𝔫𝔦𝔩(Π_{𝔮}(𝔭̂)) = (𝔫𝔦𝔩(𝔭̂) + 𝔫𝔦𝔩(𝔮))/ 𝔫𝔦𝔩(𝔮). As 𝔭 + 𝔫𝔦𝔩(𝔭̂) = 𝔤 with 𝔫𝔦𝔩(𝔭̂) ⊆ 𝔮, we have 𝔭 ∩ 𝔮 + 𝔫𝔦𝔩(𝔭̂) = 𝔮 and hence (𝔭 ∩ 𝔮 + 𝔫𝔦𝔩(𝔮)) + (𝔫𝔦𝔩(𝔭̂) + 𝔫𝔦𝔩(𝔮)) = 𝔮. It follows that Π_{𝔮}(𝔭) + 𝔫𝔦𝔩(Π_{𝔮}(𝔭̂)) = 𝔮_{0}.

It remains to show that Π_{𝔮}(𝔭̂) + 𝔫𝔦𝔩(Π_{𝔮}(𝔭)) = 𝔮_{0}, i.e., 𝔭̂ ∩ 𝔮 + 𝔫𝔦𝔩(𝔭) ∩ 𝔮 = 𝔮 (since 𝔫𝔦𝔩(𝔮) ⊆ 𝔭̂ ∩ 𝔮). For this, we use Corollary 3.18 to introduce a minimal Levi subalgebra 𝔨 of 𝔤 contained in the parabolic subalgebras 𝔭 and 𝔭̂ ∩ 𝔮. In particular, 𝔨 ⊆ 𝔭 ∩ 𝔭̂ and so the root space decomposition of 𝔤 associated to 𝔨 refines the direct sum decomposition 𝔤 = 𝔭̂ ⊕ 𝔫𝔦𝔩(𝔭). Since 𝔮 contains 𝔨, it is a sum of root spaces, so 𝔮 = 𝔤 ∩ 𝔮 = 𝔭̂ ∩ 𝔮 ⊕ 𝔫𝔦𝔩(𝔭) ∩ 𝔮. □

Let 𝒫_{𝔮-op}^{𝔤} be the set of all parabolic subalgebras of 𝔤 which are weakly opposite to 𝔮.

Proposition 5.7. The restriction of Π_{𝔮} to 𝒫_{𝔮-op}^{𝔤} sends parabolic subalgebras 𝔭 of type J ∈ P(I_{𝔤}) to parabolic subalgebras of type (op_{𝔤} ∘ ι_{𝔮} ∘ op_{𝔮_{0}})^{−1}(J) ∈ P(I_{𝔮_{0}}).

Proof. This is immediate from Corollary 5.5 and Lemma 5.6. □

Minimal parabolic subalgebras also have straightforward projections.

Proposition 5.8. If 𝔟 ∈ ℬ^{𝔤} then Π_{𝔮}(𝔟) ∈ ℬ^{𝔮_{0}}. Furthermore, if 𝔟_{0} ∈ ℬ^{𝔮_{0}} then there exists 𝔟 ∈ ℬ^{𝔤} weakly opposite to 𝔮 with Π_{𝔮}(𝔟) = 𝔟_{0}.

Proof. Using Corollary 3.18, let 𝔨 be a minimal Levi subalgebra contained in 𝔟 and 𝔮, and let 𝔮̂ be the opposite subalgebra to 𝔮 containing 𝔨. Then the root space decomposition of 𝔨 refines the direct sum decomposition 𝔤 = 𝔫𝔦𝔩(𝔮) ⊕ 𝔮 ∩ 𝔮̂ ⊕ 𝔫𝔦𝔩(𝔮̂), and so 𝔟 splits across this decomposition. For any root α with 𝔤_{α} ⊆ 𝔮 ∩ 𝔮̂, precisely one of 𝔤_{α} and 𝔤_{−α} is contained in 𝔟, so that 𝔟 ∩ 𝔮 ∩ 𝔮̂ is minimal in 𝔮 ∩ 𝔮̂. This is an isomorph of Π_{𝔮}(𝔟) in 𝔮_{0}.

