On the equivariant cohomology of isotropy actions

On the equivariant cohomology of

isotropy actions

Als Dissertation

dem Fachbereich Mathematik und Informatik

der Philipps–Universit¨

at Marburg vorgelegt

von M. Sc. Sam Hagh Shenas Noshari

aus Hamburg

Erstgutacher:

Prof. Dr. Oliver Goertsches

Zweitgutachter:

Prof. Dr. Augustin–Liviu Mare

Einreichungstermin:

24. Juli 2018

Pr ¨

ufungstermin:

26. September 2018

Hochschulkennziffer:

1180

Zusammenfassung

Sei𝐺 eine kompakte, zusammenh¨angende Lie Gruppe und 𝐾 ⊆ 𝐺 eine abgeschlossene Untergruppe. Wir zeigen, dass die Isotropiewirkung von𝐾 auf 𝐺/𝐾 ¨aquivariant formal ist und der Raum 𝐺/𝐾 formal im Sinne rationaler Homotopietheorie, falls es sich bei 𝐾 um die Identit¨atskomponente des Schnitts der Fixpunktmengen zweier verschiedener Involutionen auf𝐺 handelt, 𝐺/𝐾 also ein ℤ2× ℤ2–symmetrischer Raum ist. Ist𝐾 die Identit¨ats-komponente der Fixpunktmenge einer einzelnen Involution und𝐻 ⊆ 𝐺 eine abgeschlossene, zusammenh¨angen-de Untergruppe, die𝐾 enth¨alt, so zeigen wir, dass auch die Wirkung von 𝐾 auf 𝐺/𝐻 durch Linksmultiplikation ¨

aquivariant formal ist. Letztere Aussage ist ¨aquivalent zum Hauptresultat in [6], wird hier aber mit anderen Mit-teln bewiesen, n¨amlich durch Angabe eines algebraischen Modells f ¨ur die ¨aquivariante Kohomologie gewisser Wirkungen.

Abstract

Let𝐺 be a compact connected Lie group and 𝐾 ⊆ 𝐺 a closed subgroup. We show that the isotropy action of 𝐾 on 𝐺/𝐾 is equivariantly formal and that the space 𝐺/𝐾 is formal in the sense of rational homotopy theory whenever𝐾 is the identity component of the intersection of the fixed point sets of two distinct involutions on 𝐺, so that 𝐺/𝐾 is a ℤ2× ℤ2–symmetric space. If𝐾 is the identity component of the fixed point set of a single involution and𝐻 ⊆ 𝐺 is a closed connected subgroup containing 𝐾, then we show that the action of 𝐾 on 𝐺/𝐻 by left–multiplication is equivariantly formal. The latter statement is equivalent to the main result of [6], but is proved by different means, namely by providing an algebraic model for the equivariant cohomology of certain actions.

Contents

Zusammenfassung iii

Abstract v

I. Introduction 1

1. Introduction and background 1

2. Previous results 3

II. ℤ2× ℤ2–symmetric spaces 5

1. ℤ2× ℤ2–symmetric spaces 5

2. Preliminaries 6

3. Automorphisms 8

4. Normal forms for strongly orthogonal roots 11

5. Cohomology of associated subalgebras 22

6. Equivariant and ordinary cohomology of simpleℤ2× ℤ2–symmetric spaces 32

III. An algebraic model for the eqivariant cohomology of isotropy actions 43

1. g–actions 43

2. Constructingg–actions 46

3. Compatibility with existing actions 49

4. An exact sequence 50

5. Applications to smooth manifolds 52

Chapter I.

Introduction

1. Introduction and background

This thesis is concerned with𝐺–spaces, that is, topological spaces together with a continuous (left) action of a fixed (smooth) Lie group𝐺, and a certain invariant associated with such spaces, their equivariant cohomology. To motivate its definition, consider the problem of assigning to a𝐺–space 𝑋 an invariant that gives the same answer on any𝐺–space isomorphic to 𝑋 but yet discerns as many distinct isomorphism classes of 𝐺–spaces as possible. Perhaps among the easiest such invariants that one might come up with (apart from the isomorphism class of𝑋) is the cohomologyH(𝑋/𝐺) of the orbit space 𝑋/𝐺; of course, one might consider arbitrary coefficient groups, but here and thereafter we confine ourselves to singular real cohomology or to de Rham cohomology if the space under consideration happens to be a smooth manifold. In any case, it appears to be common understanding thatH(𝑋/𝐺) is a reasonable invariant if the 𝐺–action is free, but less well–behaved for actions with non–trivial isotropy. A frequently given example of an action justifying this last statement is the action of the circle𝑆1on the unit sphere𝑆2by rotation about a fixed axis. This action has exactly two fixed points, namely the poles of the rotation axis, and its orbit space is homeomorphic to the closed unit interval, hence has trivial cohomology. To overcome this difficulty one replaces𝑋 by what is now called the Borel construction and usually denoted 𝑋𝐺. Originally introduced in [2], this is the space𝑋𝐺∶= (𝐸𝐺 ×𝑋)/𝐺 obtained from a contractible space 𝐸𝐺 on which 𝐺 acts freely (from the right), such as the total space in the universal 𝐺–bundle 𝐸𝐺 → 𝐵𝐺 over the classifying space𝐵𝐺. The action of 𝐺 on 𝐸𝐺 × 𝑋 is the diagonal action, induced by the assignment 𝑔.(𝑒, 𝑥) = (𝑒𝑔−1, 𝑔𝑥) for 𝑔 ∈ 𝐺 and (𝑒, 𝑥) ∈ 𝐸𝐺×𝑋, and the equivariant cohomology then is defined as H_𝐺(𝑋) ∶= H(𝑋𝐺). Note that the 𝐺–

action on𝐸𝐺 × 𝑋 is free. Another indication that H_𝐺(𝑋) is a useful invariant is that it can actually be computed in many situations: quite generally, if𝐺 acts locally freely on a space 𝑋, then the map 𝑋𝐺 → 𝑋/𝐺 induced by the quotient map𝑋 → 𝑋/𝐺 yields an isomorphism H(𝑋/𝐺) → H_𝐺(𝑋), cf. [12, Section C.2.1]. On the other hand,H_𝐺(⋅) satisfies the axioms of a generalized cohomology theory with morphisms replaced by 𝐺–equivariant morphisms, so that, for example, an equivariant Mayer–Vietoris sequence is available. In very much the same way as the Mayer–Vietoris sequence can be used to compute the ordinary cohomology of spheres, its equivariant counterpart can be utilized to compute the𝑆1–equivariant cohomology of the action on𝑆2considered above, e. g. by means of the open cover consisting of the two open sets that one obtains by removing one of the poles of the rotation axis at a time. The conclusion now is thatH

𝑆1(𝑆2) = H(𝐵𝑆1) ⊕ H(𝐵𝑆1) in non–zero degrees, because for any Lie group𝐺 the equivariant cohomology of a single point is given by H

𝐺(∗) = H(𝐵𝐺) and 𝑆1

acts freely on𝑆2outside its fixed point set.

The previous eaxmple can be written more concisely asH

𝑆1(𝑆2) = H(𝐵𝑆1)⊗H(𝑆2) (recall that the classifying space of𝑆1 isℂ𝑃∞, whose cohomology ring is a polynomial algebra in one variable of degree2), and if one considersH

𝑆1(𝑆2) as a H(𝐵𝑆1)–module via the morphism of rings H_𝑆1(∗) → H_𝑆1(𝑆2) induced by the constant map𝑆2→ {∗}, then this equality is even valid as H(𝐵𝑆1)–modules, showing that the 𝑆1action on𝑆2is in fact equivariantly formal. This name was coined in [10] for actions of compact connected Lie groups𝐺 on topological spaces𝑋, although its defining property, the collapse of the Serre spectral sequence associated with the fibration 𝑋 ↪ 𝑋𝐺 → 𝐵𝐺 on the second page, was already investigated in [2], mostly for actions of tori and finite cyclic

groups of prime order. It is also worth pointing out that for a general fibration𝐹 ↪ 𝐸 → 𝐵 with connected fiber𝐹 and path–connected base 𝐵 of finite type the degeneration of the associated Serre spectral sequence at the 𝐸2–term is equivalent to surjectivity of the inclusion induced mapH(𝐸) → H(𝐹). In this situation, 𝐹 is traditionally said to be(totally) non–cohomologous to zero in𝐸, see [21, p. 148]. This shows the equivalence of the first two items in the following list of well–known characterizations of equivariant formality.

Proposition 1.1. Let𝐺 be a compact connected Lie group with maximal torus 𝑇 and 𝑋 a connected 𝐺–space. The following statements are equivalent.

(1) The𝐺–action on 𝑋 is equivariantly formal.

(2) Fiber inclusion of the fibration𝑋 ↪ 𝑋𝐺→ 𝐵𝐺 induces a surjection H_𝐺(𝑋) → H(𝑋). (3) The𝑇–action on 𝑋 obtained by restriction of the 𝐺–action is equivariantly formal.

(4) TheH(𝐵𝐺)–module H_𝐺(𝑋) is free.

(5) We have an equality of total Betti numbersdim H(𝑋) = dim H(𝑋𝑇), where 𝑋𝑇 is the fixed point set of the induced𝑇–action.

Actions on spaces with vanishing odd degree cohomology are equivariantly formal, as are symplectic mani-folds with a Hamiltonian action [10, Theorem 14.1]. Further examples of equivariantly formal actions areisotropy actions on symmetric spaces [6] and, more generally, on homogeneous spaces𝐺/𝐾 in which the subgroup 𝐾 is the connected component of the fixed point set of an arbitrary Lie group automorphism on𝐺, see [8]. Here, the isotropy action associated with a homogeneous space𝐺/𝐾 is the action of 𝐾 on 𝐺/𝐾 induced by left multiplica-tion, that is, by the assignment(𝑘, 𝑔𝐾) ↦ 𝑘𝑔𝐾 for all 𝑘 ∈ 𝐾, 𝑔𝐾 ∈ 𝐺/𝐾. Our main contribution with this thesis now is that we extend the list of actions which are known to be equivariantly formal by one more item.

In theorem II.1.2 below we will show that the isotropy action associated with𝐺/𝐾 is equivariantly formal if 𝐾 is the connected component of the common fixed point set of two distinct commuting involutions on 𝐺, in which case𝐺/𝐾 is said to be a ℤ2×ℤ2–symmetric space, provided that none of the automorphisms is the identity map. The proof borrows some ideas from the proof of the main result of [8], which we therefore summarize in section 2. The key step is to construct a subgroup𝐻 of 𝐺 which shares a maximal torus with 𝐾 and for which the cohomology of𝐺/𝐻 is more accessible than that of 𝐺/𝐾, as then the isotropy action associated with 𝐺/𝐻 is equivariantly formal if and only if so is the isotropy action associated with𝐺/𝐾. Since eventually we want to be able to give a description of a maximal torus of𝐾 in terms of a maximal torus of 𝐺, we thus study in section II.2 the problem of reconstructing a maximal torus of𝐺 from a fixed maximal torus 𝑆 of 𝐾. There is a general solution to this problem. Namely, upon fixing a reference torus𝑇 which is maximal in 𝐺 and contains 𝑆, one finds that the complexification of the Lie algebra of the centralizer of𝑆 in 𝐺, which abstractly is the union of all maximal tori of𝐺 containing 𝑆, is the direct sum of the complexification tℂoft and the weight spaces of all gℂ–roots that vanish ons. While it is known that no such root exists if𝐺/𝐾 is a symmetric space, certain gℂ– roots might (and in general will) restrict to zero ons if𝐺/𝐾 is ℤ2× ℤ2–symmetric, even if the automorphisms defining𝐾 are both inner. Fortunately, however, the set of all such roots is strongly orthogonal, meaning that the sum of two elements of that set is not a root (see [16, p. 396]), and already sets of orthogonal roots in irreducible root systems can be classified up to application of a Weyl group element. This we have done in section II.4.

