ON MOTIVIC DECOMPOSITIONS ARISING FROM THE METHOD OF BIAŁYNICKI-BIRULA

PATRICK BROSNAN

Abstract. Recently, V. Chernousov, S. Gille and A. Merkurjev have obtained a decomposition of the motive of an isotropic smooth projective homogeneous variety analogous to the Bruhat decomposition. Using the method of A. Bialynicki-Birula and a corollary, which is essentially due to S. del Baño, I generalize this decomposition to the case of a (possibly anisotropic) smooth projective variety homogeneous under the action of an isotropic reductive group. This answers a question of N. Karpenko.

1. Introduction

An important difference between the category of motives and the category of algebraic varieties over a field is the existence of interesting direct sum decompositions of motives. The simplest of these is the decomposition of the Chow motive M(ℙ^{n}) of n-dimensional projective space over a field k:

(1.1)  M(ℙ^{n}) = ⊕_{i=0}^{n}ℤ(n).

This is one of the elementary results in Grothendieck’s theory of motives (which will be recalled in §2).

An example of a less elementary decomposition is the following theorem due to M. Rost, which is in an important ingredient in his construction of the “Rost motive” [23, Proposition 2].

Theorem 1.1 (Rost decomposition). Let Q be a smooth, projective, ndimensional isotropic quadric over a field k of characteristic not equal to 2. Then

(1.2)  M(Q) = ℤ ⊕ M(Q′)(1) ⊕ ℤ(n)

where Q′ is a smooth, projective sub-quadric of codimension 2 in Q.

Since both projective spaces and quadrics are examples of projective homogeneous varieties, it is natural to look for decompositions generalizing (1.1) and (1.2) in the motives of such varieties. In the case that G is a split reductive group (i.e., when the base field k is separably closed), a decomposition for the motive of G/P was found by B. Köck [19]. In this case M(G/P) splits completely as a sum of Tate motives. A more general decomposition was later found by N. Karpenko [17] in the case of motives of flag varieties for classical groups, and recently Köck’s decomposition was further generalized by V. Chernousov, S. Gille and A. Merkurjev [9, Theorem 7.4] to the case of motives for isotropic projective homogeneous varieties for adjoint semi-simple groups. Both generalizations explicitly describe the factors appearing in the decomposition (which are, in general, not Tate motives) in terms of smaller projective homogeneous spaces. Moreover, Rost’s theorem appears as a special case of either generalization when the quadric Q is viewed as a homogeneous space for the group PSO(q) with q a quadratic form whose corresponding projective quadric is Q.

2000 Mathematics Subject Classification. Primary 11E04; Secondary 14C25.

Key words and phrases. correspondence, Chow groups and motives.

Theorem 3.5 of this paper applies results of Białynicki-Birula (as extended by W. Hesselink) to obtain a decomposition of the motive of any smooth, projective algebraic variety admitting an action of the multiplicative group. In the case of motives with rational coefficients (see Remark 2.1 for the distinction), the theorem is due to del Baño [10]. (I thank B. Köck for bringing this to my attention.) I include a sketch of the proof (which is similar to del Baño’s) for completeness and convenience of the reader.

As the class of varieties admitting multiplicative group actions includes both the homogeneous spaces considered by Karpenko and the isotropic homogeneous spaces considered by Chernousov, Gille and Merkurjev, we obtain a generalization of the Chernousov-Gille-Merkurjev decomposition. In the end, we can give a rather explicit decomposition of the motive M(X) of a projective homogeneous variety for a reductive group G as a sum of Tate twisted motives of certain “quasi-homogeneous schemes” X_{i} for the anisotropic kernel of G. Theorem 4.5 gives a rough form of this description which is refined in Theorem 7.4, the final theorem of the paper. In particular, the fact that the motive of a projective homogeneous variety admits a decomposition in terms of motives of quasi-homogeneous schemes for the anisotropic kernel answers the fundamental question posed by Karpenko in the introduction to [17].

1.1. Notation. All notions of Chow groups are taken from Fulton’s book on intersection theory [11]. The official reference for reductive groups is SGA3 [1], but some notation is taken from Springer’s book [25]. The main difference between these two references that will be important for this paper is that SGA3 demands that a reductive group is connected and a parabolic subgroup is smooth, while Springer does not make these assumptions. Since these are convenient assumptions for us, we will have to agree with SGA3. In several places, the symbol “k” is used to denote both the base field and an index. This does not seem to produce any confusion.

1.2. Outline. With one exception, the results in this paper build sequentially from general facts about motivic decompositions to specific information about the motivic decomposition of a projective homogeneous space in terms of double cosets of the Weyl group given in Theorem 7.4. Specifically, §2 reviews the theory of motives, §3 explains how Białynicki-Birula’s theorem yields a motivic decomposition and §4 introduces the concept of projective quasi-homogeneous schemes and studies motivic decompositions of such schemes in the presence of a 𝔾_{m}-action. The last two mathematical sections, §6 and §7, formulate the general theory in terms of root systems and reflection groups. The one exceptional section is §5 where I give a proof of a generalization of Rost’s nilpotence theorem following Chernousov, Gille and Merkurjev. The results in this section are not needed anywhere else in the paper.

2. Motives and the category of correspondences

The category of motives can be defined by first defining the category of correspondences and then applying the functor of idempotent completion. In fact, the decomposition theorems of this paper such as Theorem 3.5 will hold before (or after) taking idempotent completion, but, to make the connection with Chow motives explicit, I will describe both categories.

The category of correspondences is the category Corr_{k} whose objects are pairs (X, n) with X a smooth projective scheme over the field k and n an integer. The morphisms are given by

(2.1)  Hom_{Corr_{k}}((X, n), (Y, m)) = ⊕A_{d_{i}+n−m}X_{i} × Y

where X = ∐ X_{i} with X_{i} connected, d_{i} = dim X_{i} and A_{k} denotes the k-th Chow groups graded by dimension.

For a smooth, projective, variety X, M(X) denotes the object (X, 0). When M = (X, n) is an object, M(k) denotes the Tate twisted object (X, n + k). Let ℤ denote the object (Spec k, 0). The “twists” ℤ(k) of ℤ are called Tate objects. Clearly, Hom_{Corr_{k}}(Z(k), M(X)) = A_{k}X and Hom_{Corr_{k}}(M(X), Z(k)) = A_{d−k}X for X irreducible of dimension d.

The objects of the category Chow_{k} of Chow motives over k are triples (X, n, p) with p ∈ End(X, n) a morphism such that p^{2} = p. The morphisms in Chow_{k} are given by

Hom_{Chow_{k}}((X, n, p), (Y, m, q)) = qHom_{Corr_{k}}((X, n), (Y, m))p.

The category Chow_{k} is idempotent complete in the sense that every idempotent morphism has both a kernel and a cokernel. Clearly, there is a fully, faithful embedding Corr_{k} ↪ Chow_{k} given by (X, n) ⇝ (X, n, id). Moreover, this embedding is universal for functors from Corr_{k} with idempotent complete targets.

Both categories admit a tensor structure defined (on Chow_{k}) by

(X, n, p) ⊗ (Y, m, q) = (X × Y, n + m, p × q).

The category Chow_{k} also admits direct sums defined as follows. Let r_{k} denote the idempotent on M(ℙ^{k}) such that ℤ(k) = (ℙ^{k}, 0, r_{k}). Explicitly, it is given by the cycle [pt × ℙ^{k}] where pt denotes an arbitrary degree 1 closed point in ℙ^{k}. Then, for two motives (X, n, p) and (Y, m, q) with n ≤ m,

(X, n, p) ⊕ (Y, m, q) = (X ∐(Y × ℙ^{m−n}), n, p + (q × r_{m−n})).

The direct sum is the coproduct in the category of motives. In the category of correspondences, coproducts do not always exist. For example, it is not hard to see that the object ℤ ∐ ℤ(2) does not exist in the category of correspondences over ℂ. However, when coproducts do exist in Corr_{k}, they coincide with those in the category of Chow motives.

Remark 2.1. The categories of motives occurring in this paper (and in those papers of Rost, Karpenko, and Chernousov-Gille-Merkurjev) have integral coefficients. If we were to tensor the morphism sets with ℚ, replacing Hom_{Corr_{k}}(M(X), M(Y)) with Hom_{Corr_{k}}(M(X), M(Y)) ⊗ ℚ, we would obtain a category Corr_{k} ⊗ℚ which is closer to the categories of motives Grothendieck originally considered [16, 21]. However, we would also loose information. For example, using the fact that any quadric is totally isotropic over a finite separable extension, it is easy to see that every quadric decomposes as a direct sum of the Tate objects ℚ(i) = ℤ(i) ⊗ ℚ in Corr_{k} ⊗ℚ. On the other hand, Springer’s theorem on quadrics isotropic over an odd degree extension [20, Theorem 2.3 p. 198] implies that the integral motive M(Q) ∈ Corr_{k} of a smooth quadric contains a factor of ℤ(0) if an only if Q(k) ≠ ∅.

3. Motivic decomposition

The most general theorem in this paper on motivic decompositions is essentially a corollary of two results which I will now recall after giving one definition. As mentioned in the introduction, a version of the theorem (3.5) can also be found in Theorem 2.4 of S. del Baño’s paper, [10].

Definition 3.1. A flat morphism ϕ : X → Z is called an affine fibration (resp. an affine quasi-fibration) of relative dimension d if, for every point z ∈ Z, there is a Zariski open neighborhood U ⊂ Z such that X_{U} ≅ Z × 𝔸^{d} with ϕ : X_{U} → Z isomorphic to the projection on the first factor (resp. the fiber X_{z} of ϕ is isomorphic to 𝔸_{k(z)}^{d}).

Clearly an affine fibration is an affine quasi-fibration. It is a well-known consequence of the homotopy invariance of Chow groups that an affine quasifibration between smooth varieties of relative dimension d induces an isomorphism ϕ^{∗} : A_{i}Z → A_{i+d}X.

Theorem 3.2 (Karpenko). Let X be a smooth, projective variety over a field k with a filtration

X = X_{n} ⊃ X_{n−1} ⊃ ⋯ ⊃ X_{0} ⊃ X_{−1} = ∅

where the X_{i} are closed subvarieties. Assume that, for each integer i ∈ [0, n], there is a smooth projective variety Z_{i} and an affine fibration ϕ_{i} : X_{i} − X_{i−1} → Z_{i} of relative dimension a_{i}. Then, in the category of correspondences, M(X) = ∐_{i=0}^{n} M(Z_{i})(a_{i}).

The theorem was stated by Karpenko [17] for the special case that the maps ϕ_{i} : X_{i} − X_{i−1} → Z_{i} are vector bundle morphisms. However, in [9, Theorem 7.1], Chernousov, Gille and Merkurjev noticed that Karpenko’s proof actually applies to any affine quasi-fibration. (Del Baño gives a slightly different proof of the result in the proof of his Theorem 2.4).

The second result is the method of Białynicki-Birula which gives a natural situation where Karpenko’s theorem applies.

Theorem 3.3 (Białynicki-Birula, Hesselink, Iversen). Let X be a smooth, projective variety over a field k equipped with an action of the multiplicative group 𝔾_{m}. Then

(1) The fixed point locus X^{𝔾_{m}} is a smooth, closed subscheme of X.

(2) There is a numbering X^{𝔾_{m}} = ∐_{i=1}^{n} Z_{i} of the connected components of the fixed point locus, a filtration

X = X_{n} ⊃ X_{n−1} ⊃ ⋯ ⊃ X_{0} ⊃ X_{−1} = ∅

and affine fibrations ϕ_{i} : X_{i} − X_{i−1} → Z_{i}.

(3) The relative dimension a_{i} of the affine fibration ϕ_{i} is the dimension of the positive eigenspace of the action of 𝔾_{m} on the tangent space of X at an arbitrary point z ∈ Z_{i}. The dimension of Z_{i} is the dimension of T X_{z}^{𝔾_{m}}.

As the theorem stated is the product of several results of different authors, I will give a short history of the result (in lieu of a proof).

Iversen [15] showed that X^{𝔾_{m}} is smooth. Białynicki-Birula [2] showed, under the assumption that k is algebraically closed, that X is a union of locally closed subschemes X_{i}^{+} with affine fibrations ϕ_{i} : X_{i}^{+} → Z_{i} where the Z_{i} are the connected components of X^{𝔾_{m}}. In fact, this was shown with the assumption that X is projective replaced with the assumption that X is complete. Shortly thereafter, Białynicki-Birula showed that, when X is projective, there is a filtration of X and an ordering of the connected components as in the theorem such that X_{i}^{+} = X_{i}−X_{i−1} [3]. Thus, in the case that k is algebraically closed, the theorem as stated was proved by Białynicki-Birula and Iversen. Note that the existence of a filtration is deduced by embedding X equivariantly in a projective space with a diagonalized 𝔾_{m}- action. It is then easy to construct a filtration on the projective space and see that that it restricts to one on X. (However, for X smooth and complete but not projective, there are examples where no filtration satisfying X_{i}^{+} = X_{i} − X_{i−1} exists [8, Example 2, p. 30].)

