WITT GROUPS OF ALGEBRAIC GROUPS

NOBUAKI YAGITA

Abstract. Let G_{k} be a split reductive group over a field k of ch(k) = 0 corresponding to a simply connected Lie group G. Let T be a maximal torus of G. When k is an algebraically closed, the Balmer’s Witt group W^{∗}(G_{k}) is isomorphic to KO^{2∗−1}(G/T) but not to KO^{2∗−1}(G).

1. Introduction

Let X be a (quasi projective) smooth variety over a field k with k ⊂ ℂ. The Witt group W (X) is the quotient of the Grothendieck group of vector bundles with quadratic forms over X, by the subgroup generated by bundles V with quadratic forms which admit Lagrangian subbundles E. The generalized Witt group W^{∗}(X; L) is defined by Balmer [Ba] for ∗ ∈ ℤ/4 and for a line bundle L on X so that W^{0}(X; O_{X}) = W^{0}(X) = W (X). (Moreover if L = L′ in P ic(X)/2, there is an isomorphism W^{∗}(X; L) ≅ W^{∗}(W; L′).) We can define a natural map [Ya3], [Zi]

q^{∗} : W^{∗}(X) → KO^{2∗}(X(ℂ))/KU^{2∗}(X(ℂ)) →^{×η} KO^{2∗−1}(X(ℂ))

where KO^{∗}(−) and KU^{∗}(−) are (toplogical) real and complex K-theories, and 0 ≠ η ∈ KO^{−1}(pt.) ≅ ℤ/2.

Let G be a compact connected Lie group and T a maximal torus of G and B be the Borel subroup with T ⊂ B. Let us denote by G_{k} (resp. T_{k}, B_{k}) the split reductive group (resp. its maximal torus, the Borel subgroup) over k which corresponds G (resp. T, B). Let T_{k}^{1} ⊂ ... ⊂ T_{k}^{ℓ} = T_{k} be a sequence of tori of G_{k} where T_{k}^{i} ≅ (𝔸^{1} − {0})^{×i}. Recently Calmès and Fasel [Ca-Fa] proved that W^{∗}(G_{k}/B_{k}; L) = 0 whenever 0 ≠ L ∈ P ic(G_{k}/T_{k})/2. (Note that W^{∗}(G_{k}/B_{k}; L) ≅ W^{∗}(G_{k}/T_{k}; L) for all L.) There is a localization exact sequence in the Witt theory

→^{δ} W^{∗−1}(G_{k}/T_{k}^{i}; t_{i}) →^{j_{∗}} W^{∗}(G_{k}/T_{k}^{i}) → W^{∗}(G_{k}/T_{k}^{i−1}) →^{δ} ...

where (t_{1}, ..., t_{n}) is a basis of P ic(G/T)/2. Here we see W^{∗−1}(G_{k}/T_{k}^{i}; t_{i}) = 0 from the result by Calmès and Fasel. By induction on i, we easily prove

Theorem 1.1. If rank_{2}P ic(G_{k}/T_{k})/2 = rank(T_{k}) = ℓ (e.g., G is a simply connected and k is algebraically closed), then W^{∗}(G_{k}) ≅ W^{∗}(G_{k}/T_{k}).

Now let k be algebraically closed with ch(k) = 0. Also recently, Zibrowius showed [Zi] that the above map q^{∗} is always an isomorphism for each cellular variety X when k = ℂ, and it can be generalized to any algebraically closed field. Of course, the flag variety G_{k}/B_{k} is cellular. Hence we have the isomorphism

q^{∗} : W^{∗}(G_{k}) ≅ KO^{2∗}(G/T)/KU^{2∗}(G/T) ≅ KO^{2∗−1}(G/T).

2000 Mathematics Subject Classification. Primary 19G12, 55N15; Secondary 55N20.

Key words and phrases. algebraic group, Witt group, K-theory .

Let us consider the Atiyah-Hirzebruch spectral sequence converges to KO^{∗}(G/T). It is known that the differential d_{2} is the Steenrod squareing Sq^{2} mod(2). Kishimoto, Kono and Ohsita [Ki-Ko-Oh], [Ki-Oh], [Oh] computed this Atiyah-Hirzebruch spectral sequence and showed that it collapses from the E_{3}^{∗,∗′}-term for simply connected simple groups except for E_{7} and E_{8}. Moreover the above KO^{2∗−1}(G/T) is isomorphic to

H^{∗}(G/T; Sq^{2}) = H^{∗}(H^{2∗}(G/T; ℤ/2); Sq^{2})

for these groups.

On the other hand, there are spectral sequences E(P)_{r}^{∗,∗′} ( by Gille and Pardon [Gi], [Pa]) and E(BW)_{r}^{∗,∗′} (by Balmer-Walter [Ba-Wa]) such that E(P)_{r}^{∗,∗′} converges to grE(BW)_{2}^{∗,∗′} and E(BW)_{r}^{∗,∗′} converges to W^{∗}(G_{k}).

By the Borel theorem, the cohomolgy H^{∗}(G; ℤ/2) is isomorphic to a tensor algebra of a polynomial algebra generated by even dimensional elements y_{i} and an exterior algebra generated by x_{j} of odd dimensional. We consider the graded algebra grH^{∗}(G; ℤ/2) defined by the filtration w(y_{i}) = 0 and w(x_{j}) = 1. Then Sq^{2} acts on grH^{∗}(G; ℤ/2) as a differential. Let us write its homology by

H^{∗}(G; Sq^{2}) = H^{∗}(grH^{∗}(G; ℤ/2); Sq^{2}).

Then we can prove that E(P)_{3}^{∗,∗′} ≅ H^{∗}(G; Sq^{2}) with

deg(y_{i}) = (1/2|y_{i}|, 1/2|y_{i}|) and deg(x_{j}) = (1/2(|x_{j}| − 1), 1/2(|x_{j}| + 1)).

In fact, by Totaro [To], it is known that d_{2} = Sq^{2} on E(P)_{2}^{∗,∗′} when ∗ = ∗′. Moreover we will see

Theorem 1.2. Let k be an algebraically closed field in ℂ. For each simply connected simple Lie group G, we have isomorphisms

W^{∗}(G_{k}) ≅ H^{∗}(G/T; Sq^{2}) ≅ Λ(z_{1}, ..., z_{s}) deg(z_{i}) = odd,

which is also isomorphic to

\begin{cases}E(P)_{4}^{∗,∗′} ≅ H^{∗}(H^{∗}(G; Sq^{2}); d_{3}) for G = E_{6}, E_{7}, E_{8}, \\ E(P)_{3}^{∗,∗′} ≅ H^{∗}(G; Sq^{2}) otherwise. \end{cases}

The explicit value of deg(z_{i}) is given in §7 below.

I would like to thank Burt Totaro and Marcus Zibrouius who corrected many errors in the first version of this paper.

2. the total Witt group

We can consider the generalized Witt group W^{i}(X; L) for i ∈ ℤ/4 and for a line bundle L on X such that the usual Witt group W^{i}(X) = W^{i}(X; O_{X}) (see [Ba], [Ba-Ca2], [Zi] for example). We also note if L = L′ mod(2), then we have a (non canonical) isomorphism W^{∗}(X; L) ≅ W^{∗}(X; L′). Let us write the total Witt group

W^{total}(X) = ⊕_{i∈ℤ/4, L∈P ic(X)/2}W^{i}(X; L).

The Gysin and the boundary maps are defined as the maps for W^{total}(−). (Gysin maps are constructed by Calmès and Hornbostel in [Ca-Ho], also see §4.1 in [Ne].) Let g : Z ⊂ X be a regular embedding of codim = c, and U = X − Z. Let ω_{g} be the relative canonical bundle (for the definition see [Ba-Ca1]). Then we have the natural exact sequence ((11) in [Ba-Ca1])

→ W^{∗−c}(Z, ω_{g} ⊗ L|_{Z}) →^{g_{∗}} W^{∗}(X, L) →^{f^{∗}} W^{∗}(U, L|_{U}) →^{δ} ...

Here we note that the Witt group with the coefficient L is written by using Thom space T h(L) of the bundle L ([Ne], [Zi]), namely, W^{∗+1}(X; L) ≅ W̃^{∗}(T h(L)) where W̃^{∗}(−) is the reduced theory. Then we also have for a vector bundle V on X

W^{∗+c}(X; det(V)) ≅ W̃^{∗}(T h(V)) for dim(V) = c.

Let G_{k} be a split reductive group over a field k of ch(k) = 0 corresponding to a compact Lie group G. Let T_{k} be a maximal (split) torus of G_{k}. We consider the sequence of tori

T_{k}^{1} ⊂ T_{k}^{2} ⊂ ... ⊂ T_{k}^{ℓ} = T_{k} where T_{k}^{i} ≅ (𝔸^{1} − {0})^{×i} ≅ (𝔾_{m})^{×i}.

Here we assume that rank_{ℤ/2}(P ic(G_{k}/T_{k})/2) = rank(T) = ℓ and

ℤ/2{t_{1}, ..., t_{ℓ}} ≅ P ic(G_{k}/T_{k})/2.

We consider the 𝔾_{m}-bundle

𝔾_{m} → G/T_{k}^{i−1} → G/T_{k}^{i}

where we take the base t_{i} corresponding the above 𝔾_{m}-bundle. Let E(G_{k}/T_{k}^{i}) be the line bundle corresponding to the above 𝔾_{m}-bundle so that G/T_{k}^{i−1} is an open subset of E(G_{k}/T_{k}^{i}). Then we have maps

Z = G_{k}/T_{k}^{i} ⊂^{g} X = E(G_{k}/T_{k}^{i}) ⊃^{f} U = X − Z = G_{k}/T_{k}^{i−1}.

Thus we have its localization exact sequence

→ W^{∗−1}(G_{k}/T_{k}^{i}; t_{i} + L) →^{g_{∗}} W^{∗}(G_{k}/T_{k}^{i}; L) →^{f^{∗}} W^{∗}(G_{k}/T_{k}^{i−1}, L) →^{δ} ...

Let B_{k} be the Borel subgroup of G_{k}. Then we have the fibering

U_{k} → G_{k}/T_{k} → G_{k}/B_{k}

for the unipotent group U_{k}. Hence we have an isomorphism W^{∗}(G_{k}/T_{k}; L) ≅ W^{∗}(G_{k}/B_{k}; L). We recall the result Calmès and Fasel

Lemma 2.1. ([Ca-Fa]) If L ≠ 0 ∈ P ic(G_{k}/T_{k})/2, then

W^{∗}(G_{k}/T_{k}; L) ≅ W^{∗}(G_{k}/B_{k}; L) = 0.

For an algebraically closed field k, we will give a topological proof of this fact in §4 below.

Lemma 2.2. Let L_{i} = e_{1}t_{1} + ... + e_{i}t_{i} ≠ 0 ∈ P ic(G_{k}/T_{k})/2, e_{i} = 0 or 1. Then we have W^{∗}(G_{k}/T_{k}^{i}; L_{i}) = 0 for 1 ≤ i ≤ ℓ.