What makes this classification particularly useful is that in the present situation the maximal torus𝑆 of 𝐾 is the intersection of the kernels of all roots vanishing ons and the fixed point set on𝑇 of one of the automorphisms defining𝐾. All of this data can be formulated in terms of the root system of gℂand the list of possible sets of roots vanishing ons is further constrained by the requirement that the automorphisms defining𝐾 be involutive. At this point, one could thus go through the list of all possible candidates for𝑆 and verify that the subalgebra 𝑆 acts in an equivariantly formal fashion on 𝐺/𝑆. We proceed differently and show that we may sequentially modify the automorphisms defining𝐾 so as to almost always assume that one of them is an inner automorphism and that the semisimple part of the fixed point set of this inner automorphism realizes a subdiagram of the Dynkin diagram ofgℂ. Homogeneous spaces arising from such subgroups have tractable cohomology, which we determine in section II.5. Building on these results, in section II.6 we finally traverse the list of simple Lie groups, determine in each case the desired subgroup𝐻, and show that the isotropy action of 𝐻 on 𝐺/𝐻 is equivariantly formal.

Our second contribution, which actually is equivalent to the main theorem of [6], is theorem III.5.10. The statement here is that for every compact connected Lie group𝐺 and the connected component 𝐾 of the fixed

point set of any involution on𝐺 the action of 𝐾 on 𝐺/𝐻 by left–multiplication is equivariantly formal whenever 𝐻 is a closed connected subgroup of 𝐺 that contains 𝐾. Of course, the novelty is not the statement itself, but rather its proof, as it relies on an algebraic model for the equivariant cohomology of the𝐾–action on 𝐺/𝐻 which is solely built out of the Lie algebras of𝐺, 𝐻, and 𝐾, and the inclusions of the latter two into the former. We note that such a model has been realized only very recently in [4, Sect. 3.1] using methods from rational homotopy theory, while our model is established by quite elementary means using the Cartan model for equivariant coho-mology. The drawback of our method is that it only captures theA

k–module structure ofH𝐾(𝐺/𝐻), Ak⊆ S(k∗)

the space ofk–invariant polynomials on k∗, whereas the model given in [4] is isomorphic toH_𝐾(𝐺/𝐻) via an isomorphism ofA

k–algebras. To explain this deficiency, consider an action of a compact connected Lie group𝐺

on a smooth manifold𝑀. The basic observation we exploit to construct our model is that there is a sequence of vector subspacesΩ(𝑀)𝐺,i_gΩ(𝑀)𝐺,(i_g)2Ω(𝑀)𝐺, … whose sum is stable under the differential on Ω(𝑀); here, Ω(𝑀)𝐺 is the space of𝐺–invariant forms on 𝑀 and i_gdenotes the image of the operatori ∶ g → End(Ω(𝑀)), 𝑋 ↦ i_𝑋, contracting a form with the vector field induced by𝑋 ∈ g. This leads to an additive, quasi–isomorphic model ofΩ(𝑀) and hence to a model of H_𝐺(𝑀) which is isomorphic as an A

g–module.

Despite the lack of a ring structure our proof of theorem III.5.10, in contrast to the original proof in [6], does not rely on any classification result. Again, it has to be noted that a classification–free proof of the main theorem of [6] and even of [8, Theorem 1.1] was already achieved in [4, Theorem 7.8]. However, the proof presented in [4] uses𝐾–theory and relies on a reduction to the case when 𝐺 is simple, while our proof works equally well for simple and non– simple Lie groups and only uses the decomposition ofg into the eigenspaces of the involution defining𝐾.

2. Previous results

Starting with this section we will almost exclusively consider isotropy actions on homogeneous spaces and be concerned with the question when such an action is equivariantly formal. It thus seems appropriate to make the following definition: given a compact connected Lie group𝐺 and a closed connected subgroup 𝐾, we say that the pair(𝐺, 𝐾) is equivariantly formal if the action of 𝐾 on 𝐺/𝐾 by left–multiplication is equivariantly formal; we also say that(𝐺, 𝐾) is formal or a Cartan pair if the homogeneous space 𝐺/𝐾 is formal in the sense of rational homotopy theory, which means that there exist commutative differential gradedℝ–algebras 𝐴1, … , 𝐴𝑛 and a

chain of morphismsΩ(𝐺/𝐾) → 𝐴1 ← 𝐴2 → … → 𝐴𝑛 ← H(𝐺/𝐾), each of which induces an isomorphism

on the level of cohomology. While this definition is valid for arbitrary (connected) manifolds, not just𝐺/𝐾, we prefer to use the following equivalent characterization of formality which is available in this particular situation: we recall from [11] that the spaceΩ(g)gofg–invariant forms on g is an exterior algebra over an oddly graded subspace𝑃_g ⊆ Ω(g)gof dimensionrank g, called primitive space of g, and that the Samelson subspace 𝑃 of the pair(g, k) is the graded subspace of 𝑃_gwhose elements, considered as elements ofH(g), are contained in the image of the inclusion induced mapΩ(g, k) → Ω(g). Then we have dim 𝑃 ≤ rank 𝑔 − rank k, cf. [11, Theorem V, sect. 10.4], and the pair(𝐺, 𝐾) is formal if and only if the previous inequality is actually an equality; see [11, Theorem VIII, sect. 10.4] for this and various other reformulations of formality.

These preliminary notions being introduced, we briefly summarize the proof of the main result in [8] and show how [8] is related to [7].

Theorem 2.1 ([8, Theorem 1.1]). Let𝐺 be a compact connected Lie group and 𝐾 ⊆ 𝐺 the identity component of the fixed point set of an automorphism on𝐺. Then the pair (𝐺, 𝐾) is (equivariantly) formal.

Note that according to [4, Theorem A] an equivariantly formal pair(𝐺, 𝐾) with both 𝐺 and 𝐾 connected is necessarily formal as well. That formality of a pair(𝐺, 𝐾) does not necessarily enforce equivariant formality of (𝐺, 𝐾) is shown in [8, Example 3.7].

The proof of theorem 2.1 given in [8] can be divided into two major steps: the first step is to show that it suffices to consider pairs (𝐺, 𝐾) satisfying the assumptions of theorem 2.1 and for which 𝐺 is simple. In the

second step one actually proves theorem 2.1 for simple groups𝐺. Both steps crucially rely on the following general principle.

Theorem 2.2 ([4, Theorem 2.2]). Let𝐾 and 𝐻 be equal rank closed connected subgroups of a compact connected Lie group𝐺 and such that 𝐻 ⊆ 𝐾. Then (𝐺, 𝐾) is equivariantly formal if and only if so is (𝐺, 𝐻).

A proof of theorem 2.2 is also contained in [8, Proposition 3.5] under the additional hypothesis that the pairs(𝐺, 𝐾) and (𝐺, 𝐻) are formal. Since by [22, p. 212] the pair (𝐺, 𝐾) is formal if and only if so is (𝐺, 𝐻), it follows from [4, Theorem A] that this seemingly more restrictive setting is actually equivalent to the general situation considered in theorem 2.2; the proof of the first item of [8, Proposition 3.5], which essentially states that formality of(𝐺, 𝐾) is equivalent to that of (𝐺, 𝐻), is erroneous though1.

The most important consequence of theorem 2.2 is that whenever𝐻 and 𝐾 are closed connected subgroups of a compact connected Lie group𝐺 and 𝑇 is a maximal torus of both 𝐻 and 𝐾, then the pair (𝐺, 𝐾) is equivariantly formal if and only if(𝐺, 𝐻) is equivariantly formal, because this property is satisfied by either one of the pairs if and only if it is satisfied by the pair(𝐺, 𝑇). In this way one can reduce the question of equivariant formality of pairs(𝐺, 𝐾) as in theorem 2.1 and with 𝐺 simple to pairs for which 𝐾 is the identity component of the fixed point set of a finite–order automorphism. The homogeneous space𝐺/𝐾 arising from such a pair (𝐺, 𝐾) is called a𝑘–symmetric space (𝑘 ≥ 0 the order of the automorphism defining 𝐾) or generalized symmetric space, and the question whether or not(𝐺, 𝐾) is equivariantly formal was already answered affirmatively in [7]. In fact, by [7, Proposition 3.7]𝐾 shares a maximal torus with a subgroup 𝐻 dubbed “folded subgroup” in [7], because its Dynkin diagram is obtained from the Dynkin diagram of𝐺 by a process commonly called folding, and it was observed in [7, Theorem 5.5] that𝐻 is (totally) non–cohomologous to zero in 𝐺, that is, the fiber inclusion in the fibration𝐻 ↪ 𝐺 → 𝐺/𝐻 induces a surjection in cohomology. That (𝐺, 𝐻) is formal then is a classical result (cf. [11, Corollary I, sect. 10.19]) and equivariant formality follows from

Proposition 2.3 ([7, Proposition 2.6]). Let𝐺 be a compact connected Lie group, 𝐾 a closed connected subgroup. If𝐾 is totally non–cohomologous to zero in 𝐺, then (𝐺, 𝐾) is equivariantly formal.

The question of (equivariant) formality being settled for pairs in which the ambient group is simple, we return to the general situation considered in theorem 2.1. One now observes that whenever(𝐺, 𝐾) and (𝐺′, 𝐾′) are two pairs of compact and connected Lie groups such that there is an isomorphism of Lie algebra pairs (g, k) → (g′_{, k}′_{), then (𝐺, 𝐾) is (equivariantly) formal if and only if so is (𝐺}′_{, 𝐾}′_{), cf. [7, Corollary 2.4]. Thus, we}

call a Lie algebra pair(u0, h0) equivariantly formal if there exists a compact connected Lie group 𝑈 and a closed connected subgroup𝐻 such that (𝑈 , 𝐻) is equivariantly formal and (u, h) is isomorphic to (u0, h0), for then any other compact connected Lie group pair with matching Lie algebras is equivariantly formal as well. Passing to the level of Lie algebras, we denote by𝜎 the automorphism on g whose fixed point set is k. Then g decomposes as a direct sum of𝜎–invariant subalgebras g1, … , g𝑛which are minimal in the sense that none of them contains a non–trivial proper𝜎–invariant subalgebra, k decomposes accordingly as the direct sum of the subalgebras g₁∩ k, … , g_𝑛∩ k, and it only remains to check that each of the pairs (g_𝑖, g_𝑖∩ k) is (equivariantly) formal. This is indeed the case: the pair(g𝑖, g𝑖∩ k) is isomorphic to a Lie algebra pair (u ⊕ … ⊕ u, Δ(f)), where u is a compact simple Lie algebra andΔ(f) is the diagonal embedding of the fixed point set f of an automorphism on u, and Δ(u) is totally non–cohomologous to zero in g; these two facts together imply that (g𝑖, g𝑖∩ k) is (equivariantly)

formal, see [8, Section 5] for more details.

1

Namely, instead of the displayed equation in the proof of the first part of [8, Proposition 3.5] one has to consider an equation of the form 𝜏(𝜔)|t= ∑_𝑖𝑓𝑖|t⋅ 𝑔𝑖with𝑓_𝑖polynomials in the image of the transgression and𝑔_𝑖non–constant polynomials invariant under the Weyl group of𝐻. Averaging both sides over the Weyl group of 𝐾 gives the desired conclusion.

Chapter II.