For the second part, let 𝔟_{0} = 𝔠/ 𝔫𝔦𝔩(𝔮), where 𝔠 is a minimal parabolic subalgebra of 𝔤. Let 𝔠̂ be a minimal parabolic subalgebra of 𝔤 opposite to 𝔠. Then 𝔠̂ is weakly opposite to 𝔮. Since both Π_{𝔮}(𝔠) and Π_{𝔮}(𝔠̂) are minimal parabolic subalgebras of 𝔮, there exists q ∈ Q such that

𝔠 = q · (𝔠̂ ∩ 𝔮 + 𝔫𝔦𝔩(𝔮)) = (q · 𝔠̂) ∩ 𝔮 + 𝔫𝔦𝔩(𝔮);

whence 𝔟 := q · 𝔠̂ is weakly opposite to 𝔮 with Π_{𝔮}(𝔟) = 𝔟_{0}. □

Proposition 5.9. If 𝔟, 𝔠 ∈ ℬ^{𝔤} are both weakly opposite to 𝔮, then 𝔟 ∈ Q · 𝔠.

Proof. Using Corollary 5.5, let 𝔟̂ and 𝔠̂ be respectively minimal parabolic subalgebras of 𝔤 opposite to 𝔟 and 𝔠, and costandard with 𝔮. Then 𝔟̂/ 𝔫𝔦𝔩(𝔮) and 𝔠̂/ 𝔫𝔦𝔩(𝔮) are minimal parabolic subalgebras of 𝔮_{0}, so there exists q ∈ Q such that 𝔠̂ = q · 𝔟̂. It follows that q^{−1} · 𝔠 is opposite to 𝔟̂, hence conjugate to 𝔟 by an element q′ ∈ exp(𝔫𝔦𝔩(𝔟̂)) by Remarks 3.15. Since 𝔫𝔦𝔩(𝔟̂) ⊆ 𝔮, 𝔟 and 𝔠 are in the same Q-orbit. □

Corollary 5.10. Any two parabolic subalgebras of 𝔤 in the same adjoint orbit and weakly opposite to 𝔮 are in the same Q-orbit.

Proof. This follows from Theorem 4.10 and Propositions 4.5 and 5.9. □

Let ν_{𝔮} := op_{𝔤} ∘ι_{𝔮} ∘op_{𝔮_{0}} : I_{𝔮_{0}} → I_{𝔤}, and observe that the direct and inverse image operations in induced by ν_{𝔮} may be used to pull back or push forward presheaves from I_{𝔤} to I_{𝔮_{0}} or vice versa. In particular, ν_{𝔮}^{∗}𝒫_{𝔮-op}^{𝔤}(J) := 𝒫_{𝔮-op}^{𝔤}(ν_{𝔮}(J)) is isomorphic to the flag complex of the incidence system ν_{𝔮}^{∗}Γ_{𝔮-op}^{𝔤} := ⨆_{i∈I_{𝔮_{0}}} 𝒫_{𝔮-op}^{𝔤}(ν_{𝔮}(i)) (with the obvious type function), while

ν_{𝔮∗}𝒫^{𝔮_{0}}(J) := 𝒫^{𝔮_{0}}(ν_{𝔮}^{−1}(J)) defines a chamber system ν_{𝔮∗}ℬ^{𝔮_{0}} over I_{g} with vertices 𝒫^{𝔮_{0}}(I_{𝔮_{0}}). These operations are adjoint functors, so that morphisms from ν_{𝔮}^{∗}𝒫_{𝔮-op}^{𝔤} to 𝒫^{𝔮_{0}} over I_{𝔮_{0}} correspond bijectively to morphisms from 𝒫_{𝔮-op}^{𝔤} to ν_{𝔮∗}𝒫^{𝔮_{0}}. Parabolic projection is such a morphism.

Theorem 5.11. Π_{𝔮} defines an incidence morphism ψ_{𝔮}: ν_{𝔮}^{∗}Γ_{𝔮-op}^{𝔤} → Γ^{𝔮_{0}} and a chamber morphism φ_{𝔮}: ℬ_{𝔮-op}^{𝔤} → ν_{𝔮∗}ℬ^{𝔮_{0}}.

