The Equivariant Cohomology of Isotropy Actions

on Symmetric Spaces

Oliver Goertsches

Received: September 9, 2011

Communicated by Eckhard Meinrenken

Abstract. We show that for every symmetric space G/K of compact type with K connected, the K-action on G/K by left translations is equivariantly formal.

2010 Mathematics Subject Classification: 53C35, 55N25, 57S15

1 Introduction

Given compact connected Lie groups K ⊂ G of equal rank, it is well-known that the K-action on the homogeneous space G/K is equivariantly formal because the odd de Rham cohomology groups of G/K vanish. (See for example [7] for an investigation of the equivariant cohomology of such spaces.) If however the rank of K is strictly smaller than the rank of G, then the isotropy action is not necessarily equivariantly formal, and in general it is unclear when this is the case.^{1} Restricting our attention to symmetric spaces of compact type, we will prove the following theorem.

Theorem. Let (G, K) be a symmetric pair of compact type, where G and K are compact connected Lie groups. Then the K-action on the symmetric space M = G/K by left translations is equivariantly formal.

For symmetric spaces of type II, i.e., compact Lie groups, this result is already known, see Section 4.3. More generally, in the case of symmetric spaces of split rank (rank G = rank K + rank G/K), the fact that all K-isotropy groups have maximal rank implies equivariant formality, see Section 4.5. However, for the general case we have to rely on an explicit calculation of the dimension of the cohomology of the T -fixed point set M^{T}, where T ⊂ K is a maximal torus, in order to use the characterization of equivariant formality via the condition dim H^{∗}(M^{T}) = dim H^{∗}(M). With the help of the notion of compartments introduced in [1] and several results proven therein we will find in Section 4.1 a calculable expression for this dimension, and after reducing to the case of an irreducible simply-connected symmetic space in Section 4.2 we can invoke the classification of such spaces to show equivariant formality in each of the remaining cases by hand. On the way we obtain a formula for the number of compartments in a fixed K-Weyl chamber, see Proposition 4.14.

1A sufficient condition for equivariant formality of the isotropy action was introduced in [17], see Remark 4.2 below. If K belongs to a certain class of subtori of G this condition is in fact an equivalence, see [18].

Documenta Mathematica 17 (2012) 79–94

Acknowledgements. The author wishes to express his gratitude to Augustin-Liviu Mare for interesting discussions on a previous version of the paper.

2 Symmetric spaces

Let G be a connected Lie group and K ⊂ G a closed subgroup. Then K is said to be a symmetric subgroup of G if there is an involutive automorphism σ : G → G such that K is an open subgroup of the fixed point subgroup G^{σ}. We will refer to the pair (G, K) as a symmetric pair, and G/K is a symmetric space. Given a symmetric pair (G, K) with corresponding involution σ : G → G, then the Lie algebra 𝔤 decomposes into the (±1)-eigenspaces of σ: 𝔤 = 𝔨 ⊕ 𝔭, and the usual commutation relations hold: [𝔨, 𝔨] ⊂ 𝔨, [𝔨, 𝔭] ⊂ 𝔭 and [𝔭, 𝔭] ⊂ 𝔨. The rank of G/K is by definition the maximal dimension of an abelian subalgebra of 𝔭. Then clearly rank G − rank K ≤ rank G/K, and if equality holds, then we say that G/K is of split rank. A symmetric pair (G, K) is called (almost) effective if G acts (almost) effectively on G/K. Given a symmetric pair (G, K), then the kernel N ⊂ G of the Gaction on G/K is contained in K, and (G/N, K/N) is an effective symmetric pair with (G/N)/(K/N) = G/K. An almost effective symmetric pair (G, K) (and the corresponding symmetric space G/K) will be called of compact type if G is a compact semisimple Lie group. In this paper only symmetric spaces of compact type will occur. If (G, K) is effective, then G can be regarded as a subgroup of the isometry group of G/K with respect to any G-invariant Riemannian metric on G/K. If (G, K) is additionally of compact type, then this inclusion is in fact an isomorphism between G and the identity component of the isometry group.

3 Equivariant formality

The equivariant cohomology of an action of a compact connected Lie group K on a compact manifold M is by definition the cohomology of the Borel construction

H_{K}^{∗}(M) = H^{∗}(EK ×_{K} M);

we use real coefficients throughout the paper. The projection EK ×_{K} M → EK/K = BK to the classifying space BK of K induces on H_{K}^{∗}(M) the structure of an H^{∗}(BK)-algebra. An action of a compact connected Lie group K on a compact manifold M is called equivariantly formal in the sense of [3] if H_{K}^{∗}(M) is a free H^{∗}(BK)- module. If the K-action on M is equivariantly formal then automatically

H_{K}^{∗}(M) = H^{∗}(M) ⊗ H^{∗}(BK)  (1)

as graded H^{∗}(BK)-modules, see [2, Proposition 2.3]. In the following proposition we collect some known equivalent characterizations of equivariant formality.