ℤ

× ℤ

–symmetric spaces

1. ℤ₂× ℤ₂–symmetric spaces

There is yet another generalization of symmetric spaces that also incorporates the notion of𝑘–symmetric spaces, the so–calledΓ–symmetric spaces introduced in [19].

Definition 1.1. LetΓ be a finite Abelian group, 𝐺 a connected Lie group, and 𝐾 ⊆ 𝐺 a closed subgroup. The homogeneous space𝐺/𝐾 is called Γ–symmetric if there exists an injective group homomorphism Γ ↪ Aut(𝐺) such that(𝐺Γ)0⊆ 𝐾 ⊆ 𝐺Γ, where𝐺Γis the common fixed point set of the automorphismsΓ ⊆ Aut(𝐺).

Since every finite Abelian group is a product of cyclic groups, the above definition can be rephrased by saying that a homogeneous space𝐺/𝐾 with 𝐺 connected and 𝐾 ⊆ 𝐺 closed is Γ = ℤ𝑘

1× … × ℤ𝑘𝓁–symmetric if there exist𝓁 distinct commuting automorphisms 𝜎1, … , 𝜎𝓁 of𝐺, with 𝜎𝑖of order𝑘𝑖, such that

(𝐺𝜎1_{∩ … ∩ 𝐺}𝜎𝓁_{)0⊆ 𝐾 ⊆ (𝐺}𝜎1_{∩ … ∩ 𝐺}𝜎𝓁_).

Theorem 1.2. Let𝐺 be a compact connected Lie group, 𝜎₁and𝜎2two involutions on𝐺, and suppose that 𝐺/𝐾 is aℤ2× ℤ2–symmetric space, where𝐾 = (𝐺𝜎1∩ 𝐺𝜎2)0. Then the pair(𝐺, 𝐾) is (equivariantly) formal.

We note that the classification ofℤ2×ℤ2–symmetric spaces𝐺/𝐾 with 𝐺 a simple Lie group was achieved in [1] and [17], but while we do make use of the classification of simple Lie algebras and finite–order automorphisms thereon, our proof of theorem 1.2 does not rely on the classification ofℤ2× ℤ2–symmetric spaces.

Recall (cf. [14, p. 130]) that a Lie algebrag is compact, if so is the connected subgroup ofAut(g) with Lie algebra{ad𝑋 | 𝑋 ∈ g}. According to [14, Corollary 6.7, chap. II] this is the case if and only if there is a compact Lie group with Lie algebra (isomorphic to)g. If g is compact and semisimple, then every connected Lie group with Lie algebrag is compact (see [14, Theorem 6.9, chap. II]), and we call a subalgebra h⊆ g compact, if the connected subgroup𝐻 ⊆ 𝐺 with Lie algebra h is compact, where 𝐺 is the simply–connected Lie group with Lie algebrag. For the sequel and for the proof of theorem 1.2 it will be convienent to introduce the following relation on the set of all compact subalgebras of a compact semisimple Lie algebrag: two such subalgebras h, k ⊆ g are related, if there exists a sequence of compact subalgebrasm0, … , m_𝑘+1ofg such that m0 = h, m_𝑘+1 = k and if for all𝑖 = 0, … , 𝑘 the subalgebras m𝑖 andm𝑖+1share a common maximal torus, that is, if there exists a maximal toruss ⊆ m𝑖 which also is maximal torus of m𝑖+1. This defines an equivalence relation and we denote the equivalence class of a subalgebrak by[k]f. Note that ifk ⊆ g is a compact subalgebra, then the pair (g, k) is (equivariantly) formal if and only if there exists a subalgebrah∈ [k]fsuch that(g, h) is so. Now theorem 1.2 will be a consequence of

Theorem 1.3. In addition to the hypotheses of theorem 1.2 assume that𝐺 is simple. Then there exists a compact subalgebrah∈ [k]_fwhich is totally non–cohomologous to zero ing.

Proof of theorem 1.2 using theorem 1.3. Let[g, g] = g1⊕ … ⊕ g𝑚 be the decomposition of the semisimple part ofg into its simple ideals and consider the subgroupΓ = {id_g, 𝜎1, 𝜎2, 𝜎1𝜎2} inside the group of Lie algebra automorphisms ofg. It is isomorphic toℤ2×ℤ2and acts naturally on ∶= {g1, … , g𝑚}. Moreover, as was already observed in [8, Section 5], it will suffice to check that for each𝑖 the pair (m, m ∩ k), where m = ∑_𝛾∈Γ𝛾(g𝑖), is (equivariantly) formal.

Seth∶= g𝑖and choose representatives𝛾1Γ_h, … , 𝛾𝑝Γ_hfor each class inΓ/Γ_h, whereΓ_his the isotropy subgroup ath of the action ofΓ on , 𝑝 = |Γ/Γ_h|, and 𝛾1= id_g. Then an isomorphism of Lie algebras is given by the map

Φ∶ h ⊕ … ⊕ h → m, (𝑋1, … , 𝑋𝑝) ↦ 𝛾1(𝑋1) + … + 𝛾𝑝(𝑋𝑝),

because𝛾𝑠(h) and 𝛾𝑡(h) are distinct ideals of [g, g] for 𝑠 ≠ 𝑡 and m = ⨁𝑝_𝑠=1𝛾𝑠(h). Moreover, if f ⊆ h is the common fixed point set of all elements inΓ_h, thenΦ maps Δ(f), the diagonal embedding of f, isomorphically onto m ∩ k: in fact, any element𝛾 ∈ Γ permutes Γ/Γ_h, so there exist a permutation𝜋 on {1, … , 𝑝} and elements 𝛾_𝑠′∈ Γ_hfor each𝑠 such that 𝛾𝛾𝑡 = 𝛾_𝜋(𝑡)𝛾_𝑡′for all𝑡. Then we have, for all 𝑋 ∈ f:

𝛾(Φ(𝑋, … , 𝑋)) = ∑𝑝

𝑠=1𝛾𝛾𝑠(𝑋) = 𝑝

∑

𝑠=1𝛾𝜋(𝑠)(𝑋) = Φ(𝑋, … , 𝑋).

To prove the converse inclusion, note that ifΦ(𝑋1, … , 𝑋𝑝) is fixed by some 𝛾𝑖, then𝑋𝑖 = 𝑋1, because we chose 𝛾1 = id and because 𝛾𝑖𝛾𝑗(𝑋𝑗) ∈ h only holds if 𝑖 = 𝑗. Hence, if Φ(𝑋1, … , 𝑋𝑝) is fixed by all elements of Γ, then

𝑋1= 𝑋2= … = 𝑋𝑝and also𝑋1∈ f, because every 𝛾 ∈ Γ_hleavesh invariant.

Thus, it will suffice to check that(⨁𝑝_𝑠=1h, Δ(f)) is (equivariantly) formal. But an orbit of Γ is either of length 1, 2, or 4, and if 𝑝 = 1, then f is just the common fixed point set of 𝜎1and𝜎2, whence the pair in question is (equivariantly) formal by theorem 1.3. If𝑝 = 2, then Γ_hcontains one non–trivial element𝜎, so f = h𝜎is the fixed point set of an involution, and it was observed in [8, Section 5] that(h ⊕ h, Δ(f)) is (equivariantly) formal in this case as well: indeed, if we choosen∈ [f]fto be totally non–cohomologous to zero inh, which is possible by [8, Section 4] or [7, Theorem 5.5], thenΔ(n) is totally non–cohomologous to zero in h ⊕ h as well and Δ(n) ∈ [Δ(f)]f. Finally, if𝑝 = 4, then Γ_h is trivial, whencef= h. As is well–known, Δ(h) is totally non–cohomologous to zero inh⊕ h ⊕ h ⊕ h.

2. Preliminaries

Let𝐺 be a compact connected Lie group and 𝜎 a finite–order automorphism on 𝐺. It follows from [14, Lemma 5.3, chap. X], that the centralizerZ_g(s) in g of any maximal torus s of g𝜎 is a maximal torus ofg, and hence the unique maximal torus ofg containing s. Thus, if𝜎1, … , 𝜎𝓁 are commuting automorphisms of𝐺, then there is a maximal torus ofg which is invariant for all𝜎𝑖,𝑖 = 1, … , 𝓁. In fact, put 𝜎𝓁+1= id𝐺 and suppose that for some𝑖, 1 ≤ 𝑖 ≤ 𝓁, t𝑖is a maximal torus ofk𝑖, where

k_𝑖 ∶= g𝜎𝑖∩ … ∩ g𝜎𝓁+1,

and thatt𝑖 is invariant under𝜎1, … , 𝜎𝓁+1; such a torus exists for𝑖 = 1, because k1is the common fixed point set of𝜎1, … , 𝜎𝓁, whence any maximal torus ofk1is fixed by each𝜎𝑗. Since all𝜎𝑗 commute,𝜎𝑖 then restricts to a finite–order automorphism𝜎𝑖∶ k𝑖+1 → k𝑖+1with fixed point setk𝑖. Ask𝑖+1is the common fixed point set of 𝜎𝑖+1, … , 𝜎𝓁 and thus the Lie algebra of a compact Lie group, we conclude thatt𝑖+1= Z_k

𝑖+1(t𝑖) is a maximal torus ofk𝑖+1. By definition,t𝑖+1is fixed by𝜎𝑖+1, … , 𝜎𝓁, and if𝑗 ≤ 𝑖, then 𝜎𝑗(t𝑖+1) is a maximal torus of k𝑖+1containing t_𝑖, hence must be equal tot𝑖+1. Continuing in this way, we eventually obtain a maximal torust𝓁+1ofk𝓁+1 = g with𝜎𝑗(t𝓁+1) = t𝓁+1for all𝑗 = 1, … , 𝓁.

Proposition 2.1. Let𝐺 be a compact connected Lie group, a ⊆ g an Abelian subalgebra, and t a maximal torus ofg containing a. Denote byΔ ⊆ (tℂ)∗the set of roots with respect to the Cartan subalgebratℂofgℂand by Γ ⊆ Δ the set of roots vanishing on a. Then, as a vector space,

Ngℂ(a) = Zgℂ(a) = tℂ⊕ ⨁ 𝛼∈Γg

ℂ 𝛼.

Proof. Thattℂis contained inZ

definition, for every𝑌 ∈ a:

[𝑌, 𝑋] = ±𝛼(𝑌)𝑋 = 0, hencegℂ_±𝛼is contained inZ

gℂ(a). Conversely, let 𝑁 ∈ Ngℂ(a), and write

𝑁 = 𝑋0+ ∑ 𝛼∈Δ𝑋𝛼,

where𝑋0∈ tℂand𝑋𝛼 ∈ gℂ_𝛼. For𝑌 ∈ a we have

tℂ⊇ a ∋ [𝑌, 𝑁 ] = ∑

𝛼∈Δ𝛼(𝑌)𝑋𝛼 ∈ ⨁𝛼∈Δg ℂ 𝛼,

which is only possible if𝛼(𝑌)𝑋𝛼 = 0 for all 𝛼 ∈ Δ. Hence, if 𝑋𝛼 ≠ 0, then a ⊆ ker 𝛼 and 𝛼 ∈ Γ. We have shown: tℂ⊕ ⨁ 𝛼∈Γg ℂ 𝛼 ⊆ Zg(a) ⊆ Ng(a) ⊆ tℂ⊕ ⨁ 𝛼∈Γ0g ℂ 𝛼.

For the remainder of this section we fix a compact connected Lie group𝐺, two commuting involutions 𝜎1and 𝜎2on𝐺 (not necessarily different), and an Ad–invariant negative definite inner product ⟨⋅, ⋅⟩ on g for which 𝜎1 and𝜎2are isometries. Note that any negative definiteAd–invariant inner product (⋅, ⋅) on g gives rise to such an inner product: just take

(⋅, ⋅) + 𝜎₁∗(⋅, ⋅) + 𝜎₂∗(⋅, ⋅) + (𝜎1𝜎2)∗(⋅, ⋅).