5.2. Geometric configurations and their projections. We end this paper by returning full circle to Section 1 and the motivation from incidence geometry. We have noted in Proposition 4.12 that any incidence system Γ has an associated chamber system Δ = CΓ whose chambers are the full flags of Γ. In fact C is functorial and has an adjoint [31].

Definition 5.12. Let Δ be a chamber system over I. For i ∈ I, an i-coresidue of Δ is a connected component of the graph obtained from Δ by removing all edges with label i.

Proposition 5.13. For a chamber system Δ over I, let EΔ be the graph with type function t_{EΔ} : |EΔ| → I whose vertices of type i are the i-coresidues v of Δ with edges v—w if and only if v and w contain a common chamber. Then EΔ is an incidence system.

This is immediate: if two i-coresidues contain a common chamber, they are equal.

If Δ = CΓ is the chamber system of full flags of a incidence system Γ then all the full flags in an i-coresidue have the same element of type i; the map assigning this element to the i-coresidue is in fact an incidence morphism ECΓ → Γ (the co-unit of the adjunction), and is an isomorphism if and only if Γ is:

• flag regular, i.e., any element belongs to a full flag; and

• residually connected, i.e., for any i ∈ I, any two full flags containing a common element of type i have the same i-coresidue.

The parabolic incidence system Γ^{𝔤} has both properties: identifying full flags with minimal parabolics using Theorem 4.10, we first note that any maximal (proper) parabolic contains a minimal parabolic, which implies flag regularity. Now if minimal parabolic 𝔟, 𝔠 ⊆ 𝔮 for a maximal parabolic 𝔮, then 𝔟/ 𝔫𝔦𝔩(𝔮) and 𝔠/ 𝔫𝔦𝔩(𝔮) are chambers in ℬ^{𝔮_{0}}, which is connected. Thus 𝔟, 𝔠 have the same G · 𝔮-coresidue, and so Γ^{𝔤} is residually connected.

If Δ^{W} is a homogeneous thin chamber system over I with structure group W generated by I ↪ W, then its i-coresidues are the orbits of the subgroup of W generated by I \ {i}. If W = W (𝔤) then the image of an injective chamber morphism Δ^{W (𝔤)} → ℬ^{𝔤} is an apartment, and such a labelled apartment thus determines an incidence morphism Γ^{W (𝔤)} → Γ^{𝔤}. Definition 5.14. Let 𝔤 be a reductive Lie algebra. A geometric configuration in Γ^{𝔤} with combinatorial type (or combinatorics) Γ is an incidence morphism Γ → Γ^{𝔤}. A standard configuration in Γ^{𝔤} is an incidence morphism ψ: Γ^{W (𝔤)} → Γ^{𝔤} corresponding to a labelled apartment Δ^{W (𝔤)} ↪ ℬ^{𝔤}.

Standard configurations are rather special, but can be used to obtain more general configurations by parabolic projection. Suppose then that 𝔮 is parabolic in 𝔤, define ν_{𝔮}: I_{𝔮_{0}} → I_{𝔤} as in the previous section, and let ψ: Γ^{W (𝔤)} → Γ^{𝔤} be a standard configuration weakly opposite to 𝔮, i.e., with image in Γ_{𝔮-op}^{𝔤}. Then ψ immediately defines a configuration ν_{𝔮}^{∗}ψ: ν_{𝔮}^{∗}Γ^{W (𝔤)} → ν_{𝔮}^{∗}Γ_{𝔮-op}^{𝔤} by restricting along ν_{𝔮} to elements with types in I_{𝔮_{0}}. Definition 5.15. For 𝔮 parabolic in 𝔤, the parabolic projection of a standard configuration ψ weakly opposite to 𝔮 is the geometric configuration ψ_{𝔮} ∘ ν_{𝔮}^{∗}ψ: ν_{𝔮}^{∗}Γ^{W (𝔤)} → Γ^{𝔮_{0}}.