Proposition 3.1. Consider an action of a compact connected Lie group K on a compact manifold M, and let T ⊂ K be a maximal torus. Then the following conditions are equivalent:

1. The K-action on M is equivariantly formal.

2. The T -action on M is equivariantly formal.

3. The cohomology spectral sequence associated to the fibration ET ×_{T} M → BT collapses at the E_{2}-term.

4. We have dim H^{∗}(M) = dim H^{∗}(M^{T}).

5. The natural map H_{T}^{∗} (M) → H^{∗}(M) is surjective.

Proof. For the equivalence of (1) and (2) see [5, Proposition C.26]. The Borel localization theorem implies that the rank of H_{T}^{∗} (M) as an H^{∗}(BT)-module always equals dim H^{∗}(M^{T}). Then [5, Lemma C.24] implies the equivalence of (2), (3), and (4); see also [10, p. 46]. For the equivalence to (5), see [13, p. 148].

Note that by [10, p. 46] the inequality dim H^{∗}(M^{T}) ≤ dim H^{∗}(M) holds for any T -action on M. Condition (5) in the proposition shows that

Corollary 3.2. If a compact connected Lie group K acts equivariantly formally on a compact manifold M, then so does every connected closed subgroup of K.

Applying the gap method to the spectral sequence in Item (3) of Proposition 3.1 we obtain the following well-known sufficient condition for equivariant formality.

Proposition 3.3. Any action of a compact Lie group K on a compact manifold M with H^{odd}(M) = 0 is equivariantly formal.

4 Isotropy actions on symmetric spaces of compact type

Let G be a compact connected Lie group and K ⊂ G a compact connected subgroup. Because an equivariantly formal torus action always has fixed points, the only tori T ⊂ G that can act equivariantly formally on G/K by left translations are those that are conjugate to a subtorus of K. On the other hand, if a maximal torus T of K acts equivariantly formally on G/K, then we know by Corollary 3.2 that all these tori do in fact act equivariantly formally. In the following, we will prove that this indeed happens for symmetric spaces of compact type. More precisely:

Theorem 4.1. Let (G, K) be a symmetric pair of compact type, where G and K are compact connected Lie groups. Then the K-action on the symmetric space G/K by left translations is equivariantly formal.

Remark 4.2. The pair (G, K) is a Cartan pair in the sense of [4], see [4, p. 448]. Therefore, [17, Theorem A] shows that a sufficient condition for the K-action on G/K to be equivariantly formal is that the map H^{∗}(G/K)^{N_{G}(K)} → H^{∗}(G) induced by the projection G → G/K, where N_{G}(K) acts on G/K from the right, is injective. It would be interesting to know whether a symmetric pair always satisfies this condition.

4.1 The fixed point set of a maximal torus in K

Let (G, K) be a symmetric pair of compact type, where G and K are compact connected Lie groups. Denote by σ : G → G the corresponding involutive automorphism. Then M = G/K is a symmetric space of compact type. We fix maximal tori T_{K} ⊂ K and T_{G} ⊂ G such that T_{K} ⊂ T_{G}. Let 𝔤 = 𝔨 ⊕ 𝔭 be the decomposition of the Lie algebra 𝔤 into eigenspaces of σ. In order to prove Theorem 4.1 we can without loss of generality assume that the symmetric pair (G, K) is effective: if N ⊂ K is the kernel of the G-action on G/K, then clearly the K-action on G/K = (G/N)/(K/N) is equivariantly formal if and only if the K/N-action is equivariantly formal. (This follows for example from Proposition 3.1 because the fixed point sets of appropriately chosen maximal tori in K and K/N coincide.)

Lemma 4.3. The T_{K}-fixed point set in M is N_{G}(T_{K})/N_{K}(T_{K}).

Proof. An element gK ∈ M is fixed by T_{K} if and only if g^{−1}T_{K}g ⊂ K (i.e., g^{−1}T_{K}g is a maximal torus in the compact Lie group K), which is the case if and only if there is some k ∈ K with k^{−1}g^{−1}T_{K}gk = T_{K}. Thus, (G/K)^{T_{K}} = N_{G}(T_{K})/N_{G}(T_{K}) ∩ K = N_{G}(T_{K})/N_{K}(T_{K}).

Lemma 4.4 ([15, Proposition VII.3.2]). T_{G} is the unique maximal torus in G containing T_{K}.

Lemma 4.4 implies that the Lie algebra 𝔱_{𝔤} of T_{G} decomposes according to the decomposition 𝔤 = 𝔨 ⊕ 𝔭 as 𝔱_{𝔤} = 𝔱_{𝔨} ⊕ 𝔱_{𝔭}. (In fact, this statement is the first part of the proof of [15, Proposition VII.3.2].)

Proposition 4.5. Each connected component of M^{T_{K}} is a torus of dimension rank G − rank K.

Proof. Because of Lemma 4.4, the abelian subalgebra 𝔱_{𝔭} ⊂ 𝔭 is the space of elements in 𝔭 that commute with 𝔱_{𝔨}. Thus, Lemma 4.3 implies that the component of M^{T_{K}} containing eK is T_{G}/(T_{G} ∩ K) = T_{G}/T_{K} (note that the centralizer of T_{K} in K is exactly T_{K}), i.e., a rank G−rank K-dimensional torus. Because the fixed set M^{T_{K}} is a homogeneous space, all components are diffeomorphic.

We therefore understand the structure of the T_{K}-fixed point set M^{T_{K}} if we know its number of connected components, which we denote by r. In view of condition (4) in Proposition 3.1, we are mostly interested in the dimension of its cohomology. Proposition 4.5 implies immediately:

Proposition 4.6. We have dim H^{∗}(M^{T_{K}}) = 2^{rank G−rank K} · r.

In order to get a calculable expression for r we will use several results from [1, Sections 5 and 6] which we now collect. Denote by Δ_{G} = Δ_{𝔤} the root system of G with respect to the maximal torus T_{G}, i.e., the set of nonzero elements α ∈ 𝔱_{𝔤}^{∗} such that the corresponding eigenspace 𝔤_{α} = {X ∈ 𝔤^{ℂ} | [W, X] = iα(W)X for all W ∈ 𝔱_{𝔤}} is nonzero. Then we have the root space decomposition

𝔤^{ℂ} = 𝔱_{𝔤}^{ℂ} ⊕ ⨁_{α∈Δ_{𝔤}} 𝔤_{α}.  (2)

The 𝔤-Weyl chambers are the connected components of the set 𝔱_{𝔤}\⋃_{α∈Δ_{𝔤}} ker α. Because of Lemma 4.4, 𝔱_{𝔨} contains 𝔤-regular elements, hence no root in Δ_{𝔤} vanishes on 𝔱_{𝔨}. Therefore some of the 𝔤-Weyl chambers intersect 𝔱_{𝔨} nontrivially, and following [1] we will refer to these intersections as compartments. Considering as in [1] the decomposition of Δ_{𝔤} into complementary subsets Δ_{𝔤} = Δ′ ∪ Δ′′, where

Δ′ = {α ∈ Δ_{𝔤} | 𝔤_{α} ⊄ 𝔭^{ℂ}}, Δ′′ = {α ∈ Δ_{𝔤} | 𝔤_{α} ⊂ 𝔭^{ℂ}},  (3)

we have by [1, Lemma 9] that the root system Δ_{K} = Δ_{𝔨} of K with respect to T_{K} is given by

Δ_{𝔨} = {α|_{𝔱_{𝔨}} | α ∈ Δ′}.  (4)

In particular, 𝔤-regular elements in 𝔱_{𝔨} are also 𝔨-regular, and hence each compartment is contained in a 𝔨-Weyl chamber. Because of Lemma 4.4, the group N_{G}(T_{K}) is a subgroup of N_{G}(T_{G}). Both groups have the same identity component T_{G}, so we may regard the quotient group N_{G}(T_{K})/T_{G} as a subgroup of the Weyl group W (G) of G. The free action of W (G) on the 𝔤-Weyl chambers induces an action of N_{G}(T_{K})/T_{G} on the set of compartments. Because any two compartments are G-conjugate [1, Theorem 10], this action is simply transitive on the set of compartments, and it follows that the number of connected components of N_{G}(T_{K}) equals the total number of compartments in 𝔱_{𝔨}. On the other hand no connected component of N_{G}(T_{K}) contains more than one connected component of N_{K}(T_{K}).

(It is sufficient to check this for the identity component T_{G} of N_{G}(T_{K}). An element in N_{K}(T_{K}) ∩ T_{G} is an element in K centralizing T_{K}, hence already contained in T_{K}.) Because the number of connected components of N_{K}(T_{K}) equals the number of 𝔨-Weyl chambers, and each 𝔨-Weyl chamber contains the same number of compartments [1, Theorem 10], we have shown the following lemma.

Lemma 4.7. The number r of connected components of M^{T_{K}} = N_{G}(T_{K})/N_{K}(T_{K}) is the number of compartments in a fixed 𝔨-Weyl chamber. In particular it only depends on the Lie algebra pair (𝔤, 𝔨).

Let C be a 𝔤-Weyl chamber that intersects 𝔱_{𝔨} nontrivially. By [1, Lemma 8] the compartment C ∩ 𝔱_{𝔨} can be described explicitly: The involution σ : G → G permutes the 𝔤-Weyl chambers and fixes 𝔱_{𝔨}, hence it fixes C. Let B = {α_{1}, …, α_{rank G}} be the corresponding simple roots such that C is exactly the set of points where the elements of B take positive values. The involution σ acts as a permutation group on B because for any i the linear form α_{i}∘σ is again positive on C. Note that for every root α ∈ Δ_{𝔤} the linear form \frac{1}{2}(α + α ∘ σ) vanishes on 𝔱_{𝔭} and coincides with α|_{𝔱_{𝔨}} on 𝔱_{𝔨}. The set B|_{𝔱_{𝔨}} = {α_{i}|_{𝔱_{𝔨}} | i = 1, …, rank G} is a basis of 𝔱_{𝔨}^{∗} (in particular it consists of dim 𝔱_{𝔨} elements) and the compartment C ∩ 𝔱_{𝔨} is exactly the set of points in 𝔱_{𝔨} where all α_{i}|_{𝔱_{𝔨}} take positive values. It is a simplicial cone bounded by the hyperplanes ker α_{i}|_{𝔱_{𝔨}}. Any such hyperplane is either a wall of a 𝔨-Weyl chamber or the kernel of a 𝔤-root α_{i} with α_{i} ∘ σ = α_{i}, see (4). In any case, reflection along the hyperplane defines an element of N_{G}(T_{K})/T_{G} and takes C ∩𝔱_{𝔨} to an adjacent compartment. (This argument is taken from the proof of [1, Theorem 10].) It follows that the action of N_{G}(T_{K})/T_{G} on the set of compartments described above is generated by the reflections along all hyperplanes ker α|_{𝔱_{𝔨}}, where α ∈ Δ_{𝔤}. Let ⟨·, ·⟩ be the Killing form on 𝔤. The decomposition 𝔤 = 𝔨⊕𝔭 is orthogonal with respect to ⟨·, ·⟩. We identify 𝔱_{𝔤}^{∗} with 𝔱_{𝔤} and 𝔱_{𝔨}^{∗} with 𝔱_{𝔨} via ⟨·, ·⟩. For α ∈ Δ_{𝔤}, let H_{α} ∈ 𝔱_{𝔤} be the element such that α(H) = ⟨H, H_{α}⟩ for all H ∈ 𝔱_{𝔤}. Given X ∈ 𝔱_{𝔤}, we write X^{𝔨} and X^{𝔭} for the 𝔨- and 𝔭-parts of X respectively. Then H_{α}^{𝔨} corresponds to α|_{𝔱_{𝔨}} under the isomorphism 𝔱_{𝔨} ≅ 𝔱_{𝔨}^{∗}.

Lemma 4.8. Let α ∈ Δ_{𝔤} be a root with α ∘ σ ≠ α. Then either

1. ⟨H_{α}, H_{α∘σ}⟩ = 0 and |H_{α}^{𝔭}|^{2} = |H_{α}^{𝔨}|^{2} or

2. 2 · \frac{⟨H_{α},H_{α∘σ}⟩}{|H_{α}|^{2}} = −1, |H_{α}^{𝔭}|^{2} = 3|H_{α}^{𝔨}|^{2} and α + α ∘ σ ∈ Δ_{𝔤}.

Proof. We have H_{α∘σ} = H_{α}^{𝔨} − H_{α}^{𝔭}, and because Δ_{𝔤} is a root system it follows that

2 · \frac{⟨H_{α}, H_{α∘σ}⟩}{|H_{α}|^{2}} = 2 · \frac{|H_{α}^{𝔨}|^{2} − |H_{α}^{𝔭}|^{2}}{|H_{α}^{𝔨}|^{2} + |H_{α}^{𝔭}|^{2}} ∈ ℤ.

Because α and α ∘ σ are roots of equal length, this integer can only equal 0 or ±1 [12, Proposition 2.48.(d)]. Further, because α − α ∘ σ is not a root (by Lemma 4.4 no root vanishes on 𝔱_{𝔨}) and not 0, only the possibilities 0 and −1 remain, and in the latter case we also have that α+α∘σ ∈ Δ_{𝔤} [12, Proposition 2.48.(e)].

Proposition 4.9. The set Δ_{𝔤}|_{𝔱_{𝔨}} = {α|_{𝔱_{𝔨}} | α ∈ Δ_{𝔤}} is a root system in 𝔱_{𝔨}^{∗}.

Proof. It is clear that Δ_{𝔤}|_{𝔱_{𝔨}} spans 𝔱_{𝔨}^{∗}. We have to check that for all α, β ∈ Δ_{𝔤}, the quantity

2 · \frac{⟨H_{α}^{𝔨}, H_{β}^{𝔨}⟩}{|H_{α}^{𝔨}|^{2}}  (5)

is an integer. With respect to the decomposition Δ_{𝔤} = Δ′ ∪ Δ′′ (see (3)) there are four cases: If both α and β are elements of Δ′, then (5) is an integer because α|_{𝔱_{𝔨}} and β|_{𝔱_{𝔨}} are 𝔨-roots, see (4). In case α and β are elements of Δ′′, then the corresponding vectors H_{α} and H_{β} are already elements of 𝔱_{𝔨}, so H_{α}^{𝔨} = H_{α} and H_{β}^{𝔨} = H_{β}, hence (5) is an integer. Consider the case that α ∈ Δ′′ and β ∈ Δ′. Then H_{α} = H_{α}^{𝔨} ∈ 𝔱_{𝔨}, hence

2 · \frac{⟨H_{α}^{𝔨}, H_{β}^{𝔨}⟩}{|H_{α}^{𝔨}|^{2}} = 2 · \frac{⟨H_{α}, H_{β}⟩}{|H_{α}|^{2}} ∈ ℤ.

The last case to be considered is that α ∈ Δ′ and β ∈ Δ′′. In this case H_{β} = H_{β}^{𝔨} ∈ 𝔱_{𝔨}. It may happen that H_{α} ∈ 𝔱_{𝔨}, but then the claim would follow as before, so we may assume that H_{α} ∉ 𝔱_{𝔨}. It follows that α ∘ σ is a root different from α. By Lemma 4.8 we have |H_{α}^{𝔭}|^{2} = c|H_{α}^{𝔨}|^{2} with c = 1 or c = 3. We know that

2 · \frac{⟨H_{α}, H_{β}⟩}{|H_{α}|^{2}} = 2 · \frac{⟨H_{α}^{𝔨}, H_{β}^{𝔨}⟩}{|H_{α}^{𝔨}|^{2} + |H_{α}^{𝔭}|^{2}} = \frac{2}{1 + c} · \frac{⟨H_{α}^{𝔨}, H_{β}^{𝔨}⟩}{|H_{α}^{𝔨}|^{2}}

is an integer, hence multiplying with the integer 1 + c shows that (5) is an integer in this case as well. Next we have to check that for each α ∈ Δ_{𝔤} the reflection s_{α|_{𝔱_{𝔨}}} : 𝔱_{𝔨} → 𝔱_{𝔨} along ker α|_{𝔱_{𝔨}} defined by

X ↦ X − 2 · \frac{⟨H_{α}^{𝔨}, X⟩}{|H_{α}^{𝔨}|^{2}}H_{α}^{𝔨}  (6)

sends {H_{β}^{𝔨} | β ∈ Δ_{𝔤}} to itself. If H_{α} ∈ 𝔱_{𝔨} (this includes the case α ∈ Δ′′), then the reflection s_{α} : 𝔱_{𝔤} → 𝔱_{𝔤} along ker α leaves invariant 𝔱_{𝔨}, and (6) is nothing but the restriction of this reflection to 𝔱_{𝔨}. Thus, {H_{β}^{𝔨} | β ∈ Δ_{𝔤}} is sent to itself. Let α ∈ Δ′ with H_{α} ∉ 𝔱_{𝔨}. We treat the two cases that can arise by Lemma 4.8 separately: assume first that ⟨H_{α}, H_{α∘σ}⟩ = 0. In this case the two reflections s_{α} and s_{α∘σ} commute and we have, recalling that H_{α∘σ} = H_{α}^{𝔨} − H_{α}^{𝔭},

s_{α∘σ}∘s_{α}(X) = X − 2 · \frac{⟨H_{α}, X⟩}{|H_{α}|^{2}}H_{α} − 2 · \frac{⟨H_{α∘σ}, X⟩}{|H_{α∘σ}|^{2}}H_{α∘σ}

= X − 2 · \frac{⟨H_{α}, X⟩ + ⟨H_{α∘σ}, X⟩}{2|H_{α}^{𝔨}|^{2}}H_{α}^{𝔨} − 2 · \frac{⟨H_{α}, X⟩ − ⟨H_{α∘σ}, X⟩}{2|H_{α}^{𝔭}|^{2}}H_{α}^{𝔭}

= X − 2 · \frac{⟨H_{α}^{𝔨}, X⟩}{|H_{α}^{𝔨}|^{2}}H_{α}^{𝔨} + 2 · \frac{⟨H_{α}^{𝔭}, X⟩}{|H_{α}^{𝔭}|^{2}}H_{α}^{𝔭}.

In particular for each β ∈ Δ_{𝔤} the vector H_{β}^{𝔨} − 2 · \frac{⟨H_{α}^{𝔨},H_{β}^{𝔨}⟩}{|H_{α}^{𝔨}|^{2}}H_{α}^{𝔨} is the 𝔨-part of some vector H_{γ}, which shows that (6) sends {H_{β}^{𝔨} | β ∈ Δ_{𝔤}} to itself. In the second case of Lemma 4.8 we have that α + α ∘ σ ∈ Δ_{𝔤}, with ker(α + α ∘ σ) = ker α|_{𝔱_{𝔨}} ⊕ 𝔱_{𝔭}. Thus, the reflection s_{α|_{𝔱_{𝔨}}} is nothing but the restriction of s_{α+α∘σ} to 𝔱_{𝔨}; in particular it sends {H_{β}^{𝔨} | β ∈ Δ_{𝔤}} to itself.

Remark 4.10. The root system Δ_{𝔤}|_{𝔱_{𝔨}} is not necessarily reduced: if there exists a root α ∈ Δ_{𝔤} with α ∘ σ ≠ α for which the second case of Lemma 4.8 holds, then it contains α|_{𝔱_{𝔨}} as well as 2 · α|_{𝔱_{𝔨}}. This happens for instance for SU(2m + 1)/ SO(2m + 1).

Because B is the set of simple roots of Δ_{𝔤} every root α ∈ Δ_{𝔤} can be written as a linear combination of elements in B with integer coefficients of the same sign. It follows that every restriction α|_{𝔱_{𝔨}} ∈ Δ|_{𝔱_{𝔨}} is a linear combination of elements in B|_{𝔱_{𝔨}} of the same kind. We thus have proven the following lemma.

Lemma 4.11. The Δ_{𝔤}|_{𝔱_{𝔨}}-Weyl chambers are exactly the compartments. If C is a 𝔤-Weyl chamber that intersects 𝔱_{𝔨} nontrivially, with corresponding set of simple roots B ⊂ Δ_{𝔤}, then B|_{𝔱_{𝔨}} is the set of simple roots of the root system Δ_{𝔤}|_{𝔱_{𝔨}} corresponding to C ∩ 𝔱_{𝔨}.

Recall that the N_{G}(T_{K})/T_{G}-action on the set of compartments was shown to be generated by the reflections along all hyperplanes ker α|_{𝔱_{𝔨}}, where α ∈ Δ_{𝔤}. Thus, we obtain

Corollary 4.12. The N_{G}(T_{K})/T_{G}-action on the set of compartments is the same as the action of the Weyl group W (Δ_{𝔤}|_{𝔱_{𝔨}}). In particular, it is generated

by the reflections along the hyperplanes ker α_{i}|_{𝔱_{𝔨}}. Furthermore, r = \frac{|W ( Δ_{𝔤}|_{𝔱_{𝔨}})|}{|W (𝔨)|}.

Recall that whereas a reduced root system is determined by its simple roots [12, Proposition 2.66], this is no longer true for nonreduced root systems such as Δ_{𝔤}|_{𝔱_{𝔨}}, see [12, II.8]. However, the reduced elements in a nonreduced root system always form a reduced root system [12, Lemma 2.91] with the same simple roots and the same Weyl group. Using the following proposition taken from [15] we will identify this reduced root system contained in Δ_{𝔤}|_{𝔱_{𝔨}} with the root system of a second symmetric subalgebra 𝔨′ ⊂ 𝔤.

Proposition 4.13 ([15, Proposition VII.3.4]). There is an extension of σ : 𝔱_{𝔤} → 𝔱_{𝔤} to an involutive automorphism σ′ : 𝔤 → 𝔤 such that its ℂ-linear extension σ′ : 𝔤^{ℂ} → 𝔤^{ℂ} satisfies σ′|_{𝔤_{α}} = id for every root α ∈ B with α = α ∘ σ. The root system of the fixed point algebra 𝔨′ = 𝔤^{σ′} relative to the maximal abelian subalgebra 𝔱_{𝔨} has B|_{𝔱_{𝔨}} as simple roots.

The roots of 𝔨′ relative to 𝔱_{𝔨} are restrictions of certain (not necessarily all) elements in Δ_{𝔤} to 𝔱_{𝔨}; the restrictions of all elements in B occur. See [15, p. 129] for the root space decomposition of 𝔨′ with respect to 𝔨_{𝔨}. Because the sub-root system of reduced elements in Δ_{𝔤}|_{𝔱_{𝔨}} and the root system of 𝔨′ have the same simple roots, these reduced root systems coincide. In particular we obtain the following formula for r:

Proposition 4.14. We have r = \frac{|W (𝔨′)|}{|W (𝔨)|}.

Example 4.15. If rank G = rank K, i.e., if T_{K} is also a maximal torus of G, then the identity on 𝔤 satisfies the conditions of Proposition 4.13. Hence 𝔨′ = 𝔤 and the proposition says r = \frac{|W (G)|}{|W (K)|}. This however follows already from Lemma 4.3.

Example 4.16. If G/K is a symmetric space of split rank, i.e., rank G = rank K + rank G/K, then σ itself satisfies the conditions of Proposition 4.13. In fact, let α ∈ B with α = α ∘ σ. In this case α vanishes on 𝔱_{𝔭}, which implies that 𝔤_{α} is contained either in 𝔨^{ℂ} or in 𝔭^{ℂ}. But if it was contained in 𝔭^{ℂ}, then [𝔱_{𝔭}, 𝔤_{α}] = 0 and [𝔱_{𝔭}, 𝔤_{−α}] = 0, which would contradict the fact that 𝔱_{𝔭} is maximal abelian in 𝔭. Thus, we have r = 1 in the split rank case. Note that r = 1 also follows from [1, Lemma 13], combined with Lemma 4.7.

Example 4.17. The symmetric space G/K′, where K′ is the connected subgroup of G with Lie algebra 𝔨′, is not always of split rank. Assume as in Remark 4.10 that there exists a root α ∈ Δ_{𝔤} with α ∘ σ ≠ α such that α + α ∘ σ ∈ Δ_{𝔤}. Let X ∈ 𝔤_{α} be nonzero. Then [X, σ′(X)] is a nonzero element in 𝔤_{α+α∘σ}. We have σ′([X, σ′(X)]) = −[X, σ′(X)], thus [X, σ′(X)] ∈ 𝔭′, where 𝔭′ is the −1eigenspace of σ′. By definition of σ′ we have 𝔱_{𝔭} ⊂ 𝔭′, but 𝔱_{𝔭} is not a maximal abelian subspace of 𝔭′ because it commutes with [X, σ′(X)]. For example, in the case SU(2m + 1)/ SO(2m + 1) we have K′ = K although the space is not of split rank, see Subsection 4.6.2 below.

We will use below that the symmetric subalgebra 𝔨′ can be determined via the Dynkin diagram of G: σ defines an automorphism of the Dynkin diagram of G (because it is a permutation group of B), which is nontrivial if and only if rank 𝔤 > rank 𝔨. One can calculate the root system of 𝔨′ via the fact that by Proposition 4.13 the simple roots of 𝔨′ are given by B|_{𝔱_{𝔨}} = \{\frac{1}{2}(α_{i} + α_{i} ∘ σ) | i = 1, …, rank G}.

4.2 Reduction to the irreducible case

Lemma 4.18. If (G, K) and (G′, K′) are two effective symmetric pairs of connected compact semisimple Lie groups associated to the same pair of Lie algebras (𝔤, 𝔨), then the K-action on G/K is equivariantly formal if and only if the K′-action on G′/K′ is equivariantly formal.

Proof. Because K and K′ are connected, both H^{∗}(G/K) and H^{∗}(G′/K′) are given as the ℝ-algebra of 𝔨-invariant elements in Λ^{∗}𝔭, see [20, Theorem 8.5.8]. In particular dim H^{∗}(G/K) = dim H^{∗}(G′/K′). Choosing maximal tori T ⊂ K and T′ ⊂ K′, we furthermore know from Propositions 4.6 and 4.7 that dim H^{∗}((G/K)^{T}) = dim H^{∗}((G′/K′)^{T′}) because (G, K) and (G′, K′) correspond to the same Lie algebra pair. The statement then follows from Proposition 3.1.

Lemma 4.19. Given actions of compact connected Lie groups K_{i} on compact manifolds M_{i} (i = 1 … n), then the K_{1} × … × K_{n}-action on M_{1} × … × M_{n} is equivariantly formal if and only if all the K_{i}-actions on M_{i} are equivariantly formal.

Proof. Choose maximal tori T_{i} ⊂ K_{i}. Then T_{1} × … × T_{n} is a maximal torus in K_{1}×…×K_{n}. The claim follows from Proposition 3.1 because the T_{1}×…×T_{n}- fixed point set is exactly the product of the T_{i}-fixed point sets.

Lemmas 4.18 and 4.19 imply that for proving Theorem 4.1 it suffices to check it for effective symmetric pairs (G, K) of compact connected Lie groups such that G/K is an irreducible simply-connected symmetric space of compact type. Below we will make use of the classification of such spaces, see [8].

4.3 Lie groups

Given a compact connected Lie group G, the product G × G acts on G via (g_{1}, g_{2})·g = g_{1}gg_{2}^{−1}. The isotropy group of the identity element is the diagonal D(G) ⊂ G × G. In the language of Helgason [8], we obtain an irreducible symmetric pair (G × G, D(G)) of type II. The D(G)-action on (G × G)/D(G) is nothing but the action of G on itself by conjugation. But for any compact connected Lie group, the action on itself by conjugation is equivariantly formal. In fact, if T ⊂ G is a maximal torus, then the fixed point set of the T -action, G^{T}, is T itself, and thus dim H^{∗}(G^{T}) = dim H^{∗}(T) = 2^{rank G} = dim H^{∗}(G). For other ways to prove that this action is equivariantly formal see [2, Example 4.6]. For instance, equivariant formality would also follow from Proposition 4.23 below as (G × G, D(G)) is of split rank.

4.4 Inner symmetric spaces

Consider the case that the symmetric space G/K of compact type is inner, i.e., that the involution σ is inner. By [8, Theorem IX.5.6] this is the case if and only if rank G = rank K. Hence, a maximal torus T_{K} ⊂ K is also a maximal torus in G, and the T_{K}-fixed point set is by Lemma 4.3 a finite set of cardinality \frac{|W (G)|}{|W (K)|}. Because of the following classical result (see for example [4, Chapter XI, Theorem VII]), the case of inner symmetric spaces is easy to deal with.

Proposition 4.20. Given any compact connected Lie groups K ⊂ G, the following conditions are equivalent:

1. rank G = rank K.

2. χ(G/K) > 0.

3. H^{odd}(G/K) = 0.

It follows from Proposition 3.3 that the K-action on a homogeneous space G/K with rank G = rank K is always equivariantly formal. Alternatively, [2, Corollary 4.5] implies that the G-action on G/K is equivariantly formal because all its isotropy groups have rank equal to the rank of G. Then by Corollary 3.2 any closed subgroup of G acts equivariantly formally on G/K.

Proposition 4.21. If rank G = rank K, then the K-action on G/K is equivariantly formal. If T_{K} ⊂ K is a maximal torus, then the fixed point set of the induced T_{K}-action consists of exactly dim H^{∗}(G/K) = \frac{|W (G)|}{|W (K)|} points.

Remark 4.22. This is not a new result. For an investigation of the (algebra structure of the) equivariant cohomology of homogeneous spaces G/K with rank G = rank K see [7], or [9, Section 5] for an emphasis on other coefficient rings.

4.5 Spaces of split rank

Also when G/K is of split rank, i.e., rank G = rank K + rank G/K, there is a general argument that implies equivariant formality of the K-action on G/K.

Proposition 4.23. If G/K is of split rank, then the natural K-action on G/K is equivariantly formal.

Proof. We will show that every K-isotropy algebra has maximal rank, i.e., rank equal to rank 𝔨. Then equivariant formality follows from [2, Corollary 3.5]. Consider the decomposition 𝔤 = 𝔨 ⊕ 𝔭 and choose any Ad_{K}-invariant scalar product on 𝔭 that turns G/K into a Riemannian symmetric space. Then we have an exponential map exp : 𝔭 → G/K, and it is known that every orbit of the K-action on G/K meets exp(𝔞), where 𝔞 is a maximal abelian subalgebra of 𝔭. Because G/K is of split rank, there is a maximal torus T_{K} ⊂ K such that 𝔱_{𝔨} ⊕ 𝔞 is abelian. The torus T_{K} acts trivially on exp(𝔞). Thus, the K-isotropy algebra of any point in exp(𝔞) (and hence of any point in M) has maximal rank.

In the split-rank case we have r = 1 by Example 4.16. We thus have Proposition 4.24. If G/K is of split rank then dim H^{∗}(G/K) = 2^{rank G/K}. If T_{K} ⊂ K is a maximal torus, then the fixed point set of the induced T_{K}-action on G/K is a rank G/K-dimensional torus (in particular connected).

4.6 Outer symmetric spaces which are not of split rank

For the remaining cases that are not covered by any of the arguments above, i.e., irreducible simply-connected symmetric spaces of type I that are neither of equal nor of split rank, we do not have a general argument for equivariant formality of the isotropy action. Using the classification of symmetric spaces [8, p. 518], we calculate for each of these spaces the dimension of the cohomology of the T_{K}-fixed point set and show that it coincides with the dimension of the cohomology of G/K (which we take from the literature), upon which we conclude equivariant formality via Proposition 3.1. Fortunately, there are only three (series of) such symmetric spaces, namely

SU(n)/ SO(n), SO(2p + 2q + 2)/ SO(2p + 1) × SO(2q + 1), and E_{6}/ PSp(4),

where n ≥ 4 and p, q ≥ 1. We have shown with Propositions 4.6 and 4.14 that

dim H^{∗}((G/K)^{T_{K}}) = 2^{rank 𝔤−rank 𝔨} · \frac{|W (𝔨′)|}{|W (𝔨)|},

where the symmetric subalgebra 𝔨′ ⊂ 𝔤 was introduced in Proposition 4.13. Because in this section we are dealing with outer symmetric spaces, we have rank 𝔤 > rank 𝔨, so 𝔨′ ≠ 𝔤 is a symmetric subgroup of 𝔤. The orders of the appearing Weyl groups are listed in [11, p. 66].

4.6.1 SU(2m)/ SO(2m)

Let M = SU(2m)/ SO(2m), where m ≥ 2, and T ⊂ SO(2m) be a maximal torus. The only connected symmetric subgroup of SU(2m) of rank m different from SO(2m) is Sp(m). The fact that 𝔨′ = 𝔰𝔭(m) can be visualized via the Dynkin diagrams: the involution σ fixes only the middle root of the Dynkin diagram A_{2m−1} of SU(2m). Hence, after restricting, the middle root becomes a root which is longer than the other roots, and only in C_{m} there exists a root longer than the others, not in D_{m}.

We thus may calculate

r = \frac{|W (C_{m})|}{|W (D_{m})|} = \frac{2^{m} · m!}{2^{m−1} · m!} = 2;

note that for this example the number of compartments was also calculated in [1, p. 11]. It is known that dim H^{∗}(M) = 2^{m} (see for example [4, p. 493] or [14, Theorem III.6.7.(2)]), hence

dim H^{∗}(M^{T}) = 2^{2m−1−m} · r = 2^{m} = dim H^{∗}(M).

Thus, the action is equivariantly formal.

4.6.2 SU(2m + 1)/ SO(2m + 1)

Let M = SU(2m + 1)/ SO(2m + 1), where m ≥ 2, and T ⊂ SO(2m + 1) be a maximal torus. It is known that dim H^{∗}(M) = 2^{m} (see for example [4, p. 493] or [14, Theorem III.6.7.(2)]), hence

2^{m} · r = dim H^{∗}(M^{T}) ≤ dim H^{∗}(M) = 2^{m}

for some natural number r. Thus necessarily r = 1 (in fact 𝔨′ = 𝔰𝔬(2m + 1)) and the action is equivariantly formal. Note that this space is also listed as an exception in [1] as it is the only outer symmetric space which is not of split rank such that the corresponding involution fixes no root in the Dynkin diagram (and hence every compartment is a K-Weyl chamber).

4.6.3 SO(2p + 2q + 2)/ SO(2p + 1) × SO(2q + 1)

Let M = SO(2p + 2q + 2)/ SO(2p + 1) × SO(2q + 1), where p, q ≥ 1, and T ⊂ SO(2p+1)×SO(2q+1) be a maximal torus. The only connected symmetric subgroups of SO(2p + 2q + 2) of rank p + q are SO(2p′ + 1) × SO(2q′ + 1), where p′ + q′ = p + q. The involution σ fixes all roots of the Dynkin diagram D_{p+q+1} of SO(2p + 2q + 2) but two; after restricting, these two become a single root which is shorter than the others. Because A_{p+q−1}⊕A_{1} and D_{p+q} do not appear as the Dynkin diagram of any of the possible symmetric subgroups, the Dynkin diagram of 𝔨′ is forced to be B_{p+q}, which means that 𝔨′ = 𝔰𝔬(2p + 2q + 1).

We thus have

r = \frac{|W (B_{p+q})|}{|W (B_{p})| · |W (B_{q})|} = \frac{2^{p+q} · (p + q)!}{2^{p} · p! · 2^{q} · q!} = \binom{p + q}{p}.

By [4, p. 496] we have dim H^{∗}(M) = 2 · \binom{p+q}{p}, and it follows that the action is equivariantly formal because of

dim H^{∗}(M^{T}) = 2^{p+q+1−p−q} · r = 2 · \binom{p + q}{p} = dim H^{∗}(M).

Documenta Mathematica 17 (2012) 79–94

4.6.4 E_{6}/ PSp(4)

Let M = E_{6}/ PSp(4) and T ⊂ PSp(4) be a maximal torus. The only symmetric subalgebra of 𝔢_{6} of rank 4 different from 𝔰𝔭(4) is 𝔣_{4}.

We obtain

r = \frac{|W (F_{4})|}{|W (C_{4})|} = \frac{2^{7} · 3^{2}}{2^{4} · 4!} = 3.

It is shown in [19] that dim H^{∗}(M) = 12. Thus,

dim H^{∗}(M^{T}) = 2^{6−4} · r = 2^{2} · 3 = 12 = dim H^{∗}(M)

shows that the action is equivariantly formal.

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