Moreover, we put𝐾1∶= (𝐺𝜎1)0,𝐾2∶= (𝐺𝜎2)0, and choose a maximal torus𝑆 ⊆ (𝐺𝜎1∩ 𝐺𝜎2)0. According to our previous observations,𝑇1= Z𝐾

1(𝑆) then is a maximal torus in 𝐾1and𝑇 ∶= Z𝐺(𝑇1) is a maximal torus in 𝐺. Let Δ be the gℂ–roots with respect totℂ,Δ+a choice of positive roots,Γ ⊆ Δ the set of roots vanishing on s, and Γ+_{∶= Γ ∩ Δ}+. We also set𝜏𝛼 ∶= 𝛼◦𝜏 whenever 𝛼 is a root and 𝜏 is an automorphism on g leaving t invariant. Proposition 2.2. Let g= k₁⊕ p₁be the decomposition ofg into the1– and (−1)–eigenspaces of 𝜎1. Then

(1) the root spacegℂ_𝛼 is contained inpℂ₁ for all𝛼 ∈ Γ; (2) if𝛼 ∈ Γ, then 𝜎1𝛼 = 𝛼 and

(3) 𝜎2𝛼 = −𝛼;

(4) any two roots𝛼, 𝛽 ∈ Γ are strongly orthogonal, that is, neither 𝛼 + 𝛽 nor 𝛼 − 𝛽 is a root;

(5) denoting for a root𝛼 by 𝐻𝛼 ∈ it the element with ⟨𝐻𝛼, ⋅ ⟩ = 𝛼, we have tℂ= ⋂

𝛼∈Γ+ker 𝛼 ⊕ ⨁_𝛼∈Γ+ℂ𝐻𝛼,

and any two summands in this decomposition are mutually orthogonal with respect to⟨⋅, ⋅⟩.

Proof.

(1) Pick𝛼 ∈ Γ and note that 𝜎1𝛼 still vanishes on s. Thus, the space 𝑈 ∶= gℂ_𝛼 + gℂ_𝜎

1𝛼 is𝜎1–invariant, and so decomposes as the direct sum𝑈 = (𝑈 ∩ kℂ₁) ⊕ (𝑈 ∩ pℂ₁). Now proposition 2.1 implies that

𝑈 ∩ kℂ₁ ⊆ Zgℂ(s) ∩ kℂ1 = Zkℂ₁(s) = tℂ1 ⊆ tℂ,

(2) We have just seen that given𝛼 ∈ Γ the root space gℂ_𝛼 is contained inp₁ℂ. So, if we pick𝐸𝛼 ∈ gℂ_𝛼 and 𝐸−𝛼 ∈ gℂ−𝛼, then[𝐸𝛼, 𝐸−𝛼] ∈ kℂ₁. We may assume that⟨𝐸𝛼, 𝐸−𝛼⟩ = 1, and then 𝐻𝛼 = [𝐸𝛼, 𝐸−𝛼] for the element𝐻𝛼 ∈ it with ⟨𝐻𝛼, ⋅ ⟩ = 𝛼. Therefore,

𝛼 = ⟨𝐻𝛼, ⋅ ⟩ = ⟨𝜎1(𝐻𝛼), ⋅ ⟩◦𝜎1= ⟨𝐻𝛼, ⋅ ⟩◦𝜎1= 𝜎1𝛼.

(3) According to the previous item,𝐻𝛼 ∈ kℂ₁, and since𝜎1and𝜎2commute,𝜎2(𝐻𝛼) must be contained in kℂ₁ as well. Therefore,

𝐻𝛼+ 𝜎2(𝐻𝛼) ∈ tℂ∩ kℂ1 ∩ kℂ2 = sℂ.

Now for𝑌 ∈ sℂwe compute

⟨𝑌, 𝐻𝛼 + 𝜎2(𝐻𝛼)⟩ = 2⟨𝑌, 𝐻𝛼⟩ = 2𝛼(𝑌) = 0.

But⟨ ⋅, ⋅ ⟩ is non–degenerate on sℂ, hence we must have𝐻𝛼 + 𝜎2(𝐻𝛼) = 0, which is equivalent to saying that𝜎2𝛼 = −𝛼, because 𝜎2is an isometry of⟨ ⋅, ⋅ ⟩.

(4) Let𝛼, 𝛽 ∈ Γ and suppose that 𝛼 + 𝛽 was a root for a contradiction. We could choose non–zero root vectors 𝑋𝛼 ∈ gℂ𝛼 and𝑋_𝛽 ∈ gℂ

𝛽, and then[𝑋𝛼, 𝑋_𝛽] ∈ gℂ

𝛼+𝛽 would be a non–zero root vector as well. ButZ

gℂ(s) is a

Lie algebra and𝑋𝛼,𝑋_𝛽are elements ofZ

gℂ(s), so according to the first item

[𝑋𝛼, 𝑋𝛽] ∈ kℂ1 ∩ Zgℂ(s) = tℂ1 ⊆ tℂ,

which is impossible. Therefore,𝛼 + 𝛽 is not a root.

(5) Let𝛼, 𝛽 ∈ Γ+be two distinct roots. It is well known (cf. [16, Proposition 2.48, sect. II.5]) that the𝛼–string containing𝛽, that is, the subset of Δ ∪ {0} consisting of elements 𝛽 + 𝑛𝛼 with 𝑛 ∈ ℤ, has no gaps and that the integers𝑝, 𝑞 ≥ 0 such that (𝛽 + 𝑛𝛼 ∈ Δ ∪ {0}) ⟺ (−𝑝 ≤ 𝑛 ≤ 𝑞) satisfy 𝑝 − 𝑞 = 2⟨𝛼, 𝛽⟩/⟨𝛼, 𝛼⟩. Since neither𝛼 + 𝛽 nor 𝛼 − 𝛽 is a root, we hence must have

0 = ⟨𝛼, 𝛽⟩ = ⟨𝐻𝛼, 𝐻𝛽⟩.

In particular, the elements𝐻𝛼,𝛼 ∈ Γ+, are linearly independent. Now let 𝑈 = ⨁

𝛼∈Γ+ℂ𝐻𝛼 and𝑈

′_{= ⋂} 𝛼∈Γ+ker 𝛼.

Then the equation𝛼(𝑌) = ⟨𝐻𝛼, 𝑌⟩ for 𝑌 ∈ tℂshows that𝑈′ = tℂ∩ 𝑈⟂, and sotℂ= 𝑈 ⊕ 𝑈′. 3. Automorphisms

We continue to use the notation of the previous section. Given𝛼 ∈ Δ, denote by 𝑠𝐻

𝛼∶ it → it the reflection along the hyperplane orthogonal to𝐻𝛼, i.e. the map

𝑠𝐻𝛼(𝑋) = 𝑋 − 2⟨𝐻_⟨𝐻𝛼𝛼_{, 𝐻}, 𝑋⟩_𝛼⟩. Since the elements ofΓ are mutually orthogonal, we immediately have

Proposition 3.1. The members of{𝑠_𝐻𝛼| 𝛼 ∈ Γ+} commute pairwise.

Note that proposition 2.2 suggests that𝜎2acts as a product of hyperplane reflections on a certain subspace oft. This subspace will be a proper subspace in general, but if𝜎2is an inner autormophism, then it actually is

all oft. We shall show that under some mild assumptions on𝜎1the maximal toruss of k1∩ k2can in fact be recovered fromΓ.

Proposition 3.2. Suppose that𝜎₂ = 𝑐_𝑛◦𝜈 holds for some element 𝑛 ∈ 𝐺 and some automorphism 𝜈 on 𝐺 that fixest1pointwise. Then

(1) the element𝑛 is contained in Z𝐺(𝑆) ∩ N𝐺(𝑇1), (2) the maximal torust is𝜈–invariant,

(3) 𝜎2|_it= ∏_𝛼∈Γ+(𝑠𝐻

𝛼)◦(𝜈|it). Proof.

(1) By assumption,𝑇1 is contained in the1–eigenspace of 𝜈 and 𝜎2 = 𝑐𝑛◦𝜈. Since 𝑆 is contained in the 1–eigenspace of 𝜎2and𝑇1is𝜎2–invariant, the same statement is true with𝑐𝑛in place of𝜎2.

(2) Just note that𝜈(𝑇) is a maximal torus of 𝐺 containing 𝑇1, so𝜈(𝑇) = 𝑇.

(3) We already observed that𝑛 centralizes 𝑆 and it is a well–known fact (see [16, Corollary 4.51, sect. IV.5]) that centralizers of tori are connected, so, according to proposition 2.1, we may express𝑛 as 𝑛 = exp(𝑋), where𝑋 = 𝑋0+𝑋Γfor certain elements𝑋0∈ t and 𝑋Γ∈ ⨁_𝛼∈Γgℂ_𝛼. In particular, if𝑌 ∈ 𝐿, 𝐿 ∶= ⋂_𝛼∈Γ+ker 𝛼, then[𝑌, 𝑋] = 0. Thus, Ad𝑛fixes𝐿 ∩ it pointwise, as do the elements 𝑠𝐻

𝛼 with𝛼 ∈ Γ+. On the other hand, if𝛽 ∈ Γ+is arbitrary, then𝐻_𝛽 ⊆ it1by proposition 2.2, so

Ad𝑛(𝐻𝛽) = 𝜎2(𝐻𝛽) = −𝐻𝛽 = ( ∏

𝛼∈Γ+𝑠𝐻𝛼) (𝐻𝛽). Therefore,Ad𝑛restricts to∏_𝛼∈Γ+𝑠𝐻

𝛼 onit, whence the 𝜈–invariance of t implies the claim. Corollary 3.3. Suppose that𝜎₂= 𝑐_𝑛◦𝜈 and t₁⊆ g𝜈, and put𝐿 ∶= ⋂𝛼∈Γ+ker 𝛼. Then

it ∩ 𝐿 = (is) ⊕ i(t ∩ p1), it1= (is) ⊕ ⨁

𝛼∈Γ+ℝ𝐻𝛼, and rank(k1∩ k2) = rank(k1) − |Γ

+_|.

Proof. We know from proposition 2.2 thatit = (it ∩ 𝐿) ⊕ ⨁_𝛼∈Γ+ℝ𝐻𝛼 is a decomposition into two𝜎1–invariant subspaces and that⨁_𝛼∈Γ+ℝ𝐻𝛼 is entirely contained init1. Thus, we must haveit ∩ 𝐿 = (it1∩ 𝐿) ⊕ i(t ∩ p1) and it1= (it1∩ 𝐿) ⊕ ⨁𝛼∈Γ+ℝ𝐻_𝛼. Now recall that(it1)𝜎2= is, while (it1)𝜎2= it1∩ 𝐿 holds by proposition 3.2.

Ifg is simple the condition that𝜎2is a composition of an inner automorphism and an automorphism fixingt1 is not too restrictive: in fact, we will see later that if𝜎1is an outer automorphisms, then, except for Lie algebras of typeD4, we may assume that𝜎1 = 𝑐𝑡◦𝜏 and 𝜎2 = 𝑐𝑛◦𝜏 or that 𝜎1 = 𝑐𝑡◦𝜏 and 𝜎2 = 𝑐𝑛 for some involution 𝜏 ∶ 𝐺 → 𝐺 and elements 𝑡 ∈ 𝑇1,𝑛 ∈ N𝐻(𝑇1) ⋅ 𝑇, where 𝐻 = (𝐺𝜏)0.