Example 5A. Recall from Example 1A the incidence system Γ^{S} of proper nonempty subsets of an n + 1 = dim V element set S. An injective incidence system morphism Γ^{S} → Γ^{𝔤𝔩(V)} defines a simplex in P(V)—the complete configuration of all projective subspaces spanned by subsets of n + 1 distinct points in P(V); these are the standard configurations in Γ^{𝔤𝔩(V)}.

If 𝔮 is the infinitesimal stabilizer of a point L ∈ P(V), then parabolic projection defines an incidence system morphism ν_{𝔮}^{∗}Γ_{𝔮-op}^{𝔤𝔩(V)} → Γ^{𝔤𝔩(V/L)}, where Γ^{𝔤𝔩(V/L)} is an incidence system over I_{n−1} with the natural inclusion of types sending j to j + 1. Since linear subspaces of complementary dimensions are opposite, ν_{𝔮}: I_{n−1} → I_{n} is the composite j ↦ n − j ↦ n − j + 1 ↦ n + 1 − (n − j + 1) = j. Explicitly, if W ⩽ V does not contain L then the infinitesimal stabilizer of P(W) is weakly opposite to 𝔮 and is projected to (the infinitesimal stabilizer of) P((W + L)/L). The image of a simplex weakly opposite to 𝔮 is thus a complete configuration of n + 1 points in general position in P(V/L). For example, the parabolic projection of a tetrahedron in ℝP^{3} is a quadrilateral (four points and six lines) in ℝP^{2}.

Example 5B. Example 4B shows that for standard configurations in Γ^{𝔰𝔬(U,Q_{U})}, we may take the combinatorial type to be the incidence system Γ^{R,±} over I_{n} of admissible subsets of ⋃ R, where R is a set of n disjoint two element sets. This is isomorphic to the set of all faces of an n-cross polytope or n-orthoplex with the singleton subsets as vertices, or, dually, to the set of all faces of an n-cube, with the n-element subsets as vertices. A standard configuration maps the vertices of the n-cross to points in the quadric of isotropic 1-dimensional subspaces of U, such that the admissible subsets of vertices (corresponding to faces of the n-cross) span subspaces which are isotropic, i.e., lie entirely in the quadric.

If 𝔮 is the infinitesimal stabilizer of a point L on the quadric, then (the infinitesimal stabilizer of) any isotropic subspace which does not contain L and is not contained in L^{⊥} is weakly opposite to L (i.e., to 𝔮). Its projection onto the quadric of isotropic lines in L^{⊥}/L is (W ∩ L^{⊥} + L)/L. For example if U = ℝ^{4,3}, then L^{⊥}/L is isomorphic to ℝ^{3,2}, and parabolic projection maps lines and planes in a real 5-quadric Q^{5} to points and lines in a 3-quadric Q^{3}. A 3-cross is an octahedron, and so parabolic projection constructs configurations of 12 points and 8 lines in Q^{3}. This has an interpretation in Lie circle geometry [9]: points in Q^{3} parametrize oriented circles in S^{2} ≅ ℝ^{2} ∪ {∞}, which have oriented contact when the line joining the points is isotropic (i.e., lies in Q^{3}).

Given a geometric configuration ν_{𝔮}^{∗}Γ^{W (𝔤)} → Γ^{𝔮_{0}}, it is natural to ask if and when it may be obtained by parabolic projection from a standard configuration in Γ^{𝔤}. This is a lifting problem which is studied in [24] by comparing the moduli of such configurations to the moduli of those obtained by parabolic projection. Here a crucial role is played by the fundamental result that all apartments in ℬ^{𝔤} belong to the apartment complex 𝒜^{𝔤}, hence are all conjugate by G. We shall return to these ideas in subsequent work.

Appendix A. Standard Lie theory background

A.1. Semisimplicity and reducibility. Let A ⊆ End_{𝔽}(V) for a finite dimensional vector space V over a field 𝔽; then (V, A) is simple or irreducible if V has exactly two invariant subspaces 0, V ≠ 0, and semisimple or completely reducible if every A-invariant subspace of V has an A-invariant complement, i.e., V has a direct sum decomposition into irreducibles. These notions depend only on the (associative) 𝔽-subalgebra of End_{𝔽}(V) generated by A, and extend to the case that V carries a representation ρ of a group, associative algebra or Lie algebra by taking A to be be the image of ρ in End_{𝔽}(V).