Proof. Consider the localization exact sequence

→ W^{∗−1}(G_{k}/T_{k}^{ℓ}; t_{ℓ} + L_{ℓ−1}) →^{g_{∗}} W^{∗}(G_{k}/T_{k}^{ℓ}; L_{ℓ−1}) →^{f^{∗}} W^{∗}(G_{k}/T_{k}^{ℓ−1}, L_{ℓ−1}) →^{δ} ...

Since the first and the second term are zero from the result by Calmès and Fasel, the third term W^{∗}(G_{k}/T_{k}^{ℓ−1}, L_{ℓ−1}) = 0.

Next consider the localization exact sequence

→ W^{∗−1}(G_{k}/T_{k}^{i}; t_{i} + L_{i−1}) →^{g_{∗}} W^{∗}(G_{k}/T_{k}^{i}; L_{i−1}) →^{f^{∗}} W^{∗}(G_{k}/T_{k}^{i−1}, L_{i−1}) →^{δ} ...

By induction, we assume the first and the second term are zero. Then we have W^{∗}(G_{k}/T_{k}^{i−1}, L_{i−1}) = 0. □

Theorem 2.3. If rank_{2}P ic(G_{k}/T_{k})/2 = rank(T) = ℓ (e.g., G is a simply connected and k is algebraically closed), then W^{∗}(G_{k}) ≅ W^{∗}(G_{k}/T_{k}).

Proof. We consider the localization exact sequence

→ W^{∗−1}(G_{k}/T_{k}^{i}; t_{i}) →^{g_{∗}} W^{∗}(G_{k}/T_{k}^{i}) →^{f^{∗}} W^{∗}(G_{k}/T_{k}^{i−1}) →^{δ} ...

Since the first term is zero from the preceding lemma, we have an isomorphism

W^{∗}(G_{k}/T_{k}^{i}) ≅ W^{∗}(G_{k}/T_{k}^{i−1}).

When rank_{2}(P ic(G/T)/2) = ℓ, then the above isomorphism holds for i = 1 also. □

Next consider the case rank_{2}(P ic(G_{k}/T_{k})/2) + 1 = ℓ and ℤ/2{t_{2}, ..., t_{ℓ}} ≅ P ic(G_{k}/T_{k})/2. Letting L_{i} = e_{2}t_{2} + ... + e_{i}t_{i}, the arguments above also work and we have

Corollary 2.4. If rank_{2}(P ic(G_{k}/T_{k})/2) + 1 = ℓ, then we have an isomorphism

W^{∗}(G_{k}) ≅ W^{∗}(G_{k}/T_{k}) ⊗ Λ(x_{0}), |x_{0}| = 0.

Proof. We consider the localization exact sequence

→ W^{∗−1}(G_{k}/T_{k}^{1}; t_{1}) →^{g_{∗}} W^{∗}(G_{k}/T_{k}^{1}) →^{f^{∗}} W^{∗}(G_{k}) →^{δ} ...

where t_{1} ∈ L_{ℓ}. But P ic(G_{k}/T_{k}^{1}) = 0 (compare the results in §3 e.g. Theorem 3.4). So the normal bundle for G_{k}/T_{k}^{1} ⊂ E(G_{k}/T_{k}^{1}) is trivial. Hence the Gysin map g_{∗} = 0. So we have

W^{∗}(G_{k}/T_{k}^{1}; t_{1}) ≅ W^{∗}(G_{k}/T_{k}^{1}) ≅ W^{∗}(G_{k}/T_{k}).

Hence we get the desired result. □

3. cohomology theories of compact Lie group G

Let G be a compact connected Lie group. By the Borel theorem, we have the ring isomorphism

grH^{∗}(G; ℤ/2) ≅ P (y) ⊗ Λ(x_{1}, ..., x_{l}) with P (2) = ⊗_{i}ℤ/2[y_{i}]/(y_{i}^{2^{r_{i}}})

where |y_{i}| = even and |x_{j}| = odd. Moreover for each y_{i}, there is x_{j} with x_{j}^{2} = y_{i}. (In fact, H^{∗}(G; ℤ/2) is multiplicatively generated by odd dimensional elements, e.g., see [Ka].)

Let T be a maximal torus of G and BT the classifying space of T. We consider the fibering G →^{π} G/T →^{i} BT and the induced spectral sequence

E_{2}^{∗,∗′} = H^{∗}(BT; H^{∗′}(G; ℤ/2)) ⟹ H^{∗}(G/T; ℤ/2).

The cohomology of the classifying space of the torus is given by

H^{∗}(BT; ℤ/2) ≅ S(t) = ℤ/2[t_{1}, ..., t_{ℓ}] with |t_{i}| = 2,

where ℓ is also the number of the odd degree generators x_{i} in H^{∗}(G; ℤ/2). It is known that y_{i} are permanent cycles and that there is a regular sequence ([Tod],[Mi-Ni]) (b_{1}, ..., b_{ℓ}) in H^{∗}(BT; ℤ/2) such that d_{|x_{i}|+1}(x_{i}) = b_{i}. Thus we get

E_{∞}^{∗,∗′} ≅ grH^{∗}(G/T; ℤ/2) ≅ P (y) ⊗ S(t)/(b_{1}, ..., b_{ℓ}).

Since H^{∗}(G/T; ℤ) is no torsion, we get the isomorphism

H^{∗}(G/T; ℤ) ≅ ℤ[y_{1}, ..., y_{k}, t_{1}, ..., t_{ℓ}]/(f_{1}, ..., f_{k}, b̄_{1}, ..., b̄_{ℓ})

where f_{i} = y_{i}^{2^{r_{i}}} mod(Ideal(t_{1}, ..., t_{ℓ})) and b̄_{j} = b_{j} mod(2).

In particular, if |y| > 2 and |b_{1}| > 2, then P ic(G_{k}/T_{k}) ≅ ℤ/2{t_{1}, ..., t_{ℓ}} and

dim_{2}(P ic(G_{k}/T_{k})/2) = ℓ = dim(T).

Let T^{1} ⊂ ... ⊂ T^{ℓ} = T be a sequence of tori of G where T^{i} ≅ (S^{1})^{×i}. The fibering S^{1} → G/T^{i−1} → G/T^{i} induces the Gysin exact sequence

→^{δ} H^{∗−2}(G/T^{i}; ℤ/2) →^{j_{∗}=×t_{i}} H^{∗}(G/T^{i}; ℤ/2) → H^{∗}(G/T^{i−1}; ℤ/2) →^{δ} ...

From this arguments, we can compute H^{∗}(G/T^{i}; ℤ/2) from H^{∗}(G/T; ℤ/2).

Lemma 3.1. We can take tori T^{i} such that b_{1}, ..., b_{i} is regular in S(t)/(t_{i+1}, ..., t_{ℓ}), and that b_{i} = t_{i}g_{i} in S(t)/(b_{1}, ..., b_{i−1}, t_{i+1}, ..., t_{ℓ}) for some g_{i} ∈ S(t).

Proof. In §4 in [Ya2], we see that the above assumption holds for each simply connected simple Lie group G (when p = 2 in Lemma 2.1 in [Ya2]) except for G = Sp(n), E_{8}. (Note H^{∗}(Spin(n)/T; ℤ/2) ≅ H^{∗}(SO(n)/T; ℤ/2) see §3 in [Tod].) The case G = Sp(n) is almost immediate since

H^{∗}(G/T; ℤ/2) ≅ ℤ/2[t_{1}, ..., t_{n}]/(c_{1}^{2}, ..., c_{n}^{2}).

where c_{i} is the i-th elementary symmetric function on t_{1}, ..., t_{n}. In fact c_{1}^{2}, ..., c_{n}^{2} is regular and c_{i}^{2} = g_{i}t_{i} mod(t_{i+1}, ..., t_{n}).

For the case G = E_{8}, we use the result by Kono-Ishitoya, Ohsita [Ko-Is], [Oh]. It is known that grH^{∗}(G/T; ℤ/2)/(P (y)) is isomorphic to

S(t)/(b_{1}, ..., b_{ℓ=8}) ≅ S(t)/(c_{2}, c_{3}, c′_{5}, c′_{9}, I_{8}, I_{12}, I_{14}, I_{15})

in the notation in [Ko-Is] (see [Ko-Is] for details). Here c′_{9} = c_{1}(c_{8} + c_{7}c_{1} + c_{0}c_{1}^{2}) and we think b_{ℓ} = c′_{9} and t_{ℓ} = c_{1}. Next we see I_{15} = c_{8}c_{7} mod(c_{1}), and think t_{ℓ−1} = t_{8}. We see that

I_{14} = c_{7}^{2}, I_{12} = c_{6}^{2}, c′_{5} = c_{5}, I_{8} = c_{4}^{2} mod(c_{1}, c_{8}).

This shows the lemma for G = E_{8}. □

Using this lemma, we can prove the following Corollary 3.2, Theorem 3.4 and Corollary 3.5 (see also [Ya2]).

Corollary 3.2. (Lemma 2.1 in[Ya2]) We have an isomorphism

grH^{∗}(G/T^{i}; ℤ/2) ≅ H^{∗}(G/T)/(t_{i+1}, ..., t_{ℓ}) ⊗ Λ(x_{i+1}, ..., x_{ℓ}).

Examples. When G = U(n), we still know that P (y) ≅ ℤ/2 and

H^{∗}(G; ℤ/2) ≅ Λ(x_{1}, x_{3}, x_{5}, ..., x_{2n−1}) with |x_{j}| = j,
H^{∗}(G/T; ℤ/2) ≅ ℤ/2[t_{1}, ..., t_{n}]/(c_{1}, c_{2}, ..., c_{n}).

Of course b_{i} = c_{i} is regular and satisfies (by g_{i} = t_{1}...t_{i−1}) the lemma above

H^{∗}(G/T^{i}; ℤ/2) ≅ ℤ/2[t_{1}, ..., t_{i}]/(c_{1}, ..., c_{i}) ⊗ Λ(x_{2i−1}, ..., x_{2n−1}).

Let X be an algebraic variety over k. Let H^{∗,∗′}(X; ℤ/2) be the mod(2) motivic cohomology constructed by Suslin and Voevodsky [Vo1]. For nonzero element x ∈ H^{m,n}(X; ℤ/2), we define the weight degree and the different degree by

w(x) = 2n − m, d(x) = m − n.

When X is smooth, it is known that w(x) ≥ 0, d(x) ≤ dim(X) for 0 ≠ x. (For example see Corollary 2.3 in [Vo1].) Moreover from the affirmative answer by Voevodsky to the Milnor conjecture (and hence the Beilinson-Lichtenbaum conjecture) implies

H^{∗,∗′}(pt.; ℤ/2) ≅ ℤ/2[τ] ⊗ K_{∗}^{M}(k)

where 0 ≠ τ ∈ H^{0,1}(pt.; ℤ/2) ≅ ℤ/2 and the mod (2) Milnor K-theory is K_{∗}^{M}(k)/2 ≅ H^{∗,∗}(pt.; ℤ/2).

Let us denote by G_{k} the split reductive group over k corresponding to the compact Lie group G and T_{k}^{i} the spit torus. Since G_{k}/B_{k} is cellular and H^{∗,∗′}(G_{k}/B_{k}; ℤ/2) ≅ H^{∗,∗′}(G_{k}/T_{k}; ℤ/2), we have the isomorphism

H^{∗,∗′}(G_{k}/T_{k}; ℤ/2) ≅ K_{∗}^{M}(k)/2 ⊗ H^{∗,∗′}(G_{ℂ}/T_{ℂ}; ℤ/2)
≅ H^{∗,∗′}(pt.; ℤ/2) ⊗ H^{∗}(G/T; ℤ/2) with w(H^{∗}(G/T; ℤ/2)) = 0.

In particular, we note that the realization map t_{ℂ} : G_{k}/T_{k} → G_{ℂ}/T_{ℂ} induces an isomorphism P ic(G_{k}/T_{k})/2 ≅ P ic(G_{ℂ}/T_{ℂ})/2.

For the motivic theory, we have also the Thom isomorphism and hence the Gysin exact sequence

→^{δ} H^{∗−2,∗′−1}(G_{k}/T_{k}^{i}; ℤ/2) →^{×t_{i}} H^{∗,∗′}(G_{k}/T_{k}^{i}; ℤ/2) → H^{∗,∗′}(G_{k}/T_{k}^{i−1}; ℤ/2) →^{δ} ...

Since H^{2∗+1,∗}(G_{k}/T_{k}^{i}; ℤ/2) = 0 (the weight degree< 0), we have

H^{2∗−2,∗−1}(G_{k}/T_{k}^{i}; ℤ/2) →^{×t_{i}} H^{2∗,∗}(G_{k}/T_{k}^{i}; ℤ/2) → H^{2∗,∗}(G_{k}/T_{k}^{i−1}; ℤ/2) →^{δ} 0.

By descending induction on i, we easily show

H^{2∗,∗}(G_{k}/T^{i}; ℤ/2) ≅ H^{2∗}(G/T; ℤ/2)/(t_{i+1}, ..., t_{ℓ})

and there is x_{i} ∈ H^{2∗−1,∗}(G_{k}/T^{i−1}; ℤ/2) with δ(x_{i}) = b_{i}. Moreover, we have the following theorems

Theorem 3.3. (Theorem 3.1 in [Ya2]) There is an H^{∗}(pt.; ℤ/2)-module isomorphism

H^{∗,∗′}(G_{k}/T_{k}^{i}; ℤ/2) ≅ H^{∗}(G/T^{i}; ℤ/2) ⊗ H^{∗,∗′}(pt.; ℤ/2)

where the bidegree in H^{∗}(G/T^{i}; ℤ/2) is given for nonzero element u ∈ H^{∗}(G/T; ℤ/2) by w(u) = 0 and w(x_{i}) = 1.

Corollary 3.4. (Corollary 3.2 in [Ya2]) Define a filtration F′_{i} to be the H^{∗}(pt.; ℤ/2)- module generated by elements x with w(x) ≤ i. Then we have

grH^{∗,∗′}(G_{k}; ℤ/2) = ⊕F′_{i}/F′_{i−1} ≅ H^{∗,∗′}(pt.; ℤ/2) ⊗ P (y) ⊗ Λ(x_{1}, ..., x_{ℓ})

where w(P (y)) = 0 and w(x_{i}) = 1.

Proof. Since t_{ℂ} : H^{2∗,∗}(G_{k}; ℤ/2) → H^{2∗}(G; ℤ/2) is an isomorphism, we have y_{i}^{2^{r_{i}}} = 0 ∈ H^{2∗,∗}(G_{k}; ℤ/2). Let x_{i}^{2} = y_{j}. Since w(x_{i}^{2}) = 2 and t_{ℂ}(τ) = 1 and K_{+}^{M}(k) ⊂ Ker(t_{ℂ}), we see

x_{i}^{2} = τy_{j} mod(Ideal(K_{+}^{M}(k)/2)).
□

Theorem 3.5. (Theorem 3.3 in [Ya2]) Suppose that H^{∗,∗′}(X; ℤ/2) is ℤ/2[τ]-free. Then

H^{∗,∗′}(X × G_{k}/T_{k}^{i}; ℤ/2) ≅ H^{∗,∗′}(X; ℤ/2) ⊗_{H^{∗.∗′} (pt.;ℤ/2)} H^{∗,∗′}(G_{k}/T_{k}^{i}; ℤ/2).

Corollary 3.6. (Corollary 3.4 in [Ya2]) By the assumption in the preceding theorem, the Künneth formula holds for H^{∗,∗′}(G_{k}; ℤ/2). Hence it is a Hopf algebra.

4. KO-theory

We explain the KO-theory of flag manifolds G/T according to Hara, Kishimoto, Kono and Ohsita [Ha], [Ki-Ko-Oh]. Recall that the coefficient rings of the (topological) KO^{∗}-theory and KU^{∗}-theory are ( e.g., see §1 in [Ha])

KO^{∗} ≅ ℤ[μ, μ^{−1}, η, w]/(2η, η^{3}, w^{2} − 4μ, ηw),
KU^{∗} ≅ ℤ[β, β^{−1}]

with |μ| = −8, |w| = −4, |η| = −1 and |β| = −2. To compute KO^{∗}(G/T), we consider the Atiyah-Hirzebruch spectral sequence

E_{2}^{∗,∗′} ≅ H^{∗}(G/T; KO^{∗′}) ⟹ KO^{∗+∗′}(G/T).

It is well known that the first differential is ((3.1) in [Ha], [Fuj])

d_{2}(λ ⊗ x) = λη ⊗ Sq^{2}(x̄), λ ∈ KO^{∗}

where x̄ ∈ H^{∗}(G/T; ℤ/2) is the mod 2 reduction of x.

Note Sq^{2}Sq^{2} = Sq^{3}Sq^{1} from the Adem relation. So Sq^{2}Sq^{2}(x) = 0 in H^{∗}(G/T; ℤ/2) (in fact H^{odd}(G/T; ℤ/2) = 0). Let us write

H^{∗}G/T; Sq^{2}) = H(H^{2∗}(G/T; ℤ/2); Sq^{2})

the homology with the differential Sq^{2}. To compatible with W^{∗}(−), we define degree of element x ∈ H^{∗}(G/T; Sq^{2}) by half of that in H^{∗}(G/T; ℤ/2), i.e., deg(x) = 1/2|x|. Hence we have

E_{3}^{2∗,odd} ≅ E_{3}^{2∗,8∗′−1} ≅ ℤ/2{η}[μ, μ^{−1}] ⊗ H^{∗}(G/T; Sq^{2}).

Hence E_{∞}^{2∗,odd} is isomorphic to a subquotient of H^{∗}(G/T; Sq^{2}).

Kishimoto, Kono and Ohsita [Ki-Ko-Oh], [Ki-Oh] get this homology for G = U(n), Sp(n), O(n), G_{2}, F_{4}, E_{6}. For example

H^{∗}(U(2m + 1)/T; Sq^{2}) ≅ Λ(z_{3}, z_{7}, ..., z_{4m−1})

where z_{4s−1} = ∑_{i_{1}<...<i_{s}} t_{i_{1}}t_{i_{2}}^{2}...t_{i_{s}}^{2} in H^{∗}(G/T; ℤ/2), (in fact Sq^{2}(z_{4s−1}) = 0).

Since deg(d_{r}) = (r + 1, −r), we see the Atiyah-Hirzebruch spectral sequence collapses form the E_{3}^{∗,∗′}-term for the case G = U(2m + 1). The same fact happens for other groups, e.g., Kishimoto, Kono and Ohsita proved that the following assumption is satisfied for all above groups G.

Assumption 4.1. The Atiyah-Hirzebruch spectral sequence for KO^{∗}(G/T) collapses from the E_{3}^{∗,∗′}-term.

Let Y be a topological space (e.g., a finite dimensional CW -complex). We have the following well known (Bott) exact sequence ((1.1) in [Ha], (3.4) in [At])

(1) → KO^{∗+1}(Y) →^{×η} KO^{∗}(Y) →^{c} KU^{∗}(Y) →^{r·β^{−1}} KO^{∗+2}(Y) → ...

where c is the complexification map and r is the real restriction map. The map β^{−1} : KU^{∗}(Y) → KU^{∗+2}(Y) is an isomorphism. Let us write

KO^{∗}(Y)/KU^{∗}(Y) = KO^{∗}(Y)/(rKU^{∗}(Y)) = KO^{∗}(Y)/(rβ^{−1}KU^{∗}(Y)).

Then we have

KO^{∗}(Y)/KU^{∗}(Y) ≅ KO^{∗}(Y)/(Kerη) ≅ Im(η)(KO^{∗}(Y))

(which is 2-torsion since so is η), e.g., KO^{∗}/KU^{∗} ≅ ℤ/2{1, η}[μ, μ^{−1}]. Hereafter, in this paper, let us write

KO^{2∗}(Y)/KU^{2∗}(Y) = ⊕_{r∈ℤ/4}KO^{2r}(Y)/KU^{2r}(Y),

(while KO^{∗}(Y) means usually ⊕_{r∈ℤ}KO^{r}(Y)). Taking the degree modulo 4 in H^{∗}(G/T; Sq^{2}), we have a ring isomorphism ;

Corollary 4.2. The graded ring KO^{2∗}(G/T)/KU^{2∗}(G/T) is isomorphic to a subquotient of H^{∗}(G/T; Sq^{2}). Moreover if Assumption 4.1 is satisfied, then

KO^{2∗}(G/T)/KU^{2∗}(G/T) ≅ H^{∗}(G/T; Sq^{2}).

By Künneth theorem, we have

H^{∗}(X × Y; Sq^{2}) ≅ H^{∗}(X; Sq^{2}) ⊗ H^{∗}(Y; Sq^{2}).

Hereafter this section, we assume that k is algebraically closed. When X is cellular variety over ℂ, Zibrouwius shows that W^{∗}(X) ≅ KO^{2∗}(X(ℂ))/KU^{2∗}(X(ℂ)). (For the another proof, see §5 below.) This fact is easily extended to any algebraically closed field k ∈ ℂ. Hence we get

Corollary 4.3. If Assumption 4.1 holds, then KO^{∗}(G/T)/KU^{∗}(G/T) has the Künneth formula and (from Lemma 2.5)

W^{∗}(G_{k}) ≅ H^{∗}(G/T; Sq^{2})

is a Hopf algebra (for an algebraically closed field k).

Example. Let G = SU(3). Then H^{∗}(G; ℤ/2) ≅ Λ(x_{3}, x_{5}) and

H^{∗}(G/T; ℤ/2) ≅ ℤ/2[t_{1}, t_{2}, t_{3}]/(c′_{1}, c′_{2}, c′_{3}) ≅ ℤ/2[t_{1}, t_{2}]/(c_{1}^{2} + c_{2}, c_{1}c_{2})

where c′_{i} (resp. c_{i}) is the i-th elementary symmetric function of 3-variables (resp. 2-variables). Hence we see

W^{∗}(G_{k}/T_{k}) ≅ KO^{2∗}(G/T)/KU^{2∗}(G/T) ≅ H^{∗}(G/T; Sq^{2}) ≅ Λ(z_{3})

where z_{3} = t_{1}t_{2}^{2}. (SQ^{2}(z_{3}) = t_{1}^{2}t_{2}^{2} = c_{2}^{2} = 0.) Then we have W^{∗}(G_{k}) ≅ W^{∗}(G_{k}/T_{k}) and it is a primitive Hopf algebra Λ(z_{3}). On the other hand, we consider the Atiyah-Hirzebruch spectral sequence converging to KO^{∗}(G). Since Sq^{2}(x_{3}) = x_{5}, we see

E_{3}^{∗′,−1} ≅ E_{3}^{2∗,−1} ≅ H^{∗}(H^{∗}(G; ℤ/2); Sq^{2}) ≅ Λ(v_{4})

where v_{4} = x_{3}x_{5}. We easily see E_{3}^{∗,∗′} ≅ E_{∞}^{∗,∗′} from dimensional reason. Thus the map

q^{∗} : W^{∗}(G_{k}) ≅ Λ(z_{3}) → KO^{2∗}(G)/KU^{2∗}(G) ≅ Λ(v_{4})

is not injective nor surjective since deg(v_{4}) = 4 but deg(z_{3}) = 3.

We will prove the Camlès and Fasel result for an algebraically closed field k. Let L be a line bundle over X and T h(L) be its Thom class. Recall W^{∗−1}(X; L) ≅ W̃^{∗}(T h(L)).

Lemma 4.4. (Zibrowius (2.4) in [Zi])

H^{∗}(H̃^{∗}(T h(L); ℤ/2); Sq^{2}) ≅ H^{∗}(H^{∗}(X; ℤ/2); Sq^{2} + c_{1}(L)).

Proof. We recall the Steenrod algebra A_{2}-strucure of the Thom space

H̃^{∗}(T h(L); ℤ/2) ≅ H^{∗}(X; ℤ/2){c_{1}} ⊂ H^{∗}(X; ℤ/2){1, c_{1}} ≅ H^{∗}(P (L); ℤ/2)

where P (L) is the associated projective bundle and c_{1} is the first Chern class so that c_{1}^{2} = c_{1}(L)c_{1}. Hence for x ∈ H^{∗}(X); ℤ/2), we have (by Cartan formula)

Sq^{2}(xc_{1}) = Sq^{2}(x)c_{1} + xc_{1}(L)c_{1} = (Sq^{2} + c_{1}(L))xc_{1}.

Each element in H̃^{∗}(T h(L); ℤ/2) is expressed as xc_{1}, and we have the result. □

Recall that grH^{∗}(G/T; ℤ/2) ≅ P (y) ⊗ S(t)/(b_{1}, ..., b_{ℓ}) and S(t) = ℤ/2[t_{1}, ..., t_{ℓ}].

Lemma 4.5. For each 0 ≠ t ∈ ℤ/2{t_{1}, ..., t_{ℓ}}, we get

H^{∗}(H^{∗}(T h(t); ℤ/2); Sq^{2}) ≅ H^{∗}(H^{∗}(G/T; ℤ/2); Sq^{2} + t) = 0.

Proof. Let t = t_{1} and d = Sq^{2} + t. Give a weight by w(t_{1}) = 1 and w(t_{i}) = 0 for i ≥ 2 and consider the graded ring grS(t). Then d = Sq^{2} + t acts on ℤ/2[t_{i}] as Sq^{2} + 0 for i ≥ 2. Decompose grS(t) = ℤ/2[t_{1}] ⊗ B as a d-module with B = ⊗_{i≥2}ℤ/2[t_{i}] . Then

H^{∗}(grS(t); d) ≅ H^{∗}(B; d) ⊗ H^{∗}(ℤ/2[t]; d) ≅ ℤ/2 ⊗ 0 = 0

since H^{∗}(ℤ/2[t], d) = 0, (d = Sq^{2} + t : t^{even} ↦ t^{even+1}) while H^{∗}(ℤ/2[t]; Sq^{2}) ≅ ℤ/2, (Sq^{2} : t^{odd} ↦ t^{odd+1}). Hence H^{∗}(S(t); d) itself is also zero.

We use the induction on i with |x_{i}| ≤ |x_{i+1}|. The Sq^{2}-action on H^{∗}(G; ℤ/2) is a derivative (mod(Sq^{1})). Hence Sq^{2} acts on the ring generators to ring generators or zero, i.e., we can take generators x_{i} so that Sq^{2}(x_{i}) = x_{i+1} or Sq^{2}(x_{i}) = 0. (Moreover if |x_{i}| + 2 = |x_{j}|, then Sq^{2}x_{i} = x_{j} for simply connected simple Lie groups.) Let Sq^{2}(x_{1}) = x_{2}. Write d_{r}(x_{1}) = b_{1} and d_{r+2}(x_{2}) = b_{2}. Then the Cartan-Serre transgression theorem implies that

Sq^{2}(b_{1}) = Sq^{2}(d_{r}(x_{1})) = d_{r+2}(Sq^{2}(x_{1})) = d_{r+2}(x_{2}) = b_{2}.

Similarly if Sq^{2}(x_{1}) = 0, then Sq^{2}(b_{1}) = 0.

Suppose that Sq^{2}(b_{1}) = 0 (i.e., Sq^{2}(x_{1}) = 0 by the arguments above). Then S(t)/(b_{1}) is a d-module because

(Sq^{2} + t)(xb_{1}) = Sq^{2}(x)b_{1} + txb_{1} ∈ Ideal(b_{1})

by the Cartan formula. We consider the short exact sequence

0 → S(t) →^{b_{1}} S(t) → S(t)/(b_{1}) → 0

of d-modules, and consider the induced long exact sequence

→ H^{∗}(S(t); d) → H^{∗}(S(t); d) → H^{∗}(S(t)/(b_{1}); d) → ...

The first and the second terms in the above sequence are zero, and so is the third, namely H^{∗}(S(t)/(b_{1}); d) = 0.

Suppose that Sq^{2}(b_{1}) ≠ 0. Then take b_{2} with Sq^{2}(b_{1}) = b_{2} (i.e., Sq^{2}(x_{1}) = x_{2}). Then S(t)/(b_{2}) is a d-module and H^{∗}(S(t)/(b_{2}); d) = 0 by the arguments above. Next consider a short exact sequence of d-modules

0 → S(t)/(b_{2}) →^{b_{1}} S(t)/(b_{2}) → S(t)/(b_{1}, b_{2}) → 0.

Considering the induced long exact sequence of d-homology, we get H^{∗}(S(t)/(b_{1}, b_{2}); d) = 0. Similarly, we have H^{∗}(S(t)/(b_{1}, ..., b_{ℓ}); d) = 0.

Thus we see

H^{∗}(grH^{∗}(G/T; ℤ/2); d) ≅ H^{∗}(P (y); d) ⊗ H^{∗}(S(t)/(b_{1}, ..., b_{ℓ}); d) ≅ 0.
□

Theorem 4.6. (Calmès and Fasel theorem for algebraically closed field) Let k be an algebraically closed and 0 ≠ L ∈ P ic(G_{k}/T_{k})/2. Then W^{∗}(G_{k}/T_{k}; L) = 0.

Proof. We have

W^{∗−1}(G_{k}/T_{k}; L) ≅ W̃^{∗}(T h(L)) ≅ K̃O^{2∗}(T h(L))/K̃O^{2∗}(T h(L)).

The last term is isomorphic to a subquotient of H^{∗}(H̃^{∗}(T h(L); ℤ/2); Sq^{2}), which is isomorphic to H^{∗}(H^{∗}(G/T; ℤ/2); Sq + c_{1}(L)) = 0 from the preceding lemma. □

5. hermitian K-theory

The Balmer Witt group can be extended as a generalized cohomology theory for the stable 𝔸^{1}-homotopy category as follows. By Hornbostel, Schlichting, Panin and Walter [Ho], [Sch], [Pa-Wa], there is a spectrum KO in the stable 𝔸^{1}-category such that the hermitian K-theory is written as

KO^{∗,∗′}(X) ≅ Hom_{𝔸^{1}}(X, S^{∗,∗′} ∧ KO)

where S^{∗,∗′} is the sphere of deg = (∗, ∗′), e.g., S^{2,1} ≅ ℙ^{1}. Moreover the Witt group is written as

W^{i}(X) ≅ KO^{i+∗,∗}(X) for i − ∗ > 0

(namely, W^{d(∗,∗′)}(X) ≅ KO^{∗,∗′}(X) for w(∗, ∗′) < 0).

Since KO^{∗,∗′}(−) theory has a good product [Pa-Wa], so does W^{∗}(X), namely, W^{∗}(X) is a graded ring.

Let t_{ℂ} : KO^{∗,∗′}(X) → KO^{∗}(X(ℂ)) be the realization map (§3.4 in [Vo1]). So we have a natural map

W^{∗}(X) ≅ KO^{2∗−1,∗−1}(X) → KO^{2∗−1}(X).

Zibrowius proves (Theorem 2.5 in [Zi]) that the above map is an isomorphism when X is cellular and k = ℂ.

We give its (a bit different) proof here for an algebraically closed k. Let X be cellular (of dimX = n). Then by the definition, X has a filtration by closed subvarieties

∅ = Z_{−1} ⊂ Z_{0} ⊂ ... ⊂ Z_{n} = X

such that the open complement of Z_{k−1} in Z_{k} is isomorphic to ∐ 𝔸^{k}. In general, Z_{k} are not smooth. Let us write X_{i} = X − Z_{i}. So we have Z_{n} − Z_{n−1} = X_{n−1} ⊂ X_{2} ⊂ ... ⊂ X_{−1} = X. Since Z_{i} − Z_{i−1} ⊂ X_{i−1} is smooth, we have the long exact sequences

→ W^{∗}(T h(Z_{n−1} − Z_{n−2})) →^{i_{∗}} W^{∗}(X_{n−2}) →^{j^{∗}} W^{∗}(X_{n−1} = Z_{n} − Z_{n−1}) →^{δ} ...,

→ W^{∗}(T h(Z_{i} − Z_{i−1})) →^{i_{∗}} W^{∗}(X_{i−1}) →^{j^{∗}} W^{∗}(X_{i}) →^{δ} ...,

→ W^{∗}(T h(Z_{0} − Z_{−1})) →^{i_{∗}} W^{∗}(X_{−1} = X) →^{j^{∗}} W^{∗}(X_{0}) →^{δ} ...

By induction, we easily show the Zibrowius theorem from the following lemma.

Lemma 5.1. Let k be an algebraically closed field. Let U → X → T be a cofiber sequence in the stable 𝔸^{1}-homotopy category. For both Y = T, U, if the following (1),(2) are satisfied

(1) KU^{odd}(t_{ℂ}(Y)) = 0,
(2) t_{ℂ} : W^{∗}(Y) ≅ KO^{2∗}(t_{ℂ}(Y))/KU^{2∗}(t_{ℂ}(Y)) ≅ KO^{2∗−1}(t_{ℂ}(Y)),

then so are for Y = X.

Proof. Let us write KO^{∗}(t_{ℂ}(−)) by KO^{∗}(−) simply. We consider the following diagram for long exact sequences

W^{∗−1}(U) ⟶^{δ} W^{∗}(T) ⟶^{g^{∗}} W^{∗}(X) ⟶^{f^{∗}} W^{∗}(U) ⟶^{δ} W^{∗+1}(T)
t_{1}↓  ≅↓^{t_{2}}  ^{t_{3}}↓  ≅↓^{t_{4}}  ^{t_{5}}↓
KO^{2∗−2}(U) ⟶^{δ} KO^{2∗−1}(T) ⟶^{g^{∗}} KO^{2∗−1}(X) ⟶^{f^{∗}} KO^{2∗−1}(U) ⟶^{δ} KO^{2∗}(T).

First we see that the map t_{5} is injective from the following diagram

W^{∗+1}(T) ≅ KO^{2∗+1,∗}(T) ⟶_{≅}^{η} W^{∗+1}(T) ≅ KO^{2∗,∗−1}(T)
≅↓  ^{t_{5}}↓
KO^{2∗+1}(T)  ⟶^{η}  KO^{2∗}(T).

Indeed, η : KO^{odd}(T) → KO^{even}(T) is injective from the Bott exact sequence since KU^{odd}(T) = 0 (note that η : KO^{even}(T) → K^{odd}(T) is surjective.)

Next we study the map t_{1}. Since δ|KU^{2∗−2}(U) = 0 (from K^{2∗−1}(T) = 0), the map δ factors as

δ : KO^{2∗−2}(U) → KO^{2∗−2}(U)/KU^{2∗−2}(U) → KO^{2∗−1}(T).

Moreover, we have the diagram

KO^{8N+2∗−2,∗−1}(U) ⟶^{≅}  W^{∗−1}(U)  ⟶ W^{∗}(T)
t_{1}↓  ≅↓  ^{t_{2}}↓
KO^{2∗−2}(U)  ⟶ KO^{2∗−2}(U)/KO^{2∗−2}(U) ⟶ KO^{2∗−1}(T).

Then we can prove the lemma from the five lemma. If t_{3}(x) = 0, then f^{∗}(x) = 0, so there is x′ ∈ W^{∗}(T) with g^{∗}(x′) = x. Since g^{∗}(t_{2}(x′)) = 0, there is x̃ ∈ KO^{2∗−2}(U) with δ(x̃) = .t_{2}(x′). From the above diagram, we can take x′′ ∈ W^{∗−1}(U) such that t_{1}(x′′) − x̃ = 0 ∈ KO^{2∗−2}(U)/KU^{2∗−2}(U). Since t_{2}(δ(x′′)) = δ(x̃) = t_{2}(x′), we see that δx′′ = x′ and x = 0. Hence t_{3} is injective.

For y ∈ KO^{2∗−1}(X), we can take y′ ∈ W^{∗}(U) with t_{4}(y′) = f^{∗}(y). Then δ(y′) = 0 since t_{5} is injective. So there is y′′ ∈ W^{∗}(X) such that f^{∗}(t_{3}(y′′) − y) = 0. Hence there is z ∈ KO^{2∗−1}(X) with g^{∗}(z) = t_{3}(y′′)−y. Then we see t_{3}(y′′−g^{∗}t_{2}^{−1}(z)) = y. Therefore we also have the surjectivity of t_{3}. □

We will study the case which is not cellular but (1), (2) in the above lemma are satisfied.

Lemma 5.2. Let k be an algebraically closed field. Let U → X → T be a cofiber sequence in the stable 𝔸^{1}-homotopy category. For both Y = X, T (resp. Y = X, U), if (1),(2) in Lemma 5.1 are satisfied, and moreover, g^{∗} : KU^{∗}(t_{ℂ}(T)) → KU^{∗}(t_{ℂ}(X)) is injective (resp. f^{∗} : KU^{∗}(t_{ℂ}(X)) → KU^{∗}(t_{ℂ}(U)) is surjective), then (1), (2) are satisfied for Y = U (resp. Y = T).

Proof. Consider the diagram for exact sequences

W^{∗}(T) ⟶^{g^{∗}} W^{∗}(X) ⟶^{f^{∗}} W^{∗}(U) ⟶^{δ} W^{∗+1}(T) ⟶^{g_{1}^{∗}} W^{∗+1}(X)
≅↓^{t_{2}}  ^{t_{3}}↓≅  ↓^{t_{4}}  ^{t_{5}}↓  ^{t_{6}}↓
KO^{2∗−1}(T) ⟶^{g^{∗}} KO^{2∗−1}(X) ⟶^{f^{∗}} KO^{2∗−1}(U) ⟶^{δ} KO^{2∗}(T) ⟶^{g_{2}^{∗}} KO^{2∗}(X).

We want to see that t_{4} is an isomorphism. Consider the Bott exact sequence

0 → W^{∗+1}(T) ≅ KO^{2∗+1}(T) ⟶^{η} KO^{2∗}(T) ⟶ KU^{2∗}(T)
g_{1}^{∗}↓  ^{g_{2}^{∗}}↓  ^{g_{3}^{∗}}↓^{injective}
0 → W^{∗+1}(X) ≅ KO^{2∗+1}(X) ⟶^{η} KO^{2∗}(X) ⟶ KU^{2∗}(X)

Here note that ×η|KO^{odd}(−) is injective, and g_{3}^{∗} are also injective by the assumption. If x ∈ Ker(g_{2}^{∗}), then x ∈ Im(η) from the injectivity of g_{3}^{∗}. Hence we easily see (from the injectivity of ×η) that Ker(g_{2}^{∗}) ≅ Ker(g_{1}^{∗}), and t_{4} is isomorphic from the five lemma.

The second case is similarly proved by using the following diagram (to see Im(f_{2}^{∗}) ≅ Im(f_{3}^{∗})).

KU^{2∗−2}(X) ⟶ KO^{2∗−2}(X) ⟶ KU^{2∗−2}(X)/KU^{2∗−2}(X) ≅ W^{∗−1}(X)
f_{1}^{∗}↓^{surjective}  ^{f_{2}^{∗}}↓  ^{f_{3}^{∗}}↓
KU^{2∗−2}(U) ⟶ KO^{2∗−2}(U) ⟶ KU^{2∗−2}(U)/KU^{2∗−2}(U) ≅ W^{∗−1}(U)
□

Remark. When X = G/T, T = T h(t_{ℓ}) and U = G/T^{ℓ−1}, of course, t_{ℓ} : KU^{∗}(T) → KU^{∗}(X) is not injective. In fact, KU^{∗}(G/T) ≅ KU^{∗} ⊗ H^{∗}(G/T) and g_{ℓ} (given in Lemma 3.1) is in Ker(t_{ℓ}).

We consider the classifying space BG for a finite group G. First consider the case G = ℤ/2^{r}. In the stable 𝔸^{1}-homotopy category, we have the cofibering (see (6.4) in [Vo3] for details)

Bℤ/2^{r} → E →^{q_{r}} T h(E)

where E = O(−2^{r}) is the −2^{r}-th twisted line bundle (of the canonical one) of ℙ^{∞} and T h(E) is its Thom space and q_{r} is the composition map with 2^{r}-times twist on ℙ^{∞} and the quotient map.

Of course ℙ^{n} is cellular, and (see §7 in [Ya1])

W^{∗}(ℙ^{n}) ≅ \begin{cases}ℤ/2\{1, y_{n}\} for deg(y_{n}) = n if n : odd \\ ℤ/2\{1\}, otherwise. \end{cases}

Hence we have W^{∗}(ℙ^{∞}) ≅ ℤ/2. (For Witt groups of infinite spaces, see the remark after Lemma 6.1 below.) . We also see W^{∗}((ℙ^{∞})^{×n}) ≅ ℤ/2. Moreover E = O(−2^{r}) represents zero in P ic(ℙ^{∞})/2, and so we have the Thom isomorphism

W̃^{∗+1}(T h(E)) ≅ W^{∗}(E) ≅ W^{∗}(ℙ^{∞}) ≅ ℤ/2.

Therefore we have the exact sequence

→ W^{∗−1}(ℙ^{∞}) →^{×2^{r}=0} W^{∗}(ℙ^{∞}) → W^{∗}(Bℤ/2^{r}) → ...

Hence we obtain W^{∗}(Bℤ/2^{r}) ≅ Λ(x) with deg(x) = 0. (In fact x^{2} = 0 since so in E(P)_{2}^{∗,∗′}and x^{2} ≠ 1 by the naturality.)

For each space X, we have the cofibering

Bℤ/2^{r} × X → E × X →^{q_{r}×id} T h(E) × X.

By induction on i, starting (ℙ^{∞})^{×n}, we can prove with deg(x_{j}) = 0

W^{∗}((×_{s=1}^{i}Bℤ/(2^{r_{s}})) × (ℙ^{∞})^{×(n−i)}) ≅ Λ(x_{1}, ..., x_{i}).

Theorem 5.3. Let k be algebraically closed. Let G be a 2-group of rank = n i.e., G ≅ ⊕_{s=1}^{n}ℤ/(2^{r_{s}}). Then there are isomorphisms

q^{∗} : W^{∗}(BG) ≅ KO^{2∗}(BG)/KU^{2∗}(BG) ≅ Λ(x_{1}, ..., x_{n}), deg(x_{i}) = 0.

Proof. We only need to see the first isomorphism. It is well known that KU^{∗}(BG) is torsion free for each compact group G. In particular

×2^{r_{i}}|KU^{∗}((×_{s=1}^{i−1}Bℤ/(2^{r_{s}})) × (ℙ^{∞})^{×(n−i+1)})

is injective. Hence this satisfies the assumptions in the preceding lemma. Hence (2) in the lemma also satisfies for U = (×_{s=1}^{i}Bℤ/(2^{r_{s}})) × (ℙ^{∞})^{×(n−i)}. □

The isomorphisms are still given in Theorem 7.4, 7.7 in [Ya1] for G = (ℤ/2)^{⊕n}. (However it was not proved that the map q^{∗} induces these isomorphisms.)

Recall grH^{∗}(Bℤ/2^{r}) ≅ ℤ/2[y] ⊗ Λ(x), and P ic(Bℤ/2^{r})/2 ≅ ℤ/2{y}. We easily see H^{∗}(H^{∗}(ℤ/2[y] ⊗ Λ(x); Sq^{2} + y) = 0 as the proof of Lemma 4.5. The analogue of the theorem by Calmès and Fasel also holds.

Corollary 5.4. Let k be algebraically closed and let G ≅ ⊕_{s=1}^{n}ℤ/(2^{r_{s}}). Then

W^{∗}(BG; L) = 0 for 0 ≠ L ∈ P ic(BG)/2.

Let S be a 2-Sylow subgroup of a finite group G. Since we can identify the induced map g : BS → BG as a finite covering, we have the Gysin map g_{∗} in also Witt theory (so that g_{∗}g^{∗} = [G; S]).

Corollary 5.5. Let k be algebraically closed. Let G be a finite group having an abelian 2-Sylow subgroup. Then

q^{∗} : W^{∗}(BG) ≅ KO^{2∗}(BG)/KU^{2∗}(BG).

6. Gille-Pardon spectral sequence.

Balmer and Walter ((1) in [Ba-Wa]) define the Gersten-Witt complex

0 → W (k(X)) → ⊕_{x∈X^{(1)}}W (k(x)) → ... → ⊕_{x∈X^{(n)}}W (k(x)) → 0.

Let H^{∗}(W (X)) denote the cohomology group of the above cochain complex, with W (k(X)) in degree 0. Then Balmer-Walter constructed the spectral sequence (Theorem in [Ba-Wa])

E(BW)_{2}^{r,t} ≅ \begin{cases}H^{r}(W (X)) (t = 0 mod(4)) ⟹ W^{r+t}(X). \\ 0 (t ≠ 0(mod(4)) \end{cases}

By the affirmative answer to the Milnor conjecture of quadratic forms by Orlov-Vishik-Voevodsky (Theorem 4.1 in [Or-Vi-Vo]), we have an isomorphism of graded rings H_{et}^{∗}(k(x); ℤ/2) ≅ grW (k(x)). Using this fact, we can reinterpret the Pardon-Gille spectral sequence (Corollary 0.13 in [Pa] and [Gi]) as a spectral sequence

E(P)_{2}^{r,s} ≅ H_{Zar}^{r}(X; H_{ℤ/2}^{s}) ⟹ H^{r}(W (X)) ≅ E(BW)_{2}^{r,4t}

so that the differential d_{r} has degree (1, r − 1) for r ≥ 2. Here H_{ℤ/2}^{s} is the Zariski sheaf associated to the presheaf V ↦ H_{et}^{s}(V; ℤ/2) for any open subscheme V of X.

The above sheaf cohomology H_{Zar}^{r}(X; H_{ℤ/2}^{s}) is related to the motivic cohomology H^{∗,∗′}(X; ℤ/2) = ⊕_{r,s∈ℤ}H^{r,s}(X; ℤ/2) (for details, see [Vo1-3]) as follows.

Recall that τ ∈ H^{0,1}(Spec(k); ℤ/2) ≅ ℤ/2 be a generator. (For example, if k is algebraically closed, then H^{∗,∗′}(Spec(k); ℤ/2) ≅ ℤ/2[τ].) Then we get the long exact sequence from the solution of the Beilinson-Lichtenbaum conjecture (Theorem 1.3 in [To], Lemma 2.4 (5) in [Or-Vi-Vo]),

→ H^{m,n−1}(X; ℤ/2) →^{×τ} H^{m,n}(X; ℤ/2)

→ H_{Zar}^{m−n}(X; H_{ℤ/2}^{n}) → H^{m+1,n−1}(X; ℤ/2) →^{×τ} ...

All elements in the above cohomology groups are 2-torsion, and we have the following additive isomorphisms

Lemma 6.1. E(P)_{2}^{m−n,n} ≅ H_{Zar}^{m−n}(X; H_{ℤ/2}^{n}) ≅

H^{m,n}(X; ℤ/2)/(Im(τ)) ⊕ Ker(τ)|_{H^{m+1,n−1}(X;ℤ/2)}.

Remark. Let X_{n} be smooth and colim_{n}X_{n} = X. Suppose ⊕_{s}H^{∗}(X_{n}; H_{ℤ/2}^{s}) is a finite group. (Note H^{∗}(X_{n}; H_{ℤ/2}^{∗′}) = 0 for ∗ > dim(X_{n}).) Then ⊕_{m,s}E(P)_{r}^{m,s}(X_{n}) is a finite group, and so is ⊕_{m}E(BW)_{∞}^{m,0}. Therefore W^{∗}(X_{n}) is a finite group for each ∗ ∈ ℤ/4. So lim^{1} = 0, and we have

W^{∗}(X) ≅ lim_{n}W^{∗}(X_{n}).

It is well known that CH^{m}(X) ≅ H^{2m,m}(X; ℤ) (Corollary 2 in [Vo2]). Since H^{2m+1,m}(X; ℤ) = 0 (Corollary 2.3 in [Vo1]), from the Bockstein exact sequence, we have an isomorphism H^{2m,m}(X; ℤ/2) ≅ H^{2m,m}(X; ℤ)/2. In particular,

E(P)_{2}^{m,m} ≅ H_{Zar}^{m}(X; H_{ℤ/2}^{m}) ≅ H^{2m,m}(X; ℤ/2) ≅ CH^{m}(X)/2,
E(P)_{2}^{m,m+1} ≅ H_{Zar}^{m}(X; H_{ℤ/2}^{m+1}) ≅ H^{2m,m+1}(X; ℤ/2).

In H^{∗,∗′}(X; ℤ/2), we can define the cohomology operation Sq^{i} from Voevodsky (§9 in [Vo3], §3.3 in [Vo1]) (or Brosnan [Br] for CH^{∗}(X)/2). This Sq^{i} is compatible with that in the usual (topological) mod 2 cohomology via the realization map t_{ℂ} when k ⊂ ℂ (§3.4 in [Vo1]). Moreover Sq^{2}Sq^{2} = τSq^{3}Sq^{1} (Theorem 10.2 in [Vo3]). Then Totaro proved

Lemma 6.2. (Theorem 1.1 in [To]) If x ∈ E(P)_{2}^{m,m} ≅ CH^{m}(X)/2, then d_{2}(x) = Sq^{2}(x).

We assume the following assumption throughout this section

Assumption 6.3. If x ∈ E(P)_{2}^{m,m+1} ≅ H^{2m+1,m+1}(X; ℤ/2), then d_{2}(x) = Sq^{2}(x).

This assumption should be proved if the Gersten-Witt complex can be defined for each object X in the 𝔸^{1}-homotopy category (see Proposition 7.8 in [Ya1]).

Hereafter in this paper, we always assume that k is an algebraically closed field (of ch(k) = 0). Hence H^{∗,∗′}(pt.; ℤ/2) ≅ ℤ/2[τ]. Since W^{∗′′}(X) = BO^{∗,∗′}(X) has the ring structure from the results by Schlichting and Panin-Walter, W^{∗}(−) is a multiplicative cohomology theory. So W^{∗}(X) is a W^{∗}(pt.) ≅ ℤ/2-algebra, i.e., we have the cup product

W^{∗}(X) ⊗_{ℤ/2} W^{∗′}(X) → W^{∗+∗′}(X × X) →^{Δ} W^{∗+∗′}(X).

For a map X → Y in 𝔸^{1}-homotopy category, we have the long exact sequence

→ W̃^{∗}(Y/X) → W^{∗}(Y) → W^{∗}(X) →^{δ} W̃^{∗+1}(Y/X) → ...

The coboundary map δ is a derivation (over ℤ/2) in the following sense. There is a commutative diagram

W^{∗}(X) ⊗ W^{∗′}(X′) ⟶^{δ′} W^{∗+1}(Y, X) ⊗ W^{∗}(X′) ⊕ W^{∗}(X) ⊗ W^{∗′+1}(Y′, X′)
↓  ↓
W^{∗+∗′}(X × X′) ⟶^{δ}  W^{∗+∗′+1}(X × Y′ ∪ Y × X′, X × X′)

where δ′(a ⊗ b) = δ(a) ⊗ b + a ⊗ δ(b) from the standard arguments (as topological cases).

Let us consider

Z^{n} ⊂ Z^{n−1} ⊂ ... ⊂ Z^{1} ⊂ Z

series of smooth embedding with codim_{Z}(Z^{i}) = i. Then we have the diagram

W^{∗}(X; L^{0}) ⟵^{g_{∗}} W^{∗}(Z^{1}; L^{1}) ⟵^{g_{∗}} W^{∗}(Z^{2}; L^{2}) ⟵^{g_{∗}} ...
↘ f^{∗}  ↑^{δ}  ^{↘ f^{∗}}  ↑^{δ}
W^{∗}(X − Z^{1}; L^{0}) ⟶^{f^{∗}δ} W^{∗}(Z^{1} − Z^{2}; L^{1}) ⟶^{f^{∗}δ} ...

Roughly speaking, the Gersten-Witt complex is constructed by taking limit of the above diagram [Bl-Og], [Ka]. Indeed, we have the commutative diagram

W^{∗}(X − Z^{1}; L^{0}) ⟶^{f^{∗}δ} W^{∗}(Z^{1} − Z^{2}; L^{1}) ⟶^{f^{∗}δ} W^{∗}(Z^{2} − Z^{3}; L^{2}) ⟶ ...
↓  ↓  ↓
W^{∗}(k(X)) ⟶^{d} ⊕_{x∈X^{(1)}}W^{∗}(k(x)) ⟶^{d} ⊕_{x∈X^{(2)}}W^{∗}(k(x)) ⟶^{d} ...

Moreover we have a natural map

⊕_{x∈X^{(i)},x′∈X′^{(j)}}W^{∗}(k(x)) ⊗ W^{∗′}(k(x′)) → ⊕_{x∈(X×X′)^{(i+j)}}W^{∗+∗′}(k(x)).

Then we can prove that the differential d of the Gersten-Witt complex is a derivation by using that the coboundary map δ^{∗} is a derivation in the sense above.

Hence spectral sequences E(P)_{r}^{∗,∗′} and E(BW)_{r}^{∗,∗′} are that of graded rings, e.g., the differential is a derivation.

For X = G_{k}, Y = G′_{k}, we still know from Corollary 3.7,

H^{∗,∗′}(X × Y; ℤ/2) ≅ H^{∗,∗′}(X; ℤ/2) ⊗_{ℤ/2[τ]} H^{∗,∗′}(Y; ℤ/2).

Hence the spectral sequences has the same property

E(P)_{r}^{∗,∗′}(X × Y) ≅ E(P)_{r}^{∗,∗′}(X) ⊗ E(P)_{r}^{∗,∗′}(Y)
E(BW)_{r}^{∗,∗′}(X × Y) ≅ E(BW)_{r}^{∗,∗′}(X) ⊗ E(BW)_{r}^{∗,∗′}(Y)

Therefore W^{∗}(X × Y) ≅ W^{∗}(X) ⊗ W^{∗}(Y). In particular the spectral sequences for X are differential Hopf algebras. It is known (e.g., Araki [Ar],Toda [Tod]) that the image of differential of a ring generator (indecomposable element) is also a ring generator or zero. (We can identify grE_{2}^{∗,∗′} to be primitively generated, and use the fact that the image of the differential of a primitive element is primitive.)

Lemma 6.4. The Witt ring W^{∗}(G_{k}) is a Hopf algebra. So spectral sequences E(P)_{r}^{∗,∗′} and E(BW)_{r}^{∗,∗′} are differential Hopf algebras, that implies differential d_{r} works as ring generators maps to ring generators or zero.

From Corollary 3.5, we have

grH^{∗,∗′}(G_{k}; ℤ/2) ≅ ℤ/2[τ] ⊗ P (y) ⊗ Λ(x_{1}, ..., x_{ℓ})

with w(P (y)) = 0 and w(x_{i}) = 1. So we get

H^{∗,∗′}(G_{k}; ℤ/2)/(Im(τ)) ≅ P (y) ⊗ Λ(x_{1}, ..., x_{ℓ}),
and Ker(τ)|_{H^{∗,∗′} (G_{k};ℤ/2)} = 0.

Thus we get a bigraded ring isomorphism

E(P)_{2}^{∗,∗′} ≅ P (y) ⊗ Λ(x_{1}, ..., x_{ℓ})

with deg(z) = (1/2|z|, 1/2|z|) for z ∈ P (y) and deg(x_{i}) = (1/2(|x_{i}| − 1), 1/2(|x_{i}| + 1)).

On the other hand, we give a (coniveau) filtration to topological H^{∗}(G; ℤ/2) by F_{i} = {x|w(x) ≤ i} and denote by grH^{∗}(G; ℤ/2) = ⊕F_{i}/F_{i−1} the associated algebra. Then we have the ring isomorphism

grH^{∗}(G; ℤ/2) ≅ P (y) ⊗ Λ(x_{1}, ..., x_{ℓ}).

Here Sq^{2}Sq^{2} = Sq^{3}Sq^{1} = 0 and Sq^{2} define a differential of grH^{∗}(G; ℤ/2). Let us write its homology by H^{∗}(G; Sq^{2}) with deg(y_{i}) = 1/2|y_{i}| and deg(x_{i}) = 1/2(|x_{i}| − 1).

Then we have the Sq^{2}-bidegree ring isomorphism

E(P)_{2}^{∗,∗′} ≅ grH^{∗}(G; ℤ/2)

identifying w(y_{i}) = 0 and w(x_{i}) = 1. From the result of Totaro and Assumption 6.3, we have

E(P)_{3}^{∗,∗′} ≅ H^{∗}(grH^{2∗}(G; ℤ/2); Sq^{2}) = H^{∗}(G; Sq^{2}).

Recall the degree of differential deg(d_{r}) = (1, r − 1). Here we consider the assumption

Assumption 6.5. There is an algebra isomorphism

E(P)_{3}^{∗,∗′} ≅ H^{∗}(G; Sq^{2}) ≅ Λ(z_{1}, ..., z_{s}) with first.deg(z_{i}) = odd.
or E(P)_{4}^{∗,∗′} ≅ H^{∗}(H^{∗}G; Sq^{2}); d_{3}) ≅ Λ(z_{1}, ..., z_{s}),

Lemma 6.6. There is a spectral sequence

E_{2} ≅ H^{∗}(G; Sq^{2}) ⇒ H^{∗}(G/T; Sq^{2}).

Proof. (Recall the proof of Lemma 4.5 and see also §2 in [Ki-Oh].) Let us write d = Sq^{2} simply. Recall S(t) ≅ ⊗_{i=1}^{ℓ}ℤ/2[t_{i}]. We give deg(t_{i}) = 1 here to compatible with degree of W^{∗}(−). We get

H^{∗}(S(t); d) ≅ ⊗_{i}^{ℓ}H^{∗}(ℤ/2[t_{i}]; d) ≅ ℤ/2

since H^{∗}(ℤ/2[t]; Sq^{2}) ≅ ℤ/2, (Sq^{2} : t^{odd} ↦ t^{odd+1}).

Suppose that Sq^{2}(b_{1}) = 0 (i.e., Sq^{2}(x_{1}) = 0). Then S(t)/(b_{1}) is a d-module. We consider the short exact sequence

0 → S(t) →^{b_{1}} S(t) → S(t)/(b_{1}) → 0

of d-modules, and consider the induced long exact sequence

→ H^{∗}(S(t); d) →^{×b_{1}} H^{∗}(S(t); d) → H^{∗}(S(t)/(b_{1}); d) → ...

The first and the second terms in the above sequence are isomorphic to ℤ/2 and ×b_{1} = 0. Hence we have

H^{∗}(S(t)/(b_{1}); d) ≅ ℤ/2{1, b̃_{1}} ≅ Λ(b̃_{1}).

Here d(b̃_{1}) = b_{1}, which is a cycle in S(t)/(b_{1}) (but not in S(t)), and b̃_{1}^{2} ∈ Im(d) ⊂ S(t)).

Suppose that Sq^{2}(b_{1}) ≠ 0. Then can take b_{2} with Sq^{2}(b_{1}) = b_{2} (i.e., Sq^{2}(x_{1}) = x_{2}). Then S(t)/(b_{2}) is a d-module and H^{∗}(S(t)/(b_{2}); d) ≅ Λ(b̃_{2}) here note b̃_{2} = b_{1}, by the arguments above. Next consider a short exact sequence

0 → S(t)/(b_{2}) →^{b_{1}} S(t)/(b_{2}) → S(t)/(b_{1}, b_{2}) → 0.

and induced long exact sequence of d-homology,

→ Λ(b_{1}) →^{b_{1}} Λ(b_{1}) → H^{∗}(S(t)/(b_{1}, b_{2}); d) → ...

Hence we have

H^{∗}(S(t)/(b_{1}, b_{2}); d) ≅ Λ(b̃_{1,2}) d(b̃_{1,2}) = b_{1}^{2}.

Similarly, we can compute H^{∗}(S(t)/(b_{1}, ..., b_{ℓ}); d) ≅ Λ(b̄_{1}, ..., b̄_{ℓ′}).

On the other hand, we consider the homology of Λ(x_{1}, ..., x_{ℓ}). If Sq^{2}(x_{1}) = 0. then H^{∗}(Λ(x_{1}); d_{1}) ≅ Λ(x̃_{1}). If Sq^{2}(x_{1}) = x_{2}, then

H^{∗}(Λ(x_{1}, x_{2}), d) ≅ Λ(x_{1,2}) identifying x_{1,2} = x_{1}x_{2}.

Similarly we have an isomorphism H^{∗}(Λ(x_{1}, ..., x_{ℓ}); d) ≅ Λ(x̄_{1}, ..., x̄_{ℓ′}). Thus we can construct an isomorphism

H^{∗}(Λ(x_{1}, ..., x_{ℓ}); d) ≅ H^{∗}(S(t)/(b_{1}, ..., b_{ℓ}); d).

Therefore we have

H^{∗}(G; Sq^{2}) ≅ H^{∗}(P (y) ⊗ Λ(x_{1}, ..., x_{ℓ}); d)
≅ H^{∗}(P (y) ⊗ S(t)/(b_{1}, ..., b_{ℓ}); d) ≅ H^{∗}(grH^{∗}(G/T; ℤ/2); d).

Hence we have a spectral sequence

H^{∗}(G; Sq^{2}) ≅ H^{∗}(grH^{∗}(G/T; ℤ/2); d) ⇒ H^{∗}(H^{∗}(G/T; ℤ/2); d).
□

Theorem 6.7. Let k be an algebraically closed field in ℂ. For each simply connected simple Lie group G, Assumption 4.1, 6.3, 6.5 are satisfied and

W^{∗}(G_{k}) ≅ H^{∗}(G/T; Sq^{2}) ≅ Λ(z_{1}, ..., z_{s}) deg(z_{i}) = odd.

which is also isomorphic to

\begin{cases}E(P)_{4}^{∗,∗′} ≅ H^{∗}(H^{∗}(G; Sq^{2}); d_{3}) for G = E_{6}, E_{7}, E_{8} \\ E(P)_{3}^{∗,∗′} ≅ H^{∗}(G; Sq^{2}) otherwise. \end{cases}

Proof. We assume here 6.3 and 6.5 which are shown for each simple group G in the next section. From Assumption 6.5, Gill-Pardon and Balmer-Walter spectral sequences collapse because, from Lemma 6.4, the d_{r}-image of z_{i} must be some z_{j} (mod(decomposable el.)). However deg(z_{i}) − deg(z_{j}) = even but deg(d_{r}) = odd, and this means d_{r} = 0. Hence W^{∗}(G_{k}) ≅ E(P)_{4}^{∗,∗′} ≅ H^{∗}(H^{∗}(G; Sq^{2}); d_{3}).

Recall that W^{∗}(G_{k}) is a subquotient of H^{∗}(G/T; Sq^{2}) from Corollary 4.2 (and is isomorphic to it if and only if Assumption 4.1 is satisfied).

On the other hand, by the preceding lemma, we know H^{∗}(G/T; Sq^{2}) is a subquotient of H^{∗}(G; Sq^{2}). Here we also assume that

(∗) H^{∗}(G/T; Sq^{2}) is a subquotient of E(P)_{4}^{∗,∗′} ≅ H^{∗}(G; Sq^{2}); d_{3}). (Of course (∗) holds when d_{3} = 0. For the cases G = E_{6}, E_{7}, E_{8}, it is proved in the next section.) Then we get

W^{∗}(G_{k}) ≅ E(P)_{4}^{∗,∗′} ≅ H^{∗}(G/T; Sq^{2}).
□

The homology H^{∗}(G; Sq^{2}) is easier computed than H^{∗}(G/T; Sq^{2}). However, unfortunately, in this paper, we use the ring structure of H^{∗}(G/T; ℤ/2) (for each simple group) to see Lemma 3.1 (for Corollary 3.5 and Theorem 3.6) and to see Assumption 6.3. We are hoping to obtain alternative proofs that do not rely on the detailed ring structure of H^{∗}(G/T; ℤ/2).

Here we note the map of spectral sequences induced from G_{k} → G_{k}/T_{k}. Since G_{k}/B_{k} is cellular, we have isomorphisms

E(P)_{∞}^{∗,∗′}(G/T) ≅ E(P)_{3}^{∗,∗′}(G/T) ≅ E(P)_{3}^{∗,∗}(G/T) ≅ H^{∗}(G/T; Sq^{2}).

From the above theorem, we also see that the Balmer-Walter spectral sequence also collapses.

E(P)_{∞}^{∗,∗}(G/T) ≅ grE(BW)_{∞}^{∗,4∗′}(G/T) ≅ grW^{∗}(G_{k}/T_{k}).

The projection G_{k} → G_{k}/T_{k} induces the isomorphism of the Balmer-Walter spectral sequence E(BW)_{r}^{∗,∗′}(G/T) ≅ E(BW)_{r}^{∗,∗′}(G), but not for the Gille-Pardon spectral sequence, indeed

E(P)_{4}^{∗,∗}(G/T) ≅ E(P)_{4}^{∗,∗′}(G/T) → E(P)_{4}^{∗,∗′}(G)

is not injective nor surjective (while grE(BW)_{2}^{∗,∗′′} ≅ E(P)_{∞}^{∗,0}).

7. Simple Lie groups

(I) classical groups.

For classical groups, we will see Assumption 6.3 and 6.5. First, we consider the case G = U(2m + 1). Its cohomology is

H^{∗}(G; ℤ/2) ≅ Λ(x_{1}, x_{3}, ..., x_{4m+1})
≅ Λ(x_{1}) ⊗ ⊗_{s}^{m}Λ(x_{4s−1}, x_{4s+1}) with Sq^{2}x_{4s−1} = x_{4s+1}.

(Throughout this section, subscripts indicate degree, e.g, |x_{i}| = i for convenience.) Therefore we have the Sq^{2} homology

H^{∗}(G; Sq^{2}) ≅ Λ(x_{1}) ⊗ ⊗_{s=1}^{m}Λ(x_{4s−1}x_{4s+1}),

which is (changing degree so that deg(x_{i}) = 1/2(i − 1)) isomorphic to

Λ(u_{1}) ⊗ H^{∗}(G/T; Sq^{2}) ≅ Λ(z_{0}) ⊗ ⊗Λ(z_{4s−1}) deg(z_{i}) = i.

This result coincides with that given by Kishimoto, Kono and Ohsita. When G = 2m + 2, H^{∗}(G; Sq^{2}) ≅ H^{∗}(U(2m + 1); Sq^{2}) ⊗ Λ(x′_{2m+1}). This element x′_{2m+1} corresponds the element z in the notation of [Ki-Ko-Oh]. We also get H^{∗}(SU(n); Sq^{2}) ≅ H^{∗}(U(n); Sq^{2})/(z_{0}).

The case G = Sp(n) is easy. Infact, H^{∗}(G; ℤ/2) ≅ Λ(x_{3}, x_{7}, ..., x_{4n−1}) and so Sq^{2} = 0. Hence H^{∗}(G; Sq^{2}) ≅ Λ(x′_{1}, ..., x′_{2n−1}) where x′_{2i+1} corresponds x_{4i+3} in H^{∗}(G; ℤ/2).

Next we consider the case G = SO(4m + 2). Then the mod 2-cohomology is written as ( see for example [Ni])

grH^{∗}(SO(4m + 1); ℤ/2) ≅ Λ(x_{1}, x_{2}, ..., x_{4m+1})

where the multiplications are given by x_{s}^{2} = x_{2s}. We write y_{2(odd)} = x_{odd}^{2}. Let us write

grH^{∗}(G; ℤ/2) ≅ Λ(y_{2}, y_{4}, ..., y_{4m}) ⊗ Λ(x_{1}, x_{3}, ..., x_{4m+1}).

Here grP (y) ≅ Λ(y_{2}, y_{4}, ..., y_{4m}) and Sq^{2}(y_{4s+2}) = y_{4s+4}. Hence

H^{∗}(P (y); Sq^{2}) ≅ ⊗_{s}^{m}Λ(y_{4s−2}y_{4s}).

This result also coincides with the result of Kishimoto-Kono-Ohsita

H^{∗}(SO(4m + 1)/T; Sq^{2}) ≅ ⊗_{s=1}^{m}Λ(z_{4s−1}, w_{4s−1})

where z_{4s−1} (resp. w_{4s−1}) corrsponds x_{4s−1}x_{4s+1} (resp. y_{4s−2}y_{4s}). The other n cases are similar (see also [Ki-Ko-Oh]).

Here we give some note for G = Spin(n) case. The cohomology is

H^{∗}(Spin(n); ℤ/2) ≅ H^{∗}(SO(m); ℤ/2)/(x_{1}, y_{1}) ⊗ Λ(a)
≅ Λ(y_{2}, y_{4}, ..., y_{4m})/(y_{2^{s}}|s ≥ 1) ⊗ Λ(x_{3}, ..., x_{4m+1}) ⊗ Λ(a)

where |a| = 2^{t} − 1 for 2^{t−1} < m ≤ 2^{t} ([Mi-Ni], [Tod]). Then we can prove that H^{∗}(G; Sq^{2}) is isomorphic to

H^{∗}(SO(n); Sq^{2})/(z_{0}, w_{2^{s}−1}|s ≥ 1) ⊗ (⊗_{s=1}^{t−2}Λ(y′_{2^{s}−1})) ⊗ Λ(a′_{2^{t−1}−1})

where w_{2^{s}−1} (resp. y′_{2^{s−1}−1}, a′_{2^{t−1}−1}) corresponds y_{2^{s}−2}y_{2^{s}} (resp. y_{2^{s}−2}, a). Identifying y′_{2^{s}−1} = w_{2^{s}−1} and a′_{2^{t−1}−1} = w_{2^{i−1}−1}, we have an isomorphism H^{∗}(Spin(n); Sq^{2}) ≅ H^{∗}(SO(n); Sq^{2})/(z_{0}). Indeed, we still know H^{∗}(Sin(n)/T; ℤ/2) ≅ H^{∗}(SO(n)/T; ℤ/2) (see §3 in [Tod]).

Proposition 7.1. Let G be a simple classical group. Then Assumption 6.3 and 6.5 are satisfied.

Proof. Assumption 6.5 is satisfied from above arguments. We will prove Assumption 6.3 for G = SO(2m + 1). Then the other cases are shown by the naturality of maps e.g., SU(n) → SO(2n).

Let G′ = SO(2m − 3) and suppose the assumption for G′. The cohomology is written

H^{∗}(G; ℤ/2) ≅ H^{∗}(G′; ℤ/2) ⊗ Λ(y_{4m−2}, y_{4m}, x_{4m−1}, x_{4m+1})
with Sq^{2}(y_{4m−2}) = y_{4m}, Sq^{2}(x_{4m−1}) = x_{4m+1}.

In the Gille-Pardon spectral sequence if d_{2}(x_{4m−1}) ≠ x_{4m+1}, then d_{2}(x_{4m−1}) = 0 by Lemma 6.4. By th dimensional reason, x_{4m−1} is a permanent cycle. However this element is zero in H^{∗}(G/T; Sq^{2}) (there is no ring generator in dim = 4m − 1). This is a contradiction. So d_{2}(x_{4m−1}) = x_{4m+1}. Of course we know d_{2}(y_{4m−2}) = y_{4m} from Lemma 5.3. □

(II) exceptional groups G_{2} and F_{4}.

The cohomology is given

grH^{∗}(G_{2}; ℤ/2) ≅ Λ(y_{6}, x_{3}, x_{5}) with Sq^{2}x_{3} = x_{5}.

We have the natural inclusion SU(3) ⊂ G_{2}. Hence in the Gille-Pardon spectral sequence d_{2}(x_{3}) = x_{5} and hence

H^{∗}(G_{2}; Sq^{2}) ≅ Λ(z_{3}, y′_{3})

where z_{3} (resp. y′_{3}) corresponds x_{3}x_{5} (resp. y_{6}). The cohomology

H^{∗}(F_{4}; ℤ/2) ≅ H^{∗}(G_{2}; ℤ/2) ⊗ Λ(x_{15}, x_{23}).

Hence H^{∗}(F_{4}; Sq^{2}) ≅ H^{∗}(G_{2}; Sq^{2}) ⊗ Λ(x′_{7}, x′_{11}) where x′_{7} (resp. x′_{11}) corresponds x_{15} (resp. x_{23}). (See also [Ki-Oh].) Thus Assuption 6.3, 6.5 are satisfied for these cases.

(III) exceptional Lie groups E_{6}, E_{7}, E_{8}.

First consider the case G = E_{6}. Its cohomology is

grH^{∗}(E_{6}; ℤ/2) ≅ Λ(y_{6}, x_{3}, x_{5}, x_{9}, x_{15}, x_{17}, x_{23}).

where Sq^{2}x_{i} = x_{i+2}. We easily see

H^{∗}(G; Sq^{2}) ≅ Λ(y′_{3}, z_{3}, x′_{4}, z_{15}, x′_{11})

by the notation similar to (I),(II) (e.g., x′_{4} = x_{9}, z_{15} = x_{15}x_{17}, x′_{11} = x_{23}).

The spectral sequence in Lemma 6.5 does not collapse by the following reason. In H^{∗}(G/T; ℤ/2), we know Sq^{2}y_{6} = c_{4} by (§4 in [Ki-Oh], Theorem 5.9 in [Ko-Is], y_{6} is written by γ_{3} in there). So d_{3}(y′_{3}) = x′_{4}, and we have

H^{∗}(G/T; ℤ/2) ≅ Λ(z_{3}, v_{7}, z_{15}, x′_{11})

where v_{7} corresponds y_{6}x_{9}. Indeed, this result is still given in [Ki-Oh]. Now return to the Gille-Pardon spectral sequence

E(P)_{3}^{∗,∗} ≅ \begin{cases}Λ(y′_{3}, z_{3}, x′_{4}, z_{15}, x′_{11}) if d_{2}(x_{15}) = y_{17} \\ Λ(y′_{3}, z_{3}, x′_{4}, x′_{7}, x′_{8}, x′_{11}) otherwise. \end{cases}

For the second case above if d_{3}(y_{6}) ≠ x_{9}, then x_{9} = x′_{4} is a permanent cycle in the Gille-Pardon and Balmer-Walter spectral sequences by Lemma 6.4 (in fact deg(x′_{i}) − deg(x′_{4}) ≠ 1 mod(4)). This is a contradiction to the fact that W^{∗}(G) is a subquotient of H^{∗}(G/T; Sq^{2}), where there are no generator of dim = 4. For the first case, if d_{3}(y_{6}) ≠ x_{9}, then y_{6} = y′_{3} is a permanent cycle, (w(z_{3} = x_{3}x_{5}) = 2 but w(x′_{4} = x_{9}) = 1, and hence there is no differential d(z_{3}) = x′_{4}) and this is also contradicts to H^{∗}(G/T; Sq^{2}).

Thus we see d_{3}(y_{6}) = x_{9}. Then we easily prove d_{2}(x_{15}) = x_{17} by the same reason. Therefore we have Theorem 6.7 for G = E_{6}. (In fact we have showed (∗) in the proof of Theorem 6.7.)

Next we consider the case G = E_{7}. The cohomology is

grH^{∗}(E_{7}; ℤ/2) ≅ grH^{∗}(E_{6}; ℤ/2) ⊗ Λ(y_{10}, y_{18}, x_{27}).

Hence we easily see

H^{∗}(G/T; Sq^{2}) ≅ H^{∗}(E_{6}/T; ℤ/2) ⊗ Λ(y′_{5}, y′_{9}, x′_{13})
≅ Λ(v_{7}, y′_{5}, y′_{9}, z_{3}, z_{15}, x′_{11}, x′_{13})

where y′_{5} = y_{10}, y′_{9} = y_{18}, x′_{13} = x_{27}.

At last we consider the case G = E_{8}. The chomology is

grH^{∗}(E_{8}; ℤ/2) ≅ grH^{∗}(E_{7}; ℤ/2) ⊗ Λ(y_{12}, y_{24}, y_{20}, y_{30}, x_{29})
≅ Λ(y_{i}|i = 6, 12, 24, 10, 20, 18, 30) ⊗ Λ(x_{j}|j = 3, 5, 9, 15, 17, 23, 27, 29).

The Steenrod operation acts as Sq^{2}(y_{i}) = y_{i+2}, Sq^{2}(x_{i}) = x_{i+2} and Sq^{2}(y_{12}y_{10}) = y_{24} (in fact, y_{24} = y_{12}^{2}). Hence we can compute

H^{∗}(G/T; Sq^{2}) ≅ Λ(v_{7}, w_{23}, w_{19}, y′_{15}, z_{3}, z_{15}, x′_{11}, z_{27})

where w_{23} = y_{10}y_{12}y_{24}, y′_{15} = y_{30}, z_{27} = x_{27}x_{29}. Ohsita first computed this homology by using the Sq^{2}-algebra structure of H^{∗}(G/T; ℤ/2) , and of course, our result coincides with him. We can easily show d_{2}(x_{27}) = x_{29} in the Gille-Pardon spectral sequence and see Theorem 6.7.

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Department of Mathematics, Faculty of Education, Ibaraki University, Mito, Ibaraki, Japan

   E-mail address: yagita@mx.ibaraki.ac.jp