The following propositions state that in this case we may trade𝑡 ∈ 𝑇1for some element𝑡′ ∈ 𝑇 to first assume that𝑛 ∈ 𝐻 and that 𝜎2= 𝑐𝑛◦𝜏; afterwards we may replace 𝜎1by an inner automorphism.

Proposition 3.4. Suppose that𝜎₁= 𝑐_𝑡◦𝜏 and 𝜎₂= 𝑐_𝑛◦𝜈, where 𝜏 is an involution, 𝜈 = 𝜏 or 𝜈 = id_𝐺,𝑡 ∈ 𝑇1, and 𝑛 ∈ N𝐻(𝑇1) ⋅ 𝑇, with 𝐻 = (𝐺𝜏)0. Then there exist elements𝑡′ ∈ 𝑇 and ℎ ∈ N𝐻(𝑇1) such that 𝑐_𝑡′◦𝜏 and 𝑐ℎ◦𝜏 are commuting involutions whose common fixed point set hass as a maximal torus.

Proof. First suppose that𝜈 = 𝜏. Then we choose 𝑞 ∈ exp(t ∩ p1), ℎ ∈ N_𝐻(𝑇1) with 𝑛 = ℎ𝑞 and set 𝐿 ∶= ⋂_𝛼∈Γ+ker 𝛼. Note that t ∩ 𝐿 decomposes, by corollary 3.3, as t ∩ 𝐿 = s ⊕ (t ∩ p1) and that the elements of t ∩ 𝐿 are

fixed byAdℎ, because𝜎1|_t= 𝜏|_tand hence proposition 3.2 applies. So if we pick𝑌 ∈ t ∩ p1with𝑞 = exp(𝑌) and put𝑟 = exp(𝑌/2), then 𝑐𝑟−1◦𝜎2◦𝑐𝑟 is an involution,𝑞 = 𝑟2, and𝜏(𝑟) = 𝑟−1. Therefore, we have

similarly,𝑐_𝑟−1◦𝜎1◦𝑐𝑟 = 𝑐_𝑞−1𝑡◦𝜏. Thus, 𝑐_𝑞−1𝑡◦𝜏 and 𝑐_ℎ◦𝜏 are two commuting involutions. Since their common fixed point subalgebra is conjugate tok1∩ k2viaAd_𝑟−1 andAd_𝑟−1fixess, the claim follows.

Now assume that𝜈 = id𝐺. Choose a decomposition𝑛 = ℎ𝑞 as before and use corollary 3.3 to additionally find𝑠 ∈ exp(s), 𝑎 ∈ exp(⨁_𝛼∈Γ+ℝ(i𝐻𝛼)) with 𝑡 = 𝑠𝑎. We will show that 𝜇 ∶= 𝑐𝑛𝑠◦𝜏 is an involution, that 𝜇 commutes with𝜎1, and thats is a maximal torus of g𝜎1∩ g𝜇. The previous case then implies the claim, because 𝑛𝑠 ∈ N𝐻(𝑇1) ⋅ 𝑇. To begin with, we assert that (𝑐𝑠)2 = (𝑐𝑞)2; indeed,𝑐_ℎ and𝑐𝑛 coincide ont, whence we have 𝑐_ℎ(𝑎) = 𝑎−1and𝑐_ℎ(𝑠) = 𝑠 (cf. proposition 3.2), so this follows from

𝑐𝑛= 𝜎1◦𝑐𝑛◦𝜎1= 𝑐𝑡◦𝑐𝜏(𝑛)◦𝑐𝑡= 𝑐𝑠◦𝑐𝑎◦𝑐ℎ◦𝑐𝑞−1◦𝑐_𝑡= 𝑐_ℎ◦𝑐_𝑠◦𝑐_𝑎−1◦𝑐_𝑞−1◦𝑐_𝑡

together with the commutativity of𝑞, 𝑠, and 𝑎. Also note that ℎ, 𝑞, and 𝑠 commute with each other and that 𝐻 contains𝑠. These observations imply that 𝜇 is an involution commuting with 𝜎1, since

𝜇2= 𝑐𝑛𝑠◦𝜏◦𝑐𝑛𝑠◦𝜏 = 𝑐𝑛𝑠◦𝑐ℎ𝑞−1_𝑠 = (𝑐_ℎ)2◦(𝑐𝑠)2= (𝑐𝑛)2= id

and since𝑐𝑠,𝜏, and 𝑐𝑛commute with𝜎1. Finally, note that any maximal torus ofg𝜎1∩g𝜇containings is a subset ofZ_g(s) and that by propositions 2.1 and 2.2 𝜎1only fixest1onZ_g(s). Then s must be a maximal torus of g𝜎1∩g𝜇, astℂand⨁_𝛼∈Γgℂ_𝛼 are𝜇–invariant subspaces and 𝜇|_t

1= 𝜎2|t₁only fixess.

Proposition 3.5. Suppose that𝜎₁= 𝑐_𝑡◦𝜏 and that 𝜎₂= 𝑐_ℎ◦𝜏, where 𝜏 is an involution, 𝑡 is contained in 𝑇, and ℎ is an element of N𝐻(𝑇1), with 𝐻 = (𝐺𝜏)0. LetΠ_odd ⊆ Π be the set of all roots 𝛽 ∈ Π for which the integer ∑𝛼∈Γ+2⟨𝛼, 𝛽⟩/⟨𝛼, 𝛼⟩ is odd. Then 𝜏(𝛼) ≠ 𝛼 for all 𝛼 ∈ Π_odd.

Lemma 3.6. Under the assumptions of proposition 3.5 we have gℂ_𝛼 ⊆ hℂfor each root𝛼 ∈ Γ.

Proof. Observe that the requirements of proposition 3.2 are met, soℎ is an element of Z𝐺(𝑆) ∩ 𝐻 = Z𝐻(𝑆). Since Z𝐻(𝑆) is connected, we may express ℎ as ℎ = exp(𝑍) for some element 𝑍 ∈ Z_h(s) = Z_g(s) ∩ h, say 𝑍 = 𝑍0+ ∑𝛼∈Γ+𝑍_𝛼, with𝑍0∈ t and 𝑍𝛼 ∈ gℂ_𝛼 ⊕ gℂ_−𝛼for each root𝛼 ∈ Γ+. Recall that𝜎1coincides with𝜏 on t, because𝑐𝑡is the identity ont, so as𝜎1fixes each root𝛼 ∈ Γ+,𝜏 fixes each element of Γ+too. Therefore,gℂ_𝛼 and gℂ_−𝛼are eigenspaces of𝜏, whence 𝑍𝛼necessarily vanishes ifg_𝛼ℂ⊈ hℂ. However, if𝛽 ∈ Γ+was a root with𝑍_𝛽 = 0, then, as the elements ofΓ are strongly orthogonal, we also would have [𝑍, 𝐻_𝛽] = 0, and hence Ad_ℎ(𝐻_𝛽) = 𝐻_𝛽. But this is impossible, because we know from proposition 3.2 thatAd_ℎ(𝐻_𝛽) = −𝐻_𝛽. Consequently,𝑍_𝛽 ≠ 0 and gℂ_𝛽 ⊆ hℂ.

Proof of proposition 3.5. The decompositiont= s⊕s′, withs′= ⨁_𝛼∈Γ+ℝ(i𝐻𝛼)⊕(t∩p1) yields a decomposition 𝑘 = 𝑘+𝑘−for every element𝑘 ∈ 𝑇, where 𝑘+∈ exp(s) and 𝑘−∈ exp(s′). Moreover, 𝜎2restricts toid on s and to (−id) on s′, so the condition that𝑐𝑘◦𝜏 commutes with 𝜎2can be rephrased as

𝑐𝑘◦𝜏 = 𝜎2◦𝑐𝑘◦𝜏◦(𝜎2)−1⟺ 𝑐𝑘◦𝜏 = 𝑐𝜎2(𝑘)◦(𝑐ℎ)2◦𝜏 ⟺ (𝑐𝑘−)2= (𝑐ℎ)2;

but𝑐_ℎis an involution, because𝑐_ℎcommutes with𝜏 and 𝜎2is an involution, so𝑐_𝑘◦𝜏 commutes with 𝜎2if and only if𝑐_𝑘

−is an involution. In particular, if we let𝑡 = 𝑡+𝑡−, then𝑐𝑡

− is an involution. With this characterization at hand we can show that no root inΠ

oddis fixed by𝜏: let us further decompose

𝑡−as𝑡− = 𝑞𝑟, where 𝑞 ∈ exp(t ∩ p1), 𝑟 = exp(𝑍), and 𝑍 = ∑_𝛼∈Γ+𝑡𝛼i𝜋/⟨𝛼, 𝛼⟩𝐻𝛼 for certain real numbers𝑡𝛼. Recalling that each element𝛽 ∈ Γ is contained in the (−1)–eigenspace of 𝜎1, but in the fixed point set of𝜏, and thats⊕ (t ∩ p1) is the common kernel of the elements of Γ on t, we find that

−id_gℂ

𝛽 = 𝜎1|gℂ𝛽 = Ad𝑟|gℂ𝛽 = 𝑒

i𝜋𝑡𝛽_id;

so,(𝑡𝛽− 1) ∈ 2ℤ. On the other hand, if 𝛽 ∈ Π with 𝜏(𝛽) = 𝛽 is arbitrary, then Ad𝑞restricts to± id on gℂ

t∩ p₁is the(−1)–eigenspace of 𝜏 on t. Combined with the fact that 𝑐𝑡

− is an involution this gives id_gℂ

𝛽 = (Ad𝑞𝑟||gℂ𝛽)

2

= (Ad𝑟|gℂ_𝛽)

2_{= (−1)}∑𝛼∈Γ+2⟨𝛼,𝛽⟩⟨𝛼,𝛼⟩ _{⋅ id,} because2⟨𝛼, 𝛽⟩/⟨𝛼, 𝛼⟩ is an integer and (𝑡𝛼− 1) is an even number. Therefore, 𝛽 ∈ Π

odd.

Corollary 3.7. In addition to the hypotheses of proposition 3.5 assume that g is semisimple. LetΠeven= Π⧵Πodd

and choose, for each𝛼 ∈ Π

odd,𝜖𝛼 ∈ {±1} with 𝜖𝛼 = −𝜖𝜏(𝛼). There exists𝑝 ∈ exp(t ∩ p1) such that

(1) Ad𝑝is equal to(𝜖𝛼i) ⋅ id on gℂ_𝛼 and to the identity ongℂ

𝛽 for all𝛼 ∈ Π

odd,𝛽 ∈ Πeven,

(2) the automorphism𝜈 = 𝑐_{𝑝 exp(𝑋)}, where𝑋 = ∑_𝛼∈Γ+i𝜋/⟨𝛼, 𝛼⟩𝐻𝛼, is an involution, and (3) 𝜎2commutes with𝜈 and s is a maximal torus of g𝜈∩ g𝜎2.

Proof. Choose𝑌 ∈ t such that 𝛼(𝑌) = 0 for all 𝛼 ∈ Πevenand such that𝛼(𝑌) = 𝜖𝛼i𝜋/2 for all roots 𝛼 ∈ Πodd; this

is possible, because the restrictions of the elements ofΠ constitute a basis of (it)∗. Then𝑌 is necessarily contained int∩ p1, because𝛼(𝑌 + 𝜏(𝑌)) vanishes for all 𝛼 by choice of the integers 𝜖𝛽,𝛽 ∈ Π

odd. We set𝑝 ∶= exp(𝑌) and

observe thatAd𝑝indeed is equal to(𝜖𝛼i) ⋅ id on gℂ_𝛼, if𝛼 ∈ Π

odd, and toid else. Thus, for each simple root 𝛼 ∈ Π

the maps(Ad𝑝)2and(Ad_exp(𝑋))2coincide ongℂ_𝛼 and are equal toid or (−id), so 𝜈 = Ad_{𝑝 exp(𝑋)}is an involution. Moreover,𝜈 commutes with 𝜎2, because𝜈◦𝜎2= 𝜈−1◦𝜎2.