For any field extension 𝔽 ⊆ 𝔽^{c} and any 𝔽-vector space V, let V^{c} := 𝔽^{c} ⊗_{𝔽} V be the induced 𝔽^{c}-vector space, which is functorial in V (any linear map α: V → W induces α^{c}: V^{c} → W^{c}). If 𝔽 is perfect (i.e., any algebraic extension is separable) and (V, A) is semisimple, then so is (V^{c}, A^{c}). In particular, if α ∈ End_{𝔽}(V) then α (i.e., (V, {α})) is semisimple iff its minimal polynomial p_{α} has distinct irreducible factors iff α^{c} is diagonalizable (i.e., V^{c} has a basis of eigenvectors for α^{c}) in a (separable) splitting field 𝔽^{c} for p_{α}.

Lemma A.1. If α ∈ End_{𝔽}(V) is semisimple and α^{c} has eigenvalues S ⊆ 𝔽^{c} (in a separable splitting field 𝔽^{c} for p_{α}) then ad α ∈ End_{𝔽}(𝔤𝔩(V)) is semisimple and (ad α)^{c} = ad(α^{c}) has eigenvalues {λ − μ | λ, μ ∈ S}. Furthermore if, for some additive group homomorphism f: 𝔽^{c} → 𝔽^{c}, β ∈ End_{𝔽^{c}}(V^{c}) is scalar multiplication by f(λ) on the λ-eigenspace of α^{c} for all λ ∈ S, then ad β is a polynomial in ad α^{c} with no constant term.

Proof (see [2, 17]). Let V_{λ} : λ ∈ S denote the eigenspaces of α^{c} in V^{c}. Clearly (ad α)^{c} = ad(α^{c}) has eigenvalue λ − μ on the subspace consisting of all 𝔤𝔩(V^{c}) which map V_{μ} to V_{λ} and all other eigenspaces to zero. These subspaces span 𝔤𝔩(V^{c}) so ad α^{c}, hence ad α, is semisimple. Finally, ad β = q(ad α^{c}) where q(λ − μ) = f(λ) − f(μ) for all λ, μ ∈ S. Since f is additive, such a polynomial q exists by Lagrange interpolation, and has q(0) = 0. □

A.2. Jordan decomposition. Let V, W be vector spaces over a perfect field 𝔽. Any α ∈ 𝔤𝔩(V) has a unique (additive) Jordan decomposition α = α_{s} + α_{n} such that α_{s} is semisimple, α_{n} is nilpotent (i.e., α_{n}^{k} = 0 for k sufficiently large) and [α_{s}, α_{n}] = 0. Further, α_{s} and α_{n} are polynomials in α with no constant term. The following properties of Jordan decomposition, for α ∈ 𝔤𝔩(V) and β ∈ 𝔤𝔩(W), are straightforward and standard [2].

• If α ∈ 𝔠_{𝔤𝔩(V)}(A, B) for some B ⊆ A ⊆ V, then α_{s}, α_{n} ∈ 𝔠_{𝔤𝔩(V)}(A, B).

• If ϕ: V → W is a linear map with ϕ ∘ α = β ∘ ϕ then ϕ ∘ α_{s} = β_{s} ∘ ϕ and ϕ ∘ α_{n} = β_{n} ∘ ϕ.

• In 𝔤𝔩(V ⊗ W), (α ⊗ 1 + 1 ⊗ β)_{s} = α_{s} ⊗ 1 + 1 ⊗ β_{s} and (α ⊗ 1 + 1 ⊗ β)_{n} = α_{n} ⊗ 1 + 1 ⊗ β_{n}.