Białynicki-Birula’s proofs make use of the assumption that k is algebraically closed. Hesselink removed this restriction and was able to show that X is a union of locally closed X_{i}^{+} for X a smooth, proper scheme over an arbitrary base [13] provided that there is a covering of X by 𝔾_{m}-stable Zariski open affine subsets. To construct a filtration in the case X is projective, it then suffices to find a 𝔾_{m}-equivariant embedding of X into a projective space with a diagonal action of 𝔾_{m} or, equivalently, a very ample 𝔾_{m}-linearized line bundle over X. The fact that such bundles exist can be found in Mumford’s GIT [22]. Since a diagonal action of 𝔾_{m} on ℙ^{r} preserves the coordinate hyperplanes (and thus their complements), any X embedded by a 𝔾_{m}-linearized very ample line bundle automatically has a covering by 𝔾_{m}-invariant open affines. Thus Hesselink’s hypotheses are verified for X smooth and projective.

The Białynicki-Birula decomposition is explicit in the sense that the locally closed subscheme X_{i}^{+} is the set of all points x ∈ X such that lim_{t→0} tx ∈ Z_{i} where (t, x) ↦ tx is the 𝔾_{m} action. Moreover, the map ϕ_{i} : X_{i}^{+} → Z_{i} is then given by x ↦ lim_{t→0} tx.

Remark 3.4. Since X separated, lim_{t→0} tx has at most one meaning, since X is proper it has exactly one.

Theorem 3.5. Let X be a smooth, projective scheme over a field k equipped with an action of the multiplicative group 𝔾_{m}. Then, in the category Corr_{k},

(3.1)  M(X) = ∐ M(Z_{i})(a_{i})

where the Z_{i} are the connected components of X^{𝔾_{m}} and the a_{i} are determined as in Theorem 3.3 (3).

Proof. This is a corollary of the two previous theorems. □

We will refer to the decomposition of (3.1) as the motivic Białynicki-Birula decomposition. The rest of this paper will focus on the application of the theorem to the special case where X is a projective homogeneous variety. Before proceeding to the general theory, I use the theorem directly to derive Rost’s decomposition theorem.

Example 3.6. Let q : V → k be a non-degenerate quadratic form of dimension n + 2 over a field k with char k ≠ 2, and let Q be the associated n-dimensional smooth projective quadric. Suppose that q is isotropic, that is, there exists a nonzero vector v ∈ V such that q(v) = 0. Then there is a subspace W ⊂ V and two linearly independent vectors v_{1} and v_{2} such that V = kv_{1} ⊕ kv_{2} ⊕ W and

q(xv_{1} + yv_{2} + w) = xy + q′(w)

for a non-degenerate quadratic form q′ on W. (This is an easy exercise which can also be found in almost any book on quadratic forms over a field, e.g., [18, Proposition 3.7.1].) In this case, the multiplicative group 𝔾_{m} acts on Q by

[xv_{1} + yv_{2} + w] ↦ [txv_{1} + t^{−1}yv_{2} + w].

Assume that dim Q ≠ 2. The fixed point set Q^{𝔾_{m}} then has three components: the points [v_{1}] and [v_{2}] and the quadric Q′ = {[w] ∈ ℙ(W) | q′(w) = 0} which we denote by Z_{1}, Z_{2} and Z_{3} respectively. Let T_{i} denote the tangent space at an (arbitrary) point of Z_{i}. The action of 𝔾_{m} on T_{1} has only negative weights. Therefore, in the decomposition of Theorem 3.5, a_{1} = 0. The weights of T_{2} are all positive, therefore, a_{2} = n. Finally, T_{3} has weights −1 and 1 each occurring once and 0 occurring n − 2 times. Therefore a_{3} = 1. Thus we have

(3.2)  M(Q) = ℤ ∐ ℤ(n) ∐ M(Q′)(1).

It is easy to see that the above decomposition also holds when dim Q = 2, however there is a possibility that the quadric Q′ can split as a disjoint union of 2 copies of Spec k.

4. Projective homogeneous schemes

Let G denote a reductive group over a field k in the terminology of SGA3. That is, G is smooth and connected with trivial unipotent radical. Recall that a parabolic subgroup of G is a subgroup P such that G/P is projective and P is smooth over k. Let G−Sch_{k} denote the category of G-schemes over k. The objects of this category are schemes X over k equipped with a Gaction G × X → X. The morphisms are the G-equivariant scheme-theoretic morphisms. Base change induces an obvious functor G−Sch_{k} ⇝G−Sch_{L} for L an extension of k.

Let \overline{k} denote an algebraic closure of k. A G-scheme X is a projective homogeneous variety for G if X\overline{k} is isomorphic as a G\overline{k}-scheme to G\overline{k}/P for P ⊂ G\overline{k} a parabolic subgroup. It is well-known that such a projective homogeneous variety is projective over k. We will call a G-scheme X a projective quasi-homogeneous scheme if X is smooth and projective over k and the morphism ψ = (a, pr_{2}) : G × X → X × X given by (g, x) ↦ (gx, x) is smooth.

Proposition 4.1. Let X be a G-scheme over k. Then the following are equivalent.

(1) X is a projective quasi-homogeneous G-scheme.

(2) X\overline{k} is a disjoint union of projective homogeneous varieties.

(3) X is smooth, projective, and, for every geometric point x ∈ X\overline{k}, the orbit map m_{x} : G\overline{k} → X\overline{k} (given by g ↦ gx) induces a surjection dm_{x} : L(G) → T X_{x}.

Proof. Since all of the properties listed are invariant under base change of k, we can assume that k is algebraically closed.

(1) ⇒ (2): A scheme is projective quasi-homogeneous if and only if all of its connected components are projective quasi-homogeneous. (This is easy.) Therefore we can assume that X is connected. Since G × X → X × X is smooth, all orbits are open. Thus, since X is smooth and connected (hence irreducible), all orbits must intersect. It follows that there is only one orbit, namely, X itself. Thus X = G/P for some subgroup P. The smoothness of ψ then implies that P is smooth.

(2) ⇒ (3): Here we can assume X = G/P with P parabolic. The claim then follows from the assumption that P is smooth.

(3)⇒ (1): From (3), it follows that ψ : G×X → X ×X induces surjections on the tangent spaces. The claim then follows from [12, p. 270, Proposition 10.4 (iii)]. □

Remark 4.2. The motivation for considering projective quasi-homogeneous schemes in addition to projective homogeneous varieties is already apparent in the Rost decomposition of quadrics. Let q be an n-dimensional non-degenerate quadratic form, let Q be the associated n − 2 dimensional smooth projective quadric and let PSO(q) denote the special orthogonal group. Then Q is a projective homogeneous space for PSO(q) if and only if dim Q > 0. When dim Q = 0, Q can either be irreducible or a disjoint union of two copies of Spec k. In either case, it is not a projective homogeneous variety for PSO(q). However, regardless of the dimension, Q is projective quasi-homogeneous for PSO(q).

Now let X denote a projective quasi-homogeneous scheme for G, and let

𝔾_{m} →^{∼}^{λ} L ⊂ G denote the inclusion of a k-split torus in G. (The group G is called isotropic if such a split torus exists.) In this case, L acts on X and Theorem 3.5 applies to give a decomposition of M(X). The main result of this section is that, in fact, the summands appearing are themselves projective quasi-homogeneous schemes for the centralizer H = Z(λ) of L in G. A more detailed description will be obtained in §6 and §7.

Theorem 4.3. (1) H is connected, reductive and defined over k. (2) H acts on the fixed point set X^{λ}.

(3) The action map ψ_{H} : H × X^{λ} → X^{λ} × X^{λ} is smooth. Thus X^{λ} is projective quasi-homogeneous.

Proof. (1) The fact that H is connected is [Springer, 13.4.2 (i)]. The fact that it is reductive is [Springer, 7.6.4 (i)]. It is defined over k by [Springer, 13.3.1 (ii)].

(2) To see that H acts on X^{λ}, let T be a scheme over k and consider T -valued points x ∈ X^{λ}(T), h ∈ H(T) and t ∈ L(T). Then thx = htx = hx, thus, hx is in X^{λ}(T).

(3) Since the smoothness of ψ_{H} is invariant under field extension of k, we may assume that k = \overline{k}. Pick a closed point z ∈ Z. The orbit map l : G → X given by g ↦ gz, induces a surjection dl : L(G) → T X_{z} because, by assumption, X is a projective quasi-homogeneous scheme for G. The multiplicative group 𝔾_{m} acts on G via conjugation by λ, i.e., g ↦ λ(t)gλ(t^{−1}). The group 𝔾_{m} also acts on X via right multiplication, i.e., x ↦ λ(t)x. Since z is a fixed point of λ, the orbit map l is equivariant for the 𝔾_{m}-actions. Moreover, we obtain a 𝔾_{m}-action on T X_{z} compatible with the Ad-action of 𝔾_{m} on L(G).

Now L(G) ≅ L(G)_{+} ⊕ L(H) where L(G)_{+} consists of the non-zero weight space of L(G) and L(H) is the Lie algebra of H. Analogously, T X_{z} ≅ T X_{z+} ⊕ T X_{z}^{λ} where T X_{z+} is the non-zero weight space of T X_{z}. Since dl respects the weight decomposition, dl(L(H)) = T X^{λ}. Thus Proposition 4.1 (3) is satisfied. □

Corollary 4.4. Let X be a projective quasi-homogeneous scheme for an isotropic reductive group G, and let λ : 𝔾_{m} → G be the embedding of a split torus. Then, in the motivic Białynicki-Birula decomposition

M(X) = ∐ M(Z_{i})(a_{i})

of Theorem 3.5, the Z_{i} are all projective quasi-homogeneous schemes for the centralizer H of λ.

Proof. The corollary holds because the Z_{i} appearing in Theorem 3.5 are components of X^{λ}. □

4.1. Adjoint groups. For a reductive group G, let QH_{G} denote the full subcategory of G−Sch_{k} consisting of projective quasi-homogeneous schemes. If Z_{G} is the center of G, then G_{ad} = G/Z_{G} is the adjoint group of G, an adjoint semi-simple group [1, Proposition 22.4.3.5]. The restriction functor G_{ad}−Sch_{k} ⇝ G−Sch_{k} induces an equivalence of categories QH_{G_{ad}} ⇝ QH_{G}. This is because Z_{G} is smooth and acts trivially on all quasi-homogeneous schemes over G.

4.2. Anisotropic Kernels. [26] Let S denote a maximal k-split torus of G, and let Z(S) denote its centralizer. The derived subgroup DZ(S) is the semi-simple anisotropic kernel. Since Z(S) is reductive, there is an almost direct product decomposition Z(S) = DZ(S) · Z where Z is the center of Z(S)[5, Proposition 2.2]. It follows that the adjoint group of Z(S) is isomorphic to the adjoint group of the semi-simple anisotropic kernel. Thus the categories QH_{Z(S)}, QH_{DZ(S)} and QH_{Z(S)_{ad}} are all equivalent via the restriction of group functors. (Moreover, the objects are identical.)

Applying Corollary 4.4 inductively, we obtain an answer to a question of N. Karpenko [17].

Theorem 4.5. Let X be a projective quasi-homogeneous scheme for a reductive group G. Then

(4.1)  M(X) = ∐ M(Y_{i})(a_{i})

where the Y_{i} are irreducible projective quasi-homogeneous schemes for the anisotropic kernel of G (resp. for Z(S), for Z(S)_{ad}).

Remark 4.6. A projective homogeneous variety X is said to be isotropic if X = G/P for a parabolic subgroup P defined over k. Otherwise it is said to be anisotropic. X is anisotropic if and only if X(k) is empty. If X is an isotropic projective homogeneous space for a reductive group G, then there exists at least one k-split torus L in G. (See [25] or [1].) In other words, if X is isotropic then G is as well. It follows that the schemes Y_{i} appearing in (4.1) are all either anisotropic or isomorphic to Spec k.

Exercise 4.7. It is interesting to see an example of Corollary 4.4 at work on an anisotropic projective homogeneous variety for an isotropic reductive group. One such is given by the variety X of two dimensional isotropic subspaces for the quadratic form q = x_{1}^{2} + ⋯ + x_{2n}^{2} + yz over the reals with n ≥ 2. Using the methods of Example 3.6, the decomposition can be computed explicitly in terms of smaller quadrics and varieties of two dimensional isotropic subspaces. In Example 7.6, we will return to this matter, computing the decomposition using the Lie theory of PSO(q).

5. The nilpotence theorem of Chernousov, Gille and Merkurjev

As a corollary of the results of the previous section, we obtain the following theorem of Chernousov, Gille and Merkurjev [9, Theorem 8.2].

Theorem 5.1. Let X be a projective homogeneous variety for a reductive group G over a field k. Then the kernel of the map

End(M(X)) → End(M(X ⊗ \overline{k}))

consists of nilpotent endomorphisms.

The proof follows that of [9]. I include it here for the convenience of the reader and to make the point that the theorem can be obtained without the full description of the motivic decomposition obtained in [9, Theorem 7.4].

For a field extension L/k, let n_{L} denote the number of terms appearing in the decomposition

(5.1)  M(X_{L}) = ∐_{i=1}^{n_{L}} M(Z_{i})(a_{i})

of (4.1) for the projective homogeneous G_{L}-variety X_{L}. (Here the Z_{i} will depend on L.) Clearly, M ⊃ L ⇒ n_{M} ≥ n_{L}, and the maximal number of terms in the coproduct occurs precisely when each Z_{i} is Spec L. In particular, this happens when L = Spec k_{sep}.

Claim 5.2. Set N(d, n) = (d + 1)^{\overline{n}−n} with \overline{n} = n_{k_{sep}}. Then, for any morphism f ∈ End(M(X)) with f ⊗ \overline{k} = 0, f^{N(d,n_{k})} = 0.

Evidently the claim implies the theorem.

Now in the case that the maximal number of terms appears in the decomposition (i.e., n_{k} = \overline{n}), the claim is trivial because each of the objects appearing is Tate. In fact, the only morphism in End(M(X)) which vanishes in End(M(X) ⊗ \overline{k}) is the 0 morphism. Thus the claim is valid for n_{k} = \overline{n}.

Now reason by descending induction on n = n_{k}. (Properly speaking, we reason by ascending induction on \overline{n} − n_{k} starting with the case \overline{n} − n_{k} = 0.) Let f ∈ End(M(X)) be an endomorphism in the kernel of the map to End(M(X ⊗ \overline{k})) and pick a point z in one of the anisotropic components Z_{i} appearing in (5.1). (If all components are isotropic, n is maximal and the claim is already proved.) Set L = k(z). Over L, Z_{i} is isotropic. Therefore the number n_{i} = n_{L} of terms appearing in the motivic decomposition of X_{L} is greater than n. Thus the claim holds for X_{L} and f_{L}^{N(d,n_{i})} = 0. Since N(d, n_{i}) ≤ N(d, n + 1), it follows that f_{L}^{N} = 0 where N = N(d, n + 1).

I now use [7, Theorem 3.1 and Remark 3.2] in the form in which it was used by Gille, Chernousov and Merkurjev [9, Proposition 8.1].

Lemma 5.3. Let X be a smooth, projective variety over a field k and Z an r-dimensional scheme of finite type over k. Let f ∈ End(M(X)) be an endomorphism such that, for every point z ∈ Z, the morphism f_{z∗} : A_{∗}(X⊗k(z)) → A_{∗}(X⊗k(z)) vanishes. Then f_{∗}^{r+1} : A_{∗}(X×Z) → A_{∗}(X×Z) vanishes.

From the lemma and the fact that dim Z_{i} ≤ d, it follows that the composition

(5.2)  M(Z_{i})(a_{i}) →^{j_{i}} M(X) →^{f^{(d+1)N(d,n+1)}} M(X)

vanishes where the first arrow, j_{i}, is the canonical one coming from the coproduct decomposition. Thus, for each anisotropic Z_{i} in the coproduct, f^{(d+1)N} ∘ j_{i} = 0. On the other hand, if Z_{i} = Spec k, then it is easy to see that the composition in (5.2) vanishes even with f^{(d+1)N(d,n+1)} replaced by f.

Since N(d, n + 1) = (d + 1)^{\overline{n}−n_{k}−1}, the claim is proved.

Remark 5.4. Clearly the exponent (d + 1)^{\overline{n}−n_{k}} is not optimal.

6. The Weyl group and its double cosets

In this section and the next, I give an explicit description of the components Z_{i} appearing in the motivic decomposition (3.1) of a projective homogeneous variety X for an isotropic reductive group G. Roughly speaking, the geometric components are in one-to-one correspondence with certain double cosets of the Weyl group. The algebraic components correspond to equivalences classes of these double cosets under the so-called “*-action” of the absolute Galois group of the base field k (6.1).

While the language of schemes was used in the previous sections, in this section I abuse notation slightly (e.g. in the proof of lemma 6.2) and confuse points with k-valued points. This facilitates comparison with the reference [25] which is written in the language of varieties.

For T ⊂ G a maximal (but not necessarily split) torus defined over k, set W = W (G, T), the corresponding Weyl group. For a subtorus C ⊂ T, let W_{C} = W (Z_{G}(C), T) where Z_{G}(C) denotes the centralizer of C. It is a subgroup of W. Likewise, for a character ϕ : 𝔾_{m} → T, let W_{ϕ} = W (Z_{G}(ϕ), T), also a subgroup of W.

Now, in the situation of the §4, G has a cocharacter λ : 𝔾_{m} →^{≅} L ⊂ G. We can assume that L ⊂ S ⊂ T with S a maximal k-split torus and T a maximal torus with T defined over k [25, Theorem 13.3.6 and Remark 13.3.7].

If X is isotropic, there is a parabolic subgroup P such that X = G/P. Since G is reductive, we may assume that P = P (μ) for a cocharacter μ : 𝔾_{m} → S. (Roughly, P (μ) is defined as the set of g ∈ G such that lim_{t→∞} μ(t)gμ(t)^{−1} exits. See [25, §13.4.1].) Set

(6.1)  X = W_{λ}\W/W_{μ}.

Theorem 6.1. If k is separably closed, the connected components Z_{i} appearing in the motivic Białynicki-Birula decomposition of X are in one-one correspondence with the elements of X.

To begin the proof of the theorem, first note that, since k is separably closed, X is isotropic and X = G/P with P = P (μ) as above. It follows that the maximal torus T acts on G/P with fixed points corresponding to cosets in

(6.2)  Y = W/W_{μ}.

To see this, suppose that y ∈ G(k) is such that the coset yP is fixed by T. Then y^{−1}T y is a maximal torus contained in P. On the other hand, since P = P (μ) with μ : 𝔾_{m} → S ⊂ T, T is also contained in P. Since all maximal tori are conjugate within P [1, Corollary 5.7, p. 496], there is a p ∈ P (k) such that

(6.3)  p^{−1}T p = y^{−1}T y.

Thus yp^{−1} is in the normalizer N_{G}(T) of T and, thus, represents an element w ∈ W = W (G, T). If p′ is another element of P satisfying (6.3), then p′p^{−1} normalizes T. This implies that p′p^{−1} ∈ Z_{G}(μ) by the following.

Lemma 6.2. For G reductive with maximal torus T and μ ∈ X_{∗}(T), P (μ)∩ N_{G}(T) ⊂ Z_{G}(μ).

Proof. Take p ∈ P (μ) ∩ N_{G}(T) and set β(t) = pμ(t)p^{−1}. Since p normalizes T, β ∈ X_{∗}(T). Set a(t) = μ(t)pμ(t)^{−1}. Since p ∈ P (μ), a := lim_{t→0} a(t) exists. Therefore

(6.4)  lim_{t→0} β(t)μ(t)^{−1} = pa.

In particular, the limit in (6.4) exists. But, since β(t)μ(t)^{−1} is a cocharacter, this is only possible if β(t) = λ(t). □

It follows, therefore, that y and equation (6.3) determine the class π(y) of yp^{−1} in W/W_{μ}. Moreover, if y′ = yp′ is another element of G representing the coset yP, then it is easy to check that π(y) = π(y′).

Thus, there is a map π : (G/P)^{T} → Y. It is not hard to see that the map s̃: W → (G/P)^{T} given by w ↦ wP induces a map s : Y → (G/P)^{T} inverse to π. Thus (G/P)^{T} ≅ Y.

Now let yP and y′P be two points in the same component Z of (G/P)^{λ} which are both fixed by the T action. Without loss of generality, we can then assume that y and y′ normalize T. By Proposition 4.3, yP = hy′P for some h ∈ Z_{G}(λ). It follows then that h also normalizes T and is, thus, in W_{λ}. Thus π induces a map

(6.5)  q : π_{0}((G/P)^{λ}) → X .

To see that q is an isomorphism, it suffices to check that s : Y → (G/P)^{T} induces an inverse map r : X → π_{0}((G/P)^{λ}. I leave this verification, which completes the proof of Theorem 6.1, to the reader.

6.1. The Galois action. Let Γ = Gal(k_{sep}/k) denote the absolute Galois group. If X = G/P (μ) is an isotropic projective homogeneous variety, then Γ acts on G(k_{sep}), T (k_{sep}) and W (k_{sep}) stabilizing P = P (μ) and, thus, W_{μ}. It follows that Γ acts on the double coset space X. Clearly Γ also acts on (G/P)^{λ}(k_{sep}), and it is easy to see that the map r : X → π_{0}(X^{λ}) is an isomorphism of Γ-sets.

Computing the Galois action on π_{0}(X^{λ}) can be reduced to the case of isotropic X using the fact that every reductive k-group has an quasi-split inner form G_{inn} [25, Proposition 16.4.9] given by a class σ ∈ H^{1}(Γ, G_{ad}) where G_{ad} denotes the adjoint group of G. Let p : G → G_{ad} be the canonical quotient map. It is easy to see that we can arrange that T is stabilized and λ is fixed by σ. Then, in fact, σ is in the image of the map

H^{1}(Γ, Z_{G_{ad}}(p ∘ λ) ∩ N_{G_{ad}}(p(T))) → H^{1}(Γ, G_{ad}).

We have an action of G_{ad} on X and, thus, a twist X_{σ} of X with an action of λ. Under the twist, the action of the Galois group on W_{inn} = W (G_{inn}, T_{σ}) is given by the ∗-action [26]. Since G_{inn} is quasi-split, X_{σ} is isotropic and thus corresponds to a parabolic P (μ) in G_{inn}. From the previous section, we then have

π_{0}(X_{σ}^{λ}) = W_{λ}\W_{inn}/W_{μ}.

Proposition 6.3. There is an isomorphism

π_{0}(X^{λ}) ≅ π_{0}(X_{σ}^{λ}) = W_{λ}\W_{inn}/W_{μ}

of étale schemes over Spec k. (In other words, the above sets are isomorphic as Γ-sets.)

Proof. It is easy to see that π_{0}(X_{σ}^{λ}) viewed as an étale scheme over Spec k is the twist of π_{0}(X^{λ}) by σ. Since Z_{G_{ad}}(p ∘ λ) is geometrically connected, it acts trivially on the geometric points of π_{0}(X^{λ}). Thus the two schemes are isomorphic. □

7. Explicit Description and Examples

With a little extra work, we can give an explicit description of the twists and the spaces Z_{i} appearing in Corollary 4.4 and Theorem 4.5 in terms of the relevant reflection groups, Dynkin diagrams and root systems. This is a generalization of the description appearing in [9].

From the previous section, we know that the quasi-homogeneous schemes Z_{i} are in correspondence with the orbits of the ∗-action on the double cosets in W_{λ}\W_{inn}/W_{μ}. Over k_{sep}, each such Z_{i} decomposes as a disjoint union Z_{w} over the elements of the ∗-orbit. Our goal is then to describe the Tate twist associated to Z_{w} and also the projective homogeneous space Z_{w} in terms of the Dynkin diagram of Z(λ).

In obtaining our description, it will be convenient to consider the case where G is split first. Therefore, assume G is split with maximal torus T. Let R denote the set of roots of G. Choose a Borel subgroup B or, equivalently, a set R_{+} of positive roots. Then R = R_{+}∪R_{−} with R_{−} the negative roots. Let Σ denote the corresponding set of simple positive roots. The Weyl group W = W (G, T) is then generated by the reflections s_{α} in the hyperplanes defined by the α ∈ Σ. We let ℓ(w) denote the corresponding length function on W: ℓ(w) is the length l of a minimal expression w = s_{1}s_{2} ⋯ s_{l} of w in terms of the simple roots.

Now let X be a projective homogeneous variety. We have X = G/P (μ) for some cocharacter μ : 𝔾_{m} → T which is non-negative on R_{+}. (Any cocharacter can be conjugated to a non-negative one.) Let J = {α ∈ Σ | ⟨α, μ⟩ = 0}. Then W_{μ} is the subgroup of W generated by the s_{α} with α ∈ J. Accordingly, we will also write W_{J} for this subgroup. Now, if G has a non-central cocharacter λ : 𝔾_{m} → T, there is one which is non-negative on Σ. Thus, setting I = {α ∈ Σ | ⟨α, λ⟩ = 0}, we have W_{λ}\W/W_{μ} = W_{I}\W/W_{J}. Note that the correspondence X ⇝ J between isomorphism classes of projective homogeneous varieties for G and subsets J ⊂ Σ is one-to-one and onto. We will call X the homogeneous variety associated to J, and we will call J the set of roots of X.

We now use the result of an exercises in Humphrey’s book on reflection groups [14, Ex. 1 on p. 20].

Exercise 7.1. Any double coset in W_{I}\W/W_{J} has a unique element b of minimal length. The element b satisfies the following equivalent properties:

(1) ℓ(bs_{α}) = ℓ(b) + 1 for α ∈ J and ℓ(s_{α}b) = ℓ(b) + 1 for α ∈ I.

(2) bα > 0 for α ∈ J and b^{−1}α > 0 for α ∈ I.

Moreover, any element w ∈ W may be written as

(7.1)  w = abc

with a ∈ W_{I}, c ∈ W_{J} and ℓ(w) = ℓ(a) + ℓ(b) + ℓ(c).

Solution. The equivalence of the two properties is Lemma 1.6 on p. 12 of [14]. Let b be an element of minimal length in the double coset. Then ℓ(bs_{α}) ≥ ℓ(b) for all α ∈ J, and, since lengths either go up by one or down by one upon multiplying by a reflection, this implies that ℓ(bs_{α}) = ℓ(b) + 1 for all α ∈ J. Similarly, ℓ(s_{α}b) = ℓ(b)+1 for α ∈ I. Thus b satisfies both properties (1) and (2).

Now suppose w ∈ W. Write w = abc with ℓ(a) and ℓ(c) minimal. We can write a, b and c out as reduced words in the simple reflections as follows:

a = r_{1}r_{2} ⋯ r_{l},
b = s_{1}s_{2} ⋯ s_{m},
c = t_{1}t_{2} ⋯ t_{n}.

Since a ∈ W_{I} (resp. c ∈ W_{J}) and the length function on W restricts to that on W_{I} (resp. W_{J}), we can assume that the r′_{i}s are in I and the t′_{i}s are in J. So ℓ(w) ≤ l + m + n and, if ℓ(w) < l + m + n, there must be a pair of reflections in the word for w that can be deleted [14, Theorem 1.7]. This pair cannot involve one of the s′_{i}s because otherwise b would not be minimal in the double coset. On the other hand, it cannot simply involve two of the r′_{i}s because then our word for a would not be reduced. Likewise it cannot simply involve two of the t′_{i}s. Finally, the word cannot involve an r_{i} and a t_{j} because we assumed that ℓ(a) and ℓ(c) were minimal.

Now suppose b′ were another coset representative for W_{I}bW_{J} of minimal length. Then b′ = abc for a and c as above. But, since ℓ(b′) = ℓ(a) + ℓ(b) + ℓ(c), we must have a = c = 1. □

Remarks 7.2. (1) In contrast with the single coset case [14, Proposition 10.10], the elements a and c are not unique. This is clear, for example, if W = ℤ/2 × ℤ/2 and W_{I} = W_{J} is the first factor of ℤ/2. (2) In the special case where I = J, this exercise is used in [9] (see Proposition 3.4). (3) What I have stated as the “exercise” is actually the solution to Humphrey’s question which is whether or not a minimal element exists. A more complete version of the exercise is in Bourbaki [6, Exercise 1.3, p. 37].

For a subset K ⊂ Σ, let R_{K} denote the set of roots generated by K. Let R_{K}^{+} = R_{K} ∩ R^{+} (resp. R_{K}^{−} = R_{K} ∩ R^{−}). We can now give our explicit decomposition in the split case.

Proposition 7.3. Let G be a split reductive k-group with maximal torus T and simple roots Σ. Let X = G/P be a projective homogeneous variety for G with J the corresponding set of simple roots, let λ be a non-central cocharacter of G which is non-negative on Σ and vanishing precisely on I ⊂ Σ and let E be the set of minimal length coset representatives of W_{I}\W/W_{J} with W_{I} and W_{J} as above. Then, under the Białynicki-Birula decomposition for λ,

M(X) = ∐_{w∈E} M(Z_{w})(ℓ(w))

with Z_{w} the orbit of wP under Z(λ). The set I ⊂ Σ is the set of simple roots of Z(λ). Moreover, the roots of Z_{w} are

J_{w} = {α ∈ I | w^{−1}α ∈ R_{J}}.

Proof. First note that, if w ∈ E and α ∈ I, w^{−1}α ∈ R^{+}. This is condition (2) of the exercise. Now the twist a_{w} associated to the motive M(Z(λ)wP) in the motivic decomposition of Corollary 4.4 is the rank of the positive weight space of λ on T (G/P)_{wP}. This is the same as the rank r of the positive weight space of Ad(w^{−1})λ on T (G/P)_{P}. Now T (G/P)_{P} is naturally identified with L(G)/(kR_{J}^{−} ⊕ L(T) + kR^{+}). (Here I write kR for the free vector space on the set R and view it as a subspace of L(G) in the natural way.) It follows that r is the number of negative roots α not in R_{J}^{−} such that ⟨w^{−1}λ, α⟩ = ⟨λ, wα⟩ > 0. Thus

r = #{α ∈ R^{−} − R_{J} | wα ∈ R^{+} − R_{I}}
= #(R^{+} − R_{I}^{+}) ∩ w(R^{−} − R_{J}^{−})
= #(R^{+} ∩ wR^{−} − R^{+} ∩ wR_{J}^{−} − R_{I}^{+} ∩ wR^{−}).

Now R^{+} ∩ wR_{J}^{−} and R_{I}^{+} ∩ wR^{−} are both empty by part (2) of the exercise. So

r = #(R^{+} ∩ wR^{−})
= ℓ(w).

Determining the roots of Z_{w} is now easy. We have

Z_{w} = Z(λ)/(Z(λ) ∩ wP w^{−1}).

Set P_{w} = Z(λ)∩wP w^{−1}. By definition, P_{w} is a parabolic subgroup of Z(λ). Note that P_{w} is actually a standard parabolic subgroup in that it contains the Borel subgroup B_{λ} = B ∩ Z(λ) where B is the standard Borel subgroup such that L(B) = kR^{+}. This follows from the fact that w^{−1}R_{I}^{+} ⊂ R^{+}. Now, L(P) = kR_{+} ⊕ L(T) ⊕ kR_{J}^{−} and L(Z(λ)) = kR_{I}^{+} ⊕ L(T) ⊕ kR_{I}^{−}. Thus, using the exercise, we have

L(P_{w}) = (kR_{I}^{+} ⊕ L(T) ⊕ kR_{I}^{−}) ∩ w(kR_{+} ⊕ L(T) ⊕ kR_{J}^{−})
= kR_{I}^{+} ⊕ L(T) ⊕ (kR_{I}^{−} ∩ wkR_{J}^{−})
= kR_{I}^{+} ⊕ L(T) ⊕ (kR_{I}^{−} ∩ wkR_{J}).

Now it follows that

J_{w} = {α ∈ I | w^{−1}α ∈ R_{J}}.
□

7.1. The Tits Index. To every reductive k-group G, J. Tits has associated a part-algebraic, part-combinatorial object known as the Tits index of G. It consists of the Dynkin diagram for G, a graph with vertices Σ corresponding to the simple roots of a given maximal k-torus, together with a Galois action on the vertices preserving a set Σ_{0} of distinguished vertices. The subset Σ_{0} consists of the simple roots orthogonal to a maximal k-split torus. Thus Σ_{0} is the set of roots of the semi-simple anisotropic kernel. The group Gal(k_{sep}/k) acts on Σ via the ∗-action stabilizing Σ_{0}. When drawing the diagram, the Galois orbits of the vertices not in Σ_{0} are circled and vertices in the same ∗-orbit are supposed to be drawn “close together.”

Now, in this picture, projective homogeneous varieties (i.e., k-defined conjugacy classes of parabolics) are in one-to-one correspondence with ∗- invariant subset J of Σ [26, (2.5.4)]. Isotropic projective homogeneous varieties (i.e., conjugacy classes containing a k-defined parabolic) are in one-to-one correspondence with ∗-invariant subsets J of Σ containing Σ_{0}. Moreover, it is easy to see that for every ∗-invariant subset I of Σ containing Σ_{0}, there is a k-defined cocharacter λ of T which vanishes on I but is positive on Σ − I. In this case, Z(λ) is the Levi component L_{I} of the parabolic subgroup P_{I} associated to I. (The necessary argument is given in the proof of [25, Lemma 15.1.2]).

We now have the following result which follows from the previous proposition by standard methods of descent.

Theorem 7.4. Suppose I and J are ∗-invariant subsets of Σ with I ⊃ Σ_{0}. Let E be the set of minimal length coset representatives for W_{I}\W_{inn}/W_{J}, and let \overline{E} be the set of orbits of E under the ∗-action. Let X be the projective homogeneous variety associated to J. We have

M(X) = ∐_{\overline{w}∈\overline{E}} M(Z\overline{w})(ℓ(w))

where Z\overline{w} is a projective quasi-homogeneous scheme for the reductive group L_{I} (= Z(λ)). Moreover, the base change of Z\overline{w} to k_{sep} is a disjoint union

Z\overline{w} ⊗ k_{sep} = ∐_{w∈\overline{w}} Z_{w}

where Z_{w} is the projective homogeneous variety for L_{I} ⊗ k_{sep} corresponding to the subset

J_{w} = {α ∈ I | w^{−1}α ∈ R_{J}}.

Remark 7.5. If the ∗-orbit \overline{w} consists of one element w, then Z\overline{w} is the projective homogeneous variety corresponding to J_{w}. In particular, its structure as a k-variety is determined by the combinatorics. At any rate, since Z\overline{w} is a projective quasi-homogeneous variety, it is a subvariety of the variety of parabolics of the reductive group L_{I}.

Example 7.6. We now return to the example of (4.7). Here we have G = P SO(q) with q = x_{1}^{2} + ⋯ + x_{2n−2}^{2} + yz (with k = ℝ) and X the projective quasi-homogeneous scheme of two-dimensional subspaces. As long as n ≥ 3, X is a projective homogeneous variety for G. (When n = 2, X has two geometric components.)

Now the Dynkin diagram of G (decorated as in the Tits index) is the following picture.

[figure: α_{1}; α_{2}; •; •; .; .; .; •; α_{n−2}; •; •; α_{n−1}; α_{n}]

The ∗-action exchanges α_{n−1} with α_{n} and leaves all other roots fixed. When n ≥ 4, the set J ⊂ Σ corresponding to X is Σ − {α_{2}}. For n = 3, J = Σ−{α_{2}, α_{3}}. We assume n ≥ 4 at first and sketch the case where n = 3 (where the ∗-action plays a significant role) at the end of this example.

If we set I = Σ_{0} = Σ − {α_{1}}, then we are in the setting of Theorem 7.4. Write s_{i} = s_{α_{i}} for the generators of the Weyl group. From the theory of Coxeter complexes ([14, § 1.15]), we see that W_{I}\W is identified with the set of vectors of the form ±e_{i} in the real vector space ℝ^{n} = ℝe_{1} + ⋯ + ℝe_{n}. The action of W on the right is given by e_{i}s_{j} = e_{(j,j+1)i} for 1 ≤ j < n where (j, j +1) denotes the transposition in the symmetric group exchanging j and j + 1. For j = n, we have e_{i}s_{n} = −e_{(n−1,n)i}. Now it is fairly easy to see that the cosets in W_{I}\W containing elements of E are e_{1} = W_{I}1, e_{3} = W_{I}s_{1}s_{2}, and −e_{2} = W_{I}s_{1}s_{2} ⋯ s_{n−2}s_{n−1}s_{n}s_{n−2} ⋯ s_{3}s_{2}. It is also easy to see that the representatives listed are in fact the elements of E. That is,

E = {1, s_{1}s_{2}, s_{1}s_{2} ⋯ s_{n−2}s_{n−1}s_{n}s_{n−2} ⋯ s_{2}}.

Writing w_{1}, w_{2} and w_{3} for the elements listed in order, we have

ℓ(w_{1}) = 0, ℓ(w_{2}) = 2, ℓ(w_{3}) = 2n − 3.

Clearly J_{w_{1}} = I ∩ J = {α_{3}, …, α_{n}}. To compute J_{w_{2}}, note that

w_{2}^{−1}α_{2} = s_{2}s_{1}α_{2}
= s_{2}(α_{1} + α_{2})
= (α_{1} + α_{2}) − α_{2}
= α_{1} ∈ J.

Thus α_{2} ∈ J_{w_{2}}. A similar computation shows that, for i > 2, w_{2}^{−1}α_{i} ∈ J_{w_{2}} if and only if α_{i} is not connected to α_{2} in the Dynkin diagram. Thus, for n ≥ 5, J_{w_{2}} = {α_{2}, α_{4}, …, α_{n}}. However, for n = 4, J_{w_{2}} = {α_{2}}.

Finally, to compute J_{w_{3}}, note that w_{3}^{−1}α_{i} = α_{i} for 2 < i ≤ n−2, w_{3}^{−1}α_{2} = α_{1} + α_{2}, w_{3}^{−1}α_{n−1} = α_{n} and w_{3}^{−1}α_{n} = α_{n−1}. It follows that J_{w_{3}} = J_{w_{1}} = I − {α_{2}}.

Putting all of this together, we have the decomposition

(7.2)  M(X) = M(Q) ⊕ M(Y)(2) ⊕ M(Q)(2n − 3)

where Q is the motive of a quadric of isotropic lines for the quadratic form q′ = x_{1}^{2} + ⋯ + x_{2n}^{2}, and Y is isomorphic to the space of isotropic planes for q′.

When n = 3, W_{I}\W (viewed in terms of the Coxeter complex) has four W_{J} orbits containing the minimal W_{I}-cosets

e_{1} = W_{I}w_{1}, w_{1} = 1;
e_{3} = W_{I}w_{2}, w_{2} = s_{1}s_{2};
−e_{3} = W_{I}w_{3}, w_{3} = s_{1}s_{3};
−e_{2} = W_{I}w_{r}, w_{4} = s_{1}s_{2}s_{3}.

The elements w_{i} are written in reduced form so we have ℓ(w_{1}) = 1, ℓ(w_{2}) = ℓ(w_{3}) = 2 and ℓ(w_{4}) = 3. Note that w_{2} and w_{3} are conjugate under the ∗-action. Clearly J_{w_{1}} = ∅ and some computation shows that J_{w_{4}} = ∅ as well. On the other hand, J_{w_{2}} = α_{2} while J_{w_{3}} = α_{3}.

It turns out then that Z\overline{w_{1}} = Z\overline{w_{4}} = Q and Z\overline{w_{3}} = Y with Q and Y as in 7.2. In other words, we obtain the same decomposition as in the case n = 4. However, we learn that Y ⊗ ℂ = Z_{w_{2}} ∐ Z_{w_{3}} with Z_{w_{2}} = Z_{w_{3}} = ℙ^{1}.

It is also possible (and perhaps easier) to work out (7.2) directly using the geometry of the 𝔾_{m}-action on X and the weight decomposition of T X at the various fixed loci as suggested in Exercise 4.7.

8. Acknowledgments

It is a pleasure to thank Chernousov, Gille, and Merkurjev as well as P. Belkale, N. Fakhruddin and O. Gabber for extremely helpful discussions and H. Furdyna and B. Brosnan for pointing out several typos. I would also like to thank B. Shipman for sending me a reprint of her paper [24] describing the connected components of (G/B)^{T} which helped me greatly to guess Theorem 6.1, J. Carrell for sending me useful references to his paper “Torus actions and cohomology” in [4] and B. Köck for telling me about del Baño’s work.

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