Hence, it remains to show that s is a maximal torus of g𝜈 ∩ g𝜎2, and to this end it suffices to verify the maximality ofs. However, we already know that the complexification ofZ_g(s) is the sum of the 𝜎1– and𝜎2– invariant subspacestℂand⨁_𝛼∈Γgℂ_𝛼. By construction,Ad𝑝 equalsid on the latter space, because 𝜏(𝛼) = 𝛼 for 𝛼 ∈ Γ, while Ad_exp(𝑋)is just(−id) by proposition 2.2; hence 𝜈 only fixes t in Z_g(s), and the fixed point set of 𝜎2 ont is precisely s, because t1= t𝜏. Thus, onlys is fixed by both𝜈 and 𝜎2inZ_g(s).

4. Normal forms for strongly orthogonal roots 4.1. Abstract normal forms

In the previous sections we learned that for a suitable choice of Cartan subalgebra the set of roots vanishing on a maximal torus of the joint fixed point subalgebra of two commuting inner involutions is strongly orthogonal and satisfies a certain involutivity condition. The purpose of this section is to establish a normal form for all sets of roots satisfying these properties.

Recall (cf. [16, p. 149]) that an(abstract) root system(𝑉 , ⟨⋅, ⋅⟩, Δ) consists of a finite–dimensional Euclidean vector space(𝑉 , ⟨⋅, ⋅⟩) together with a non–empty set Δ ⊆ 𝑉 of non–zero vectors such that

(1) 𝑉 = span_ℝΔ,

(2) for each𝛼 ∈ Δ the reflection

𝑠𝛼∶ 𝑉 → 𝑉 , 𝑣 ↦ 𝑣 − 2⟨𝛼, 𝑣⟩_{⟨𝛼, 𝛼⟩ 𝛼,}

mapsΔ into itself, and

(3) the number2⟨𝛼, 𝛽⟩/⟨𝛼, 𝛼⟩ is an integer whenever 𝛼 and 𝛽 are elements of Δ.

A root systemΔ is reduced if 𝛼 ∈ Δ implies that 2𝛼 ∉ Δ. It is called reducible if there exists a non–trivial disjoint decompositionΔ = Δ′⊔ Δ′′such that⟨𝛼′, 𝛼′′⟩ = 0 for all 𝛼′∈ Δ′and𝛼′′∈ Δ′′. If no such decomposition exists, thenΔ is irreducible.

(1) A pair(𝑈 , Ω) is a root subsystem of Δ if

a) Ω ⊆ Δ is non–empty,

b) 𝑈 = span_ℝΩ, and c) 𝑠𝛼(Ω) ⊆ Ω for all 𝛼 ∈ Ω.

(2) Theroot subsystem ofΔ spanned by 𝑆, 𝑆 ⊆ Δ a non–empty set, is the pair (span_ℝ𝑆, Δ ∩ span_ℤ𝑆). Remark 4.2. LetΔ be a root system.

(1) If(𝑈 , Ω) is a root subsystem of Δ, then (𝑈 , ⟨⋅, ⋅⟩|𝑈 ×𝑈, Ω) is a root system. If 𝑆 ⊆ Δ is a non–empty subset, then the root subsystem spanned by𝑆 is a root subsytem in the sense of definition 4.1.

(2) Let(𝑈 , Ω) be a root subsystem of Δ. We can identify the Weyl group 𝑊 (Ω) of Ω, which by definition is a subgroup ofO(𝑈 , ⟨⋅, ⋅⟩|𝑈 ×𝑈), with a subgroup 𝑊 (Ω, Δ) of the Weyl group 𝑊 (Δ) of Δ, where

𝑊 (Ω, Δ) ∶={𝑤 ∈ 𝑊 (Δ) | 𝑤 = 𝑠𝛼1◦ … ◦𝑠𝛼𝑘,𝛼𝑖∈ Ω}⊆ O(𝑉 , ⟨⋅, ⋅⟩).

In fact, the map𝑝 ∶ 𝑊 (Ω, Δ) → 𝑊 (Ω) restricting an element 𝑤 ∈ 𝑊 (Ω, Δ) to 𝑈 is a homomorphism of groups. Moreover, if𝑤 ∈ 𝑊 (Ω), say with 𝑤 = 𝑡𝛼

1◦ … ◦𝑡𝛼𝑘, where𝛼𝑖 ∈ Ω and 𝑡𝛼

𝑖∶ 𝑈 → 𝑈 denotes

reflection along the hyperplane in𝑈 perpendicular to 𝛼𝑖, then 𝑝(𝑠𝛼1◦ … ◦𝑠𝛼𝑘) = 𝑤;

and if𝑤 ∈ ker 𝑝, then 𝑤 = id𝑉, because𝑝(𝑤) = id𝑈 and𝑤(𝑣′) = 𝑣′for all𝑣′∈ 𝑈⟂by definition.

Recall that any choice of positive rootsΔ+in a root systemΔ determines a set of simple roots Π ⊆ Δ+, and that any root𝛼 can be uniquely written as 𝛼 = ∑_𝛽∈Π𝑚𝛽𝛽 for integers 𝑚𝛽of the same sign. The number∑_𝛽∈Π𝑚𝛽is commonly referred to as thelevel of the root𝛼.

Proposition 4.3. LetΔ be a reduced irreducible root system, Δ+ ⊆ Δ a choice of positive roots, and 𝛼₀ ∈ Δ. There exists a unique root𝛿 of maximal level in the orbit 𝑊 ⋅ 𝛼0of the Weyl group𝑊 = 𝑊 (Δ), and this root satisfies⟨𝛿, 𝛼⟩ ≥ 0 for all 𝛼 ∈ Δ+.

Proof. Choose any root𝛿 of maximal level in 𝑊 ⋅ 𝛼0= {𝑤(𝛼0) | 𝑤 ∈ 𝑊 }. If 𝛼 ∈ Δ+is a root with⟨𝛿, 𝛼⟩ < 0, then𝑠𝛼(𝛿) is a root having higher level than 𝛿 and still is contained in 𝑊 ⋅ 𝛼0, which is impossible. Therefore, we have⟨𝛿, 𝛼⟩ ≥ 0 for any positive root 𝛼. In order to prove the uniqueness statement, let Π ⊆ Δ+be the simple roots associated with the given choice of positivity and note that𝛿 is positive, so we may write

𝛿 = ∑

𝛼∈Π𝑚𝛼𝛼,

with𝑚𝛼 ∈ ℤ≥0. We claim that each of the integers𝑚𝛼is non–zero. For if this was not the case, thenΠ = Π′∪Π′′ withΠ′= {𝛼 | 𝑚𝛼 = 0} and Π′′= {𝛼 | 𝑚𝛼 > 0} would be a non–trivial disjoint union. Moreover, for any 𝛽 ∈ Π′ we would have

⟨𝛿, 𝛽⟩ = ∑

𝛼∈Π′′𝑚𝛼⟨𝛼, 𝛽⟩,

and the right hand side is non–positive, because the inner product of two distinct simple roots already is non– positive. By what we have just shown, ⟨𝛿, 𝛽⟩ ≥ 0, and so ⟨𝛿, 𝛽⟩ = 0 and hence ⟨𝛼, 𝛽⟩ = 0 would have to hold for all𝛼 ∈ Π′′and𝛽 ∈ Π′. But this is impossible, because we are assumingΔ to be irreducible. Now let 𝛾 ∈ 𝑊 ⋅ 𝛼0be another root of maximal level. The same line of reasoning as before also applies to𝛾 and shows

that𝛾 = ∑_𝛼∈Π𝑛𝛼𝛼 for integers 𝑛𝛼 > 0. In particular, since there is some simple root 𝛽 ∈ Π with ⟨𝛿, 𝛽⟩ > 0, we also must have⟨𝛿, 𝛾⟩ > 0. Therefore, 𝛿 − 𝛾 is either a positive or a negative root (or 0), and since

𝛿 − 𝛾 = ∑

𝛼∈Π(𝑚𝛼 − 𝑛𝛼)𝛼,

it follows that(𝑚𝛼 − 𝑛𝛼)𝛼∈Π is either a sequence of non–negative or non–positive integers. But𝛿 and 𝛾 have the same level, that is,∑_𝛼∈Π𝑚𝛼 = ∑_𝛼∈Π𝑛𝛼, and therefore𝑚𝛼 = 𝑛𝛼 for all𝛼 ∈ Π.

LetΔ be a reduced irreducible root system and Δ+a choice of positive roots. A well–known consequence of the classification of such root systems is that any two simple roots of the same length are contained in the same Weyl group orbit. On the other hand, every root is contained in the Weyl group orbit of a simple root (see [16, Proposition 2.62, sect. II.6]), so if𝐿 is the length of a root in Δ+, then by proposition 4.3 we may unambiguously speak of the highest root (with respect to the level) of length𝐿.

Now letΠ ⊆ Δ+be the simple roots andΓ ⊆ Δ a non–empty set of (not necessarily strongly) orthogonal roots such thatΓ = (−Γ). We further suppose that all elements of Γ are of the same length 𝐿 > 0 and put Γ+= Γ ∩ Δ+. We claim that there is a way to describe the possible elements thatΓ may contain, up to application of a Weyl group element. To this end, let us introduce some notation for non–empty subsets𝐴 ⊆ Π that we will make use of in the sequel. Given such a set𝐴 we write Δ𝐴to denote the root subsystem ofΔ spanned by 𝐴 and we putΔ+

𝐴 = Δ𝐴∩ Δ+, which is a notion of positivity with simple roots𝐴. Moreover, we call 𝐴 irreducible if Δ𝐴 is irreducible, and refer to a non–empty subset𝐴′ ⊆ 𝐴 as an irreducible component of 𝐴 if 𝐴′is maximal (with respect to inclusion) among all irreducible subsets of𝐴. Note that 𝐴 decomposes as 𝐴 = 𝐴1∪ … ∪ 𝐴𝑝, where each𝐴𝑖is an irreducible component of𝐴 and the members of 𝐴𝑖are orthogonal to𝐴𝑗for all𝑖 ≠ 𝑗. Finally, if 𝐴 is irreducible and admits roots of length𝐿, then we write 𝛿(𝐴) to denote the highest root of length 𝐿 in Δ𝐴(with respect toΔ+_𝐴).

Next, we recursively define a family(𝑖)𝑖=0,…,𝑛of non–empty subsets of(Π) (the power set of Π) as follows. We put0∶= {Π} and suppose that for some 𝑘 ≥ 0 the sets 0, … , _𝑘are already defined. Then a non–empty subset𝐴 ⊆ Π is contained in _𝑘+1if and only if

(1) Δ𝐴is irreducible and admits roots of length𝐿,

(2) there exists a (possibly empty) set𝐵 ⊆ Π whose members are orthogonal to each member of 𝐴 and a set 𝑣(𝐴) ∈ 𝑘such that

𝛿(𝑣(𝐴))⟂∩ 𝑣(𝐴) = 𝐵 ∪ 𝐴;

in other words,𝐴 is an irreducible component of 𝛿(𝑣(𝐴))⟂∩ 𝑣(𝐴) that admits roots of length 𝐿. We put 𝑛 ∶= 𝑘 if no such𝐴 exists and call 0, … , 𝑛thenormal form tree for(Δ, Δ+) and 𝐿.

Remark 4.4. Closely related to the normal form tree construced above is the so–called cascade of strongly orthogonal roots defined in [18, Section 1]: indeed, if𝐴 ∈ 𝑖for some𝑖 > 1, then in the notation of [18] 𝛿(𝐴) is an offspring of𝛿(𝑣(𝐴)). If 𝐴0, … , 𝐴𝑖are such that𝐴𝑖 ∈ 𝑖, then{𝛿(𝐴0), … , 𝛿(𝐴𝑖)} is called a chain cascade in [18].

Proposition 4.5. Any two distinct sets𝑖,𝑗 are disjoint andΔ𝐴,Δ_𝐴′ are perpendicular for all𝐴, 𝐴′ ∈ 𝑘 with𝐴 ≠ 𝐴′. Moreover, for𝐴 ∈ 𝑘+1the element𝑣(𝐴) is the only set in 𝑘with𝐴 ∩ 𝑣(𝐴) ≠ ∅.

Remark 4.6. Thus, we may define a graph with vertices the elements of0∪…∪𝑛, where𝐴, 𝐴′are connected by an edge if and only if𝐴 = 𝑣(𝐴′). The resulting graph is a tree, hence the name.

Proof. We first show by induction on𝑘 = 0, … , 𝑛 that 𝑣(𝐴) is the only set in _𝑘intersecting𝐴 ∈ _𝑘+1non– trivially and that𝐴, 𝐴′ ∈ _𝑘 have non–trivial intersection only if𝐴′ = 𝐴. This is immediate if 𝑘 = 0, because 0 = {Π}, so suppose that the induction hypothesis has been established for some natural number 𝑘 ≥ 0.

Choose𝐴, 𝐴′∈ _𝑘+1arbitrarily and note that by the induction assumption𝑣(𝐴) and 𝑣(𝐴′) are the unique sets in_𝑘with𝐴 ∩ 𝑣(𝐴) ≠ ∅ and 𝐴′∩ 𝑣(𝐴′) ≠ ∅. Hence, if 𝐴 ∩ 𝐴′is non–empty, then, since𝐴 ⊆ 𝑣(𝐴) and 𝐴′⊆ 𝑣(𝐴′) holds by definition, also𝑣(𝐴)∩𝑣(𝐴′) is non–empty, so by the induction assumption we must have 𝑣(𝐴) = 𝑣(𝐴′). The defining property of𝑣(𝐴) is that 𝛿(𝑣(𝐴))⟂∩ 𝑣(𝐴) = 𝐵 ∪ 𝐴 holds for some subset 𝐵 ⊆ Π whose members are orthogonal to each member of𝐴. Therefore,

Δ𝐴′= (Δ_𝐴′∩ span_ℤ(𝐴′∩ 𝐵)) ∪ (Δ_𝐴′∩ span_ℤ(𝐴′∩ 𝐴))

is a decomposition into two sets whose members are mutually orthogonal, whence by irreducibility ofΔ_𝐴′we must haveΔ_𝐴′⊆ span_ℤ(𝐴∩𝐴′). Thus, 𝐴′∩𝐵 is empty and 𝐴′ ⊆ 𝐴. Exchanging the roles of 𝐴 and 𝐴′we conclude that𝐴 = 𝐴′, so two sets in_𝑘+1intersect non–trivially only if they are equal. To finish the induction step, just note that if𝐴 ∈ _𝑘+2is arbitrary and𝐵 ∈ _𝑘+1intersects𝐴 non–trivially, then also 𝑣(𝐴) ∩ 𝐵 ≠ ∅, because 𝐴 ⊆ 𝑣(𝐴), so by what we have just shown 𝐵 = 𝑣(𝐴).

Now suppose that𝐴, 𝐴′ ∈ _𝑘 are two distinct sets and let𝑗 ≥ 0 be the smallest integer such that 𝑣𝑗+1(𝐴) = 𝑣𝑗+1_(𝐴′_{). By definition we have 𝛿(𝑣}𝑗+1_(𝐴))⟂_{∩ 𝑣}𝑗+1_{(𝐴) = 𝐵 ∪ 𝑣}𝑗_{(𝐴) for some set 𝐵 which is perpendicular to}

𝑣𝑗(𝐴) and hence intersects 𝑣𝑗(𝐴) trivially. Since we just showed that 𝑣𝑗(𝐴) intersects 𝑣𝑗(𝐴′) trivially as well, we conclude that𝑣𝑗(𝐴′) must be contained in 𝐵. Thus, 𝑣𝑗(𝐴′) is perpendicular to 𝑣𝑗(𝐴), whence 𝐴 and 𝐴′ are perpendicular too, because𝐴 ⊆ 𝑣𝑗(𝐴) and 𝐴′ ⊆ 𝑣𝑗(𝐴′). Finally, suppose that 𝐴 is contained in _𝑘∩ _𝑘+𝑗 for integers𝑘 ≥ 0 and 𝑗 ≥ 1. Then 𝑣𝑘(𝐴) ∈ 0∩ 𝑗, whence𝑣𝑘(𝐴) = Π. This is impossible, however, because each element of𝑗is a proper subset ofΠ.

Corollary 4.7. For𝐵 ∈ _𝑘, and all𝐴 ∈ 0∪ … ∪ _𝑘such that𝐴 ≠ 𝐵 we have 𝐵 ⊆ 𝛿(𝐴)⟂.

Proof. If𝐴 ∈ 𝑘, the statement follows readily from proposition 4.5, so we suppose that𝐴 ∈ 𝑘−𝑗 for some 𝑗 ≥ 1. If 𝐴 is different from 𝑣𝑗_{(𝐵), then even 𝑣}𝑗_{(𝐵) and 𝐴 are perpendicular. If 𝐴 is equal to 𝑣}𝑗_{(𝐵), then 𝑣}𝑗−1_{(𝐵) ⊆}

𝛿(𝐴)⟂holds by definition, so𝐵 ⊆ 𝑣𝑗−1(𝐵) is perpendicular to 𝛿(𝐴).

Corollary 4.8. Let𝐵 ∈ _𝑘. Any𝑤 ∈ 𝑊 (𝐵) permutes the members of {Δ𝐴| 𝐴 ∈ 𝑚}, if 𝑚 ≤ 𝑘.

Proof. Fix some𝑗 ≥ 0 and put 𝓁 ∶= 𝑘 − 𝑗. If 𝐴 ∈ 𝓁 is different from𝑣𝑗(𝐵), then 𝐴 and 𝑣𝑗(𝐵) are perpendicular, whence so are𝐴 and 𝐵. Since 𝑤 is a product of reflections 𝑠𝛼 with𝛼 ∈ 𝐵, 𝑤 hence fixes 𝐴 and Δ𝐴in this case. On the other hand, if𝐴 = 𝑣𝑗(𝐵), but 𝑗 > 0, let 𝐶1, … , 𝐶𝑝 ⊆ Π be the irreducible components of 𝛿(𝐴)⟂∩ 𝐴. Note that𝐶𝑖is contained in𝓁+1if and only ifΔ𝐶

𝑖admits roots of length𝐿, so we may further assume that for some 𝑠 ≥ 1 the sets 𝐶1, … , 𝐶𝑠contain roots of length𝐿, while 𝐶𝑠+1, … , 𝐶𝑝do not, and that𝑣𝑗−1(𝐵) = 𝐶1. Now observe that the root subsystem spanned by𝛿(𝐴)⟂∩ 𝐴 is precisely 𝛿(𝐴)⟂∩ Δ_𝐴. Indeed, any root𝛼 ∈ Δ_𝐴is aℤ≥0– or ℤ≤0–linear combination of elements in𝐴, so if ⟨𝛿(𝐴), 𝛼⟩ = 0, then 𝛼 must be a linear combination of elements in𝛿(𝐴)⟂∩ 𝐴, because ⟨𝛿(𝐴), 𝛽⟩ ≥ 0 holds for all 𝛽 ∈ 𝐴 by proposition 4.3. Hence, we have

𝛿(𝐴)⟂∩ Δ𝐴= Δ𝐶1∪ … ∪ Δ𝐶𝑝.

Also note that𝐵 is perpendicular to 𝛿(𝐴), but contained in Δ𝐴, so𝑤 leaves 𝛿(𝐴)⟂∩ Δ𝐴invariant. Hence, since𝑤 is an isometry andΔ𝐶

𝑖 is irreducible, we must have𝑤(Δ𝐶

𝑖) ∈ {Δ𝐶1, … , Δ𝐶𝑝} for each 𝑖. Moreover, if Δ𝐶𝑖 admits roots of length𝐿, then so does 𝑤(Δ_𝐶

𝑖), whence 𝑤 even permutes the set {Δ𝐶1, … , Δ𝐶𝑠}. Theorem 4.9. There exists a Weyl group element𝑤 ∈ 𝑊 (Δ) such that

(1) 𝑤(Γ) ∩ Δ+⊆ {𝛿(𝐴) | 𝐴 ∈ 0∪ … ∪ 𝑛} and

(2) if𝛿(𝐴) is contained in 𝑤(Γ) ∩ Δ+, then either𝐴 = Π or 𝛿(𝑣(𝐴)) is contained in 𝑤(Γ).

Lemma 4.10. If𝛼 ∈ Δ_𝐴,𝐴 ∈ _𝑘, is perpendicular to𝛿(𝐴), then 𝛼 ∈ Δ_𝐴′for some irreducible component𝐴′of 𝛿(𝐴)⟂∩ 𝐴. If in addition 𝛼 is of length 𝐿, then 𝑘 < 𝑛 and 𝐴′is contained in_𝑘+1.

Proof. Express𝛼 as 𝛼 = ∑_𝛽∈𝐴𝑚_𝛽𝛽 for integers (𝑚_𝛽)_𝛽∈𝐴 of the same sign. Since ⟨𝛼, 𝛿(𝐴)⟩ = 0 holds by assumption, we conclude that only those coefficients𝑚_𝛽with⟨𝛿(𝐴), 𝛽⟩ = 0 can be non–zero, and since 𝛼 is a root, some𝑚_𝛽must be non–zero. Hence,𝛿(𝐴)⟂∩ 𝐴 is non–empty and 𝛿(𝐴)⟂∩ 𝐴 = 𝐶1∪ … ∪ 𝐶𝑝, where𝐶1, … , 𝐶𝑝 are the irreducible components. Thus, if𝛽 ∈ 𝐶𝑖for some𝑖 and some 𝛽 with 𝑚_𝛽 ≠ 0, then also 𝛼 ∈ Δ_𝐶

𝑖. Moreover, if𝛼 is of length 𝐿, then Δ_𝐶

𝑖admits roots of length𝐿, so 𝐶𝑖 ∈ _𝑘+1and𝑘 < 𝑛.

Proof of theorem 4.9. Put𝑛+1∶= ∅ and denote for each 𝑘 = −1, … , 𝑛 by 𝛿(_≤𝑘) the set {𝛿(𝐴) | 𝐴 ∈ 0∪ … ∪ _𝑘}. We inductively prove that for 𝑘 = −1, … , 𝑛 there exists an element 𝑤 ∈ 𝑊 (Δ) such that

(1) every element in(𝑤(Γ) ∩ Δ+) ⧵ 𝛿(≤𝑘) is contained in Δ𝐴for some𝐴 ∈ 𝑘+1and (2) 𝛿(𝑣(𝐴)) ∈ 𝑤(Γ) whenever 𝛼 ∈ 𝑤(Γ) ∩ Δ𝐴for some𝐴 ∈ 1∪ … ∪ _𝑘+1.

For 𝑘 = −1 the set 𝛿(≤𝑘) is empty and 0 = {Π}, so we may take 𝑤 = id in this case. Now suppose that the induction hypothesis holds for some number𝑘 ≤ 𝑛, so there exists 𝑤 ∈ 𝑊 (Δ) verifying the two properties above. In particular, there exist elements𝐴1, … , 𝐴𝑝∈ 𝑘+1such that each element ofΓ′∶= (𝑤(Γ) ∩ Δ+) ⧵ 𝛿(≤𝑘) is contained in someΔ𝐴

1, … , Δ𝐴𝑝, and we may assume𝑝 to be the minimal number of elements required to satisfy this property. Thus, we may choose an element𝛾𝑖 ∈ Γ′∩ Δ𝐴

𝑖for each𝑖 = 1, … , 𝑝. Since Δ𝐴

𝑖 is reduced and irreducible, all roots of the same length are contained in one Weyl group orbit, so there exists an element 𝑤𝑖∈ 𝑊 (Δ𝐴𝑖) such that 𝑤𝑖(𝛾𝑖) is the highest root of Δ𝐴𝑖having length𝐿, that is, 𝑤𝑖(𝛾𝑖) = 𝛿(𝐴𝑖). Now consider the element𝑤′∶= 𝑤1◦ … ◦𝑤𝑝. We know from proposition 4.5 that𝑤′leaves each of the root systemsΔ𝐴

𝑗invariant, because each𝑤𝑖is a product of root reflections𝑠𝛼 with𝛼 ∈ 𝐴𝑖. The same reasoning combined with corollary 4.7 shows that𝑤𝑖fixes𝛿(𝐴𝑗) for all 𝑖 ≠ 𝑗 and also all roots in 𝛿(_≤𝑘). Hence, 𝑤′fixes the elements in𝛿(_≤𝑘), so if we put ̃𝑤 ∶= 𝑤′◦𝑤, then the set ̃𝑤(Γ)∩Δ+fully contains𝑤(Γ)∩𝛿(≤𝑘) and all of the roots 𝛿(𝐴1), … , 𝛿(𝐴𝑝). Moreover, each root𝛼 in ( ̃𝑤(Γ) ∩ Δ+) ⧵ 𝛿(≤𝑘+1) is contained in some Δ𝐴

𝑖, because the same is true for(𝑤′)−1(𝛼) ∈ Γ′. Since the roots inΓ are pairwise orthogonal, such an 𝛼 hence is orthogonal to 𝛿(𝐴𝑖), because 𝑤′(𝛾𝑖) = 𝛿(𝐴𝑖), and therefore already contained inΔ𝐴for some𝐴 ∈ 𝑘+2by lemma 4.10; in particular, no such𝛼 exists if 𝑘 = 𝑛 − 1. It remains to verify the second property, so suppose that we are given a positive root𝛼 ∈ ̃𝑤(Γ) ∩ Δ𝐵for some 𝐵 ∈ 1∪ … ∪ 𝑘+2. We already know from the induction assumption that either𝛼 ∈ 𝛿(_≤𝑘) or 𝛼 ∈ Δ𝐴

𝑖 must

hold, and if𝛼 ∈ 𝛿(_≤𝑘), then 𝐵 must be contained in 1∪ … ∪ _𝑘 by corollary 4.7. Since 𝑤′ fixes𝛿(_≤𝑘) pointwise, the induction statement for𝑘 shows that 𝛿(𝑣(𝐵)) must be contained in ̃𝑤(Γ) if 𝛼 ∈ 𝛿(_≤𝑘). If 𝛼 ∈ Δ𝐴

𝑖 for some𝑖 and 𝐵 ∈ _{𝑘+1−𝑗}for some𝑗 ≥ 0, then Δ𝐵andΔ_{𝑣𝑗(𝐴}

𝑖)intersect non–trivially, hence𝐵 and 𝑣𝑗(𝐴𝑖) must be equal by proposition 4.5. Moreover,(𝑤′)−1(𝛼) and 𝛼 both are contained in Δ_{𝑣𝑗(𝐴}

𝑖), because𝑤′leaves invariant Δ𝐴𝑖, so by corollary 4.8(𝑤′)−1must leaveΔ_{𝑣𝑗(𝐴}

𝑖)andΔ𝐵invariant as well. Therefore,(𝑤′)−1(𝛼) is contained in 𝑤(Γ) ∩ Δ𝐵, whence by induction assumption𝛿(𝑣(𝐵)) ∈ 𝛿(≤𝑘) is contained in 𝑤(Γ) and also ̃𝑤(Γ). The final case to consider is that𝛼 is an element of some Δ𝐴

𝑖, but that𝐵 ∈ 𝑘+2. Then𝐴𝑖 = 𝑣(𝐵), and 𝛿(𝐴𝑖) is contained in ̃𝑤(Γ) by construction.

4.2. Normal forms for simply laced root systems

Withis this section, we fix a reduced irreducible root systemΔ whose roots are all of the same length, a set of positive rootsΔ+with corresponding simple rootsΠ, and a non–empty set of strongly orthogonal roots Γ ⊆ Δ. As before, we also setΓ+= Γ ∩ Δ+and we additionally suppose that the integer

𝑝(𝛼) ∶= 𝑝(Δ, Γ, 𝛼) ∶= ∑

𝛽∈Γ+ 2⟨𝛼, 𝛽⟩

⟨𝛽, 𝛽⟩

is even for all roots𝛼. Note that if 𝑤 ∈ 𝑊 (Δ) is arbitrary, then 𝑝(Δ, 𝑤(Γ), 𝛼) still is even, because this number is equal to𝑝(𝑤−1(𝛼)). Hence, we may use theorem 4.9 to assume that Γ+is contained in{𝛿𝐴| 𝐴 ∈ 0∪ … ∪ 𝑛} and that each𝛿(𝑣(𝐴)) is contained in Γ whenever 𝛿(𝐴) is an element of Γ and 𝐴 ≠ Π.

Example 4.11 (Normal form forA_𝑟). It will be convenient to associate with any reduced irreducible root system Ω with positive roots Ω+and simple rootsΦ a modified Dynkin diagram. By this we shall mean the graph with verticesΦ ∪ {𝛿𝑐}𝑐, where𝛿𝑐denotes the highest root of length𝑐 and 𝑐 ranges over all root lengths in Ω, and whose edge set is built according to the rules of an ordinary Dynkin diagram. The resulting diagram for root systems of typeA𝑟,𝑟 ≥ 1, is given in figure 1. If Δ is of type A𝑟 and we label the simple rootsΠ = {𝛼1, … , 𝛼𝑟}

𝛼1 𝛼2

𝛿

𝛼𝑟−1 𝛼𝑟

Figure 1. Modified Dynkin diagram for root systems of typeA𝑟,𝑟 ≥ 1. The highest root is 𝛿 = 𝛼1+ … + 𝛼𝑟. as in figure 1, we can immediately read off the sets0, … , 𝑛. In fact,

0= {Π}, 1={{𝛼2, … , 𝛼𝑟−1}}, … , 𝑖 ={{𝛼𝑖+1, … , 𝛼𝑟−𝑖}}, …

soΓ+ = {𝛿(𝐴0), … , 𝛿(𝐴𝑞)} for some 𝑞 < ⌈𝑟/2⌉, where 𝐴𝑖 = {𝛼𝑖+1, … , 𝛼𝑟−𝑖}. However, the constraint 𝑝(𝛼) ∈ 2ℤ can only be satisfied if𝑟 is odd and 𝑞 = (𝑟 − 1)/2, for otherwise 𝛼𝑞+1− 𝛿(𝐴𝑞) is a root and 𝑝(𝛼𝑞+1) = 1. Therefore, 𝑟 = 2𝑘 + 1 and Γ+is equal to{𝛿1, … , 𝛿𝑘+1}, where 𝛿𝑖= 𝛼𝑖+ … + 𝛼𝑟−𝑖+1.

Example 4.12 (Normal form forD_𝑟). Suppose thatΔ is of type D𝑟, 𝑟 ≥ 4, and enumerate the simple roots Π = {𝛼1, … , 𝛼𝑟} as in figure 2. We first assume that 𝑟 = 2𝑘 + 1 is odd. Then we have, for 𝑖 ≥ 1:

𝛼1 𝛼2

𝛿

𝛼𝑟−2

𝛼𝑟−1

𝛼𝑟

Figure 2. Modified Dynkin diagram for root systems of typeD𝑟,𝑟 ≥ 4. The highest root is 𝛿 = 𝛼1+ 2𝛼2+ … + 2𝛼𝑟−2+ 𝛼𝑟−1+ 𝛼𝑟.

… , 𝑖 ={{𝛼2𝑖−1}, {𝛼2𝑖+1, … , 𝛼𝑟}}, … , _𝑘−1={{𝛼_2𝑘−3}, {𝛼_2𝑘−1, 𝛼_2𝑘, 𝛼_2𝑘+1}},_𝑘={{𝛼_2𝑘−1}}. Thus, if we let𝐴𝑖 = {𝛼𝑖, … , 𝛼𝑟}, then there exists a maximal integer 1 ≤ 𝑚 ≤ 𝑘 such that Γ+contains the element 𝛿(𝐴2𝑚−1), and then Γ+will also contain𝛿(𝐴1), 𝛿(𝐴3), … , 𝛿(𝐴2𝑚−3), because 𝑣(𝐴2𝑖+1) = 𝐴2𝑖−1. No element𝛼2𝑖−1 with𝑚 < 𝑖 ≤ 𝑘 can be contained in Γ+, for otherwise we could choose𝑖 maximal with 𝛼2𝑖−1 ∈ Γ+, and then𝛼2𝑖−1is the only element ofΓ+not perpendicular to𝛼2𝑖, whence𝑝(𝛼2𝑖) = −1. Similarly, if 𝛼2𝑖−1is contained inΓ+for some 1 < 𝑖, then 𝛼2𝑖−3is contained inΓ+as well, for otherwise𝑝(𝛼2𝑖−2) = −1 would hold. On the other hand, 𝛼2𝑚−1 must be contained inΓ+to ensure𝑝(𝛼2𝑚) ∈ 2ℤ, hence Γ+is equal to{𝛼1, 𝛿(𝐴1), 𝛼3, 𝛿(𝐴3), … , 𝛼2𝑚−1, 𝛿(𝐴2𝑚−1)}, for some1 ≤ 𝑚 ≤ 𝑘. Now suppose that 𝑟 = 2𝑘. This time we have

𝑖={{𝛼2𝑖−1}, {𝛼2𝑖+1, … , 𝛼𝑟}}for𝑖 < 𝑘 − 1 and _𝑘−1={{𝛼_2𝑘−3}, {𝛼_2𝑘−1}, {𝛼_2𝑘}}.

We again let𝐴𝑖= {𝛼𝑖, … , 𝛼𝑟} and define 1 ≤ 𝑚 ≤ 𝑘 −2 to be the maximal integer such that Γ+contains𝛿(𝐴2𝑚−1). If𝑚 < 𝑘 −2, then the same argument as in the case of odd rank shows that Γ+is equal to{𝛼2𝑖−1, 𝛿(𝐴2𝑖−1) | 𝑖 ≤ 𝑚}. If𝑚 = 𝑘 −2, then an odd (in particular non–zero) number of elements of {𝛼_2𝑘−3, 𝛼_2𝑘−1, 𝛼_2𝑘} must be contained in Γ+, for otherwise𝑝(𝛼_2𝑘−2) is not even, and if 𝛼_2𝑘−3is contained inΓ+, then the same reasoning as in the previous case shows that𝛼1, … , 𝛼_2𝑘−3actually are contained inΓ+. For later reference, let us summarize all the cases we