• In 𝔤𝔩(V^{∗}), α^{T} = (α_{s})^{T} + (α_{n})^{T} is the Jordan decomposition of the transpose α^{T}. Thus Jordan decomposition is preserved in tensor representations of 𝔤𝔩(V) and the first property extends to subspaces B ⊆ A in such a tensor representation. In particular, if V has an algebra structure, and α is a derivation, so are α_{s} and α_{n}. Also recall that a linear map ρ: 𝔤 → 𝔤𝔩(V) is a representation iff for any x ∈ 𝔤, ad(ρ(x)) ∘ ρ = ρ ∘ ad(x): 𝔤 → 𝔤𝔩(V).

Proposition A.2. Let ρ: 𝔤 → 𝔤𝔩(V) be a representation and x ∈ 𝔤. Then ad(x)_{s}, ad(x)_{n} ∈ 𝔡𝔢𝔯(𝔤) (the Lie algebra of derivations of 𝔤), ad(ρ(x))_{s} = ad(ρ(x)_{s}), ad(ρ(x))_{n} = ad(ρ(x)_{n}), ad(ρ(x)_{s}) ∘ ρ = ρ ∘ ad(x)_{s} and ad(ρ(x)_{n}) ∘ ρ = ρ ∘ ad(x)_{n}.

In particular, for ρ: 𝔤 → 𝔤𝔩(V) faithful and x ∈ 𝔤 with ρ(x)_{s} = ρ(x_{s}) and ρ(x)_{n} = ρ(x_{n}) for some x_{s}, x_{n} ∈ 𝔤, it follows that ad(x_{s}) + ad(x_{n}) is the Jordan decomposition of ad(x).

Conversely, the following is the key to the proof of the abstract Jordan decomposition (or preservation of Jordan decomposition) for semisimple Lie algebras [2, 17, 21].

Proposition A.3. Let ρ: 𝔤 → 𝔤𝔩(V) be a semisimple representation over a field 𝔽 of characteristic zero, and suppose x = x_{s} + x_{n} where ad(x_{s}) is semisimple, ad(x_{n}) is nilpotent, x_{n} ∈ [𝔤, 𝔤] and [x_{s}, x_{n}] = 0. Then ρ(x)_{s} = ρ(x_{s}) and ρ(x)_{n} = ρ(x_{n}).

Proof. Since ad(x_{s}) is semisimple, ρ(x_{s})_{n} ∈ 𝔠_{𝔤𝔩(V)}(ρ(𝔤)) by Proposition A.2, hence the inverse image of its span is an ideal in 𝔤 acting nilpotently on V, so ρ(x_{s})_{n} = 0 since 𝔫𝔦𝔩_{ρ}(𝔤) = ker ρ; thus ρ(x_{s}) is semisimple. Similarly, since ad(x_{n}) is nilpotent, ρ(x_{n})_{s} ∈ 𝔠_{𝔤𝔩(V)}(ρ(𝔤)), hence ρ(x_{n})_{s}^{c} ∈ 𝔠_{𝔤𝔩(V^{c})}(ρ(𝔤)^{c}) for any field extension 𝔽 ⊆ 𝔽^{c}. Since ρ(x_{n})_{s} is a polynomial in ρ(x_{n}), ρ(x_{n})_{s}^{c} preserves any 𝔤-invariant subspace of V^{c}; but also ρ(x_{n})_{s} = ρ(x_{n}) − ρ(x_{n})_{n}, where the first term is in [ρ(𝔤), ρ(𝔤)] and the second term is nilpotent, so ρ(x_{n})_{s}^{c} has vanishing trace on any such subspace. If 𝔽^{c} is the algebraic closure of 𝔽, ρ(x_{n})_{s}^{c} then vanishes on any minimal invariant subspace of V^{c}. Since ρ is semisimple, ρ(x_{n})_{s} = 0, i.e., ρ(x_{n}) is nilpotent. Now [ρ(x_{s}), ρ(x_{n})] = ρ([x_{s}, x_{n}]) = 0, so the result follows. □

A.3. Invariant forms. An invariant form on a Lie algebra 𝔤 is a symmetric bilinear form (x, y) ↦ ⟨x, y⟩ such that ⟨[z, x], y⟩ + ⟨x, [z, y]⟩ = 0 for all x, y, z ∈ 𝔤. For a subspace 𝔰 ⊆ 𝔤, we define 𝔰^{⊥} = {x ∈ 𝔤 | ∀y ∈ 𝔰, ⟨x, y⟩ = 0}. For 𝔰, 𝔱 ⊆ 𝔤, 𝔰 ⊆ 𝔱^{⊥} iff 𝔱 ⊆ 𝔰^{⊥}, and we say 𝔰, 𝔱 are orthogonal, written 𝔰 ⊥ 𝔱. If 𝔰 ⊥ 𝔰, we say 𝔰 is isotropic; for any subspace 𝔰, 𝔰 ∩ 𝔰^{⊥} is isotropic. Invariance means that (x, y, z) ↦ ⟨[x, y], z⟩ is a 3-form on 𝔤, and hence for any subspaces 𝔯, 𝔰, 𝔱 ⊆ 𝔤, 𝔯 ⊥ [𝔰, 𝔱] iff 𝔰 ⊥ [𝔱, 𝔯] iff 𝔱 ⊥ [𝔯, 𝔰]. Thus 𝔯 ⊆ [𝔰, 𝔱]^{⊥} iff [𝔯, 𝔰] ⊆ 𝔱^{⊥} iff 𝔯 ⊆ 𝔠_{𝔤}(𝔰, 𝔱^{⊥}), and so 𝔠_{𝔤}(𝔰, 𝔱^{⊥}) = [𝔰, 𝔱]^{⊥} = [𝔱, 𝔰]^{⊥} = 𝔠_{𝔤}(𝔱, 𝔰^{⊥}).

Proposition A.4. Let ⟨·, ·⟩ be an invariant form on 𝔤 and let 𝔰 be a subspace of 𝔤.

(1) 𝔫_{𝔤}(𝔰) ⊆ 𝔠_{𝔤}(𝔰, 𝔰^{⊥⊥}) = [𝔰, 𝔰^{⊥}]^{⊥} = 𝔫_{𝔤}(𝔰^{⊥}), 𝔠_{𝔤}(𝔰) ⊆ 𝔠_{𝔤}(𝔰, 𝔤^{⊥}) = [𝔰, 𝔤]^{⊥} and 𝔷(𝔤) ⊆ 𝔠_{𝔤}(𝔤, 𝔤^{⊥}) = [𝔤, 𝔤]^{⊥} with equality (in each containment) if ⟨·, ·⟩ is nondegenerate.

(2) If 𝔰 ⩽ 𝔤, then [𝔰, 𝔰^{⊥}] ⊆ 𝔰^{⊥}, hence also 𝔰 ∩ 𝔰^{⊥} ⊴ 𝔰; in particular, 𝔤^{⊥} is an isotropic ideal in 𝔤 (which is zero iff ⟨·, ·⟩ is nondegenerate).

(3) If 𝔰 ⊴ 𝔤, then 𝔰^{⊥} ⊴ 𝔤, hence also 𝔰 ∩ 𝔰^{⊥} ⊴ 𝔤, [𝔰, 𝔰^{⊥}] ⊴ 𝔤 and 𝔰 + 𝔰^{⊥} ⊴ 𝔤.

(4) 𝔤 ⊥ [𝔰, 𝔰] if and only if [𝔤, 𝔰] ⊥ 𝔰. In particular, if ⟨·, ·⟩ is nondegenerate, any isotropic ideal in 𝔤 is abelian.

Proposition A.5. Suppose 𝔤 is a Lie algebra with a nondegenerate invariant form and no nontrivial abelian ideals. Then 𝔤 is semisimple.

Proof (see [12]). We induct on the dimension of 𝔤. Let 𝔩 be a minimal nontrivial ideal in 𝔤; then 𝔩 ∩ 𝔩^{⊥} is an isotropic ideal in 𝔤, hence abelian by Proposition A.4 (4). Since 𝔩 is nonabelian (hence simple) by assumption, 𝔩 ∩ 𝔩^{⊥} = 0. Hence 𝔤 is the orthogonal direct sum of 𝔩 and 𝔩^{⊥}, and the latter is semisimple by induction. □

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  E-mail address: D.M.J.Calderbank@bath.ac.uk

Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.

  E-mail address: passawan@su.ac.th

Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand.