ON THE TORSION OF CHOW GROUPS OF TWISTED SPIN-FLAGS

S. BAEK, K. ZAINOULLINE, AND C. ZHONG

Abstract. In the present paper we provide a uniform bound for the annihilators of the torsion of the Chow groups of the variety of Borel subgroups of a strongly inner linear algebraic group of orthogonal type.

1. Introduction

Let X be the variety of Borel subgroups of a semisimple linear algebraic group G over an arbitrary field k. One way to study the geometry of X is to study its Chow groups CH^{d}(X) of algebraic cycles modulo the rational equivalence relation. Simple transfer arguments together with the Bruhat-Tits decomposition imply that the Chow groups CH^{d}(X; ℚ) with rational coeffcients of any variety of Borel subgroups X can be identified with CH^{d}(G/B; ℚ) for some split G. Since CH^{d}(G/B) is a free Abelian group of rank equal to the number of cells of codimension d in G/B, the problem of determining CH^{d}(X) is reduced to determining its torsion part.

The latter seems to be a highly nontrivial question. Only very few partial results are known and most of them concern small codimensions (d ≤ 4) or the dimensionzero cases: for strongly inner groups and d = 2, 3 we refer to [15], [4] and for d = 4 to [1]; for inner groups of type A and d = 2 see [8]; for quadrics and d = 2, 3, 4 see [9], [10]. In particular, in [10] it was shown that the torsion of CH^{4} can be infinitely generated.

In the present paper we provide a uniform bound for the annihilator of the torsion of CH^{d}(X) for any d and strongly inner orthogonal group G. Namely, we prove the following:

1.1. Theorem. Let G be a strongly inner group of an orthogonal type of rank n over a field of characteristic different from 2. Let X be the respective variety of Borel subgroups. Then for all 2 ≤ d ≤ n − 1 the integer

M_{d} = (d − 1)! ∏_{i=2}^{d}(i − 1)! · [i/2]! · 2^{i+1}

annihilates the torsion part of CH^{d}(X).

We would like to stress that our bound M_{d} doesn’t depend on the rank n of G but only on the codimension d. Note also that for groups of types A and C there are no non-split strongly inner forms, so the torsion part of CH^{d}(X) is trivial. Hence, the only interesting case to consider (of groups that form an infinite series) is the orthogonal case.

The only estimate for the annihilator of the torsion of CH^{d}(X) known in the literature is given by the torsion index of G (see [5]). For strongly inner orthogonal groups, i.e. for Spin groups, it was computed in [16, Thm.01] and has the property that t_{G} → ∞ as the rank of G grows.

The paper is organized as follows. In sections 2 and 3 we prove several technical facts (Proposition 2.9 and Corollary 3.4) concerning the ideals of generalized invariants and symmetric functions. These facts are used in section 4 to relate the kernel of the characteristic map with the ideal of invariants (Propositions 4.5 and 4.7). In section 5 we extend the results of the paper [1] by providing a uniform upper bound for all the exponents of the Weyl group action (Proposition 5.6). In the last section we combine the obtained results to prove the main theorem (Theorem 6.1).

2. Divided differences and ideals of invariants

In the present section we provide several basic facts concerning the ring of symmetric polynomials over an arbitrary commutative ring and the associated invariant ideals. We refer to [7] and [11] for details.

2.1. Consider a polynomial ring R = A[e_{1}, …, e_{n}] over a commutative ring A. The symmetric group S_{n} acts on R by permutations of variables {e_{1}, …, e_{n}}. The subring of invariants R^{S_{n}} is a polynomial ring in elementary symmetric functions [17, Thm. 1, 2]

s_{1} = e_{1} + … + e_{n}, s_{2} = ∑_{i<j} e_{i}e_{j}, s_{3} = ∑_{i<j<k} e_{i}e_{j}e_{k}, …, s_{n} = ∏_{i=1}^{n} e_{i}.

Let J = (s_{1}, s_{2}, …, s_{n}) denote the ideal of R generated by symmetric functions. We denote by ϵ: R → A the augmentation map e_{i} ↦ 0. Observe that ϵ restricts to ϵ: R^{S_{n}} → A.

2.2. Following [7, §0] consider divided difference operators Δ_{σ}, σ ∈ S_{n}. Each of them is an A-linear operator Δ_{σ}: R^{(m)} → R^{(m−l(σ))} decreasing the degree m of a homogeneous polynomial in e_{1}, …, e_{n} by the length l(σ) of permutation σ. It is defined as follows:

We set Δ_{1} = id. If m < l(σ), then we set Δ_{σ} = 0. For a (non-trivial) transposition τ = (ij), we set Δ_{τ} (f) = (f − f^{τ})/(e_{i} − e_{j}) for f ∈ R^{(m)}, m ≥ 1. If σ is a product of transpositions, we define Δ_{σ} to be the composite of the respective Δ_{τ}. This doesn’t depend on the choice of a reduced decomposition of σ. Observe that if s is a symmetric function, then

Δ_{σ}(s · f) = s · Δ_{σ}(f).

By definition we have

R^{S_{n}} = {f ∈ R | Δ_{τ} (f) = 0 for all non-trivial transpositions τ}.

2.3. Definition. Following [7, p.239] we define the ideal of generalized invariants I as

I = {f ∈ R | Δ_{σ}(f) = 0 ∀σ ∈ S_{n} with l(σ) = deg(f)}.

2.4. Lemma. We have J ⊆ I.

Proof. Follows from the fact that Δ_{σ}(s · f) = s · Δ_{σ}(f) = 0 for any s · f ∈ J^{(m)} and l(σ) = m. □

2.5. Definition. Following [11, §1] we define the ideal of stable invariants J_{∞} inductively as:

Set R_{1} := R and R_{m+1} := R_{m} ⊗_{R_{m}^{Sn}} A for m ≥ 1, where A is the R_{m}^{S_{n}}-module via the augmentation ϵ. Observe that S_{n} acts on R_{m+1} via the action on R_{m} and there is a canonical S_{n}-equivariant surjection R_{m} → R_{m+1}. Let J_{m} denote the kernel of the composite R_{1} → … → R_{m} → R_{m+1}, i.e. R_{m+1} = R/J_{m}. We then set

J_{∞} := ⋃_{m≥1} J_{m}.

2.6. Remark. The ideal J_{m} can be also defined inductively as follows:

J_{m} = \{ \begin{matrix}(0) & if m = 0, \\ \{f ∈ R | Δ_{σ}(f) ∈ J_{m−1}, ∀σ ∈ S_{n}\} & if m ≥ 1. \end{matrix}

2.7. Lemma. (cf. [11, Lemma 3.2]) We have I ⊆ J_{∞}.

Proof. We show by induction on m that I^{(m)} ⊆ J_{m}.

If m = 1 and f ∈ I^{(1)}, then Δ_{τ} (f) = 0 for all τ ≠ 1, implies that f ∈ R^{S_{n}} ∩ker ϵ. Therefore, f ∈ J_{1}.

Suppose that I^{(m)} ⊆ J_{m}. For m+1 and f ∈ I^{(m+1)}, if Δ_{σ}(f) = 0 for any reduced decomposition σ = τ_{1}τ_{2} … τ_{m+1}, then Δ_{τ_{1}...τ_{m}}Δ_{τ_{m+1}}(f) = 0. So by induction Δ_{τ_{m+1}}(f) ∈ I^{(m)} ⊆ J_{m} and, therefore, Δ_{τ} (f) ∈ J_{m} for all τ ≠ 1. By the remark above, f ∈ J_{m+1}. □

The ideal J_{∞} is universal in the following sense

2.8. Lemma. (cf. [11, Lemma 2.1]) Let J′ ⊂ R be an S_{n}-stable ideal with ϵ(J′) = 0. If (R/J′)^{S_{n}} = A, then J_{∞} ⊆ J′.

Proof. We prove J_{m} ⊂ J′ by induction on m. If m = 1, since (R/J′)^{S_{n}} = A, the compositions R^{S_{n}} ↪ R → R/J′ and R^{S_{n}} →^{ϵ} A ↪ R/J′ coincide, hence, there is a map R/J_{1} = R ⊗_{R^{Sn}} A → R/J′, which shows that J_{1} ⊂ J′.

Now assume that J_{m} ⊂ J′. Repeating the above arguments after replacing R (resp. J′) by R_{m} = R/J_{m} (resp. by the ideal J′_{m} = (J′) in R_{m}), we see that J_{m+1} ⊂ J. This finishes the proof. □

2.9. Proposition. We have J = I = J_{∞} in the polynomial ring R = A[e_{1}, …, e_{n}].

Proof. Since R^{S_{n}} = A ⊗ ℤ[e_{1}, …, e_{n}]^{S_{n}}, (R/J)^{S_{n}} = A. Therefore, by Lemma 2.8, J_{∞} ⊆ J. The proposition then follows by combining Lemmas 2.4 and 2.7. □

3. Elementary symmetric functions and power sums

In the present section we prove several technical lemmas which will be used in the subsequent section.

Let α = (α_{1}, ..., α_{m}) denote a partition with 1 ≤ α_{1} ≤ α_{2} ≤ … ≤ α_{m} of an integer |α| = α_{1} + … + α_{m}. We set |(0)| = 0. Let s_{α} = ∏_{i=1}^{m} s_{α_{i}} denote the product of respective elementary symmetric functions (here s_{0} = 1).

3.1. Lemma. Consider a homogeneous polynomial of degree d ≤ n with integer coefficients

P_{m} = ∑_{{α | d−m≤|α|≤d}} f^{α}s_{α}, where 0 ≤ m < d and deg(f^{α}) = d − |α|.

If M | P_{m} for some positive integer M, then for each α there exists a homogeneous polynomial v^{α} of degree d − |α| such that

∑_{{α | d−m≤|α|≤d}} v^{α}s_{α} = 0 and M | (f^{α} + v^{α}).

Proof. We proceed by induction on m ≥ 0.

If m = 0, then P_{0} = ∑_{|α|=d} f^{α}s_{α}, f^{α} ∈ ℤ. If M | P_{0}, then M | f^{α} for each α as {s_{α}}_{|α|=d} are linearly independent in A[e_{1}, …, e_{n}] for A = ℤ/Mℤ. So we can take v^{α} = 0 for each α.

Assume it is true for m − 1, m ≥ 1. Now we prove it for m. We use β to denote those partitions α with |α| = d − m. Observe that deg(f^{β}) = m. Apply to P_{m} a divided difference operator Δ of length m, we obtain:

M | Δ(P_{m}) = ∑_{β} Δ(f^{β})s_{β}, where Δ(f^{β}) ∈ ℤ.

Similar to m = 0 case this implies that M | Δ(f^{β}) for every Δ of length m. By Definition 2.3 this means that f^{β} ∈ I, where I is the ideal of generalized invariants for A = ℤ/Mℤ. By Proposition 2.9 we have J = I and, therefore,

f^{β} ≡ ∑_{j=1}^{m} g_{m−j}^{β}s_{j} mod M for some polynomials g_{m−j}^{β} of degree m − j.

Plugging it into the original expression for P_{m}, we obtain

P_{m} ≡ ∑_{{α|d−m+1≤|α|≤d}}(f^{α} + g_{d−|α|}^{β(α)})s_{α} ≡ 0 mod M,

where β(α) is the unique β such that s_{α} = s_{|α|−|β(α)|}s_{β(α)}. By induction, for each α such that d − m + 1 ≤ |α| there exists a polynomial v^{α} such that

∑_{{α|d−m+1≤|α|≤d}} v^{α}s_{α} = 0 and M | (f^{α} + g_{d−|α|}^{β(α)} + v^{α}).

Now we set ṽ^{β} = − ∑_{j=1}^{m} g_{m−j}^{β}s_{j} and ṽ^{α} = g_{d−|α|}^{β(α)} + v^{α} for |α| ≥ d − m + 1. Then ∑_{d−m≤|α|≤d} ṽ^{α}s_{α} = 0. So these ṽ^{α} satisfy the condition of the lemma. This finishes the proof. □

3.2. Let q_{i} = e_{1}^{i} + e_{2}^{i} + … + e_{n}^{i}, i ≥ 1 denote the power sum symmetric function. Given a partition α = (α_{1}, …, α_{m}), let q_{α} = ∏_{i=1}^{m} q_{α_{i}} denote the product of respective power sum functions.

According to [12, Ch I. 2.11] the elementary symmetric function s_{i} can be written

in terms of q_{j}, j ≤ m as s_{i} = \frac{1}{i!} ∑_{|α|=i} a_{α}q_{α}, a_{α} ∈ ℤ. Since i! = max ; ∑ i_{j}=i\{i_{j}!\}, we

have

s_{α} = \frac{1}{|α|!} ∑_{\{β, |β|=|α|\}} a_{α,β}q_{β} for some a_{α,β} ∈ ℤ.

Multiplying by the respective denominators we obtain the following version of Lemma 3.1 for power sum functions:

3.3. Corollary. Assume that 1 ≤ d ≤ n and d! | M. Consider a homogeneous polynomial P = ∑_{i=1}^{d} f_{d−i}q_{i} of degree d with integer coefficients (deg(f_{d−i}) = d−i).

If M | P, then there exist f̂_{d−i}, i = 1 … d such that

∑_{i=1}^{d} f̂_{d−i}q_{i} = P and \frac{M}{d!} | f̂_{d−i}.

Proof. Let q_{i} = ∑_{|α|=i} c_{α}s_{α}, c_{α} ∈ ℤ, so we get M | P = ∑_{|α|≥1} c_{α}f_{d−|α|}s_{α}. By Lemma 3.1 there exist v^{α} with P = ∑(c_{α}f_{d−|α|} + v^{α})s_{α} and (c_{α}f_{d−|α|} + v^{α}) all divisible by M. Expressing s_{α} in terms of q_{β} using the formula from 3.2 we obtain

P = \frac{1}{d!} ∑_{α}(c_{α}f_{d−|α|} + v^{α}) ∑_{β} a_{α,β}q_{β} = \frac{1}{d!} ∑_{i=1}^{d} f̃_{d−i}q_{i} for some f̃_{d−i}.

Now all these f̃_{d−i} are integral polynomials in c_{α}f_{d−|α|}+v^{α} and, hence, are divisible

by M. Therefore, f̂_{d−i} =^{def} \frac{1}{d!} f̃_{d−i} is divisible by \frac{M}{d!} . □

Restricting to power sums of even degree only we obtain

3.4. Corollary. Assume that 1 ≤ d ≤ 2n + 1 and let [d/2]! | M. Consider a homogeneous polynomial P = ∑_{i=1}^{[d/2]} f_{d−2i}q_{2i} of degree d with integer coefficients.

If M | P, then there exist f̂_{d−2i}, i = 1, …, [d/2] such that

∑_{i=1}^{[d/2]} f̂_{d−2i}q_{2i} = P and \frac{M}{[d/2]!} | f̂_{d−2i}.

Proof. For each i we express f_{d−2i} as a linear combination

f_{d−2i} = ∑_{δ} e^{δ}f_{d−2i}^{δ},

where δ = (δ_{1}, …, δ_{n}) with δ_{i} = 0, 1, e^{δ} = ∏_{i=1}^{n} e_{i}^{δ_{i}} and f_{d−2i}^{δ} is a linear combination of even monomials e_{1}^{2i_{1}}e_{2}^{2i_{2}} … e_{n}^{2i_{n}} only. Denote |δ| = ∑ δ_{j}. Collecting the terms with e^{δ} we obtain

P = ∑_{δ} e^{δ} ∑_{i=1}^{[d/2]} f_{d−2i}^{δ}q_{2i} ≡ 0 mod M.

This implies that M | ∑_{i=1}^{[d/2]} f_{d−2i}^{δ}q_{2i} for each δ. We apply Corollary 3.3 to the polynomial P_{δ} = ∑_{i=1}^{[d/2]} f_{d−2i}^{δ}q_{2i} viewed as a polynomial in variables e_{j}^{2} of degree d_{δ} = \frac{d−|δ|}{2}. We obtain polynomials f̂_{d−2i}^{δ} divisible by \frac{M}{d_{δ}!} such that ∑_{i=1}^{[d/2]} f_{d−2i}^{δ}q_{2i} =

∑_{i=1}^{[d/2]} f̂_{d−2i}^{δ}q_{2i}.

We then set

f̂_{d−2i} = ∑_{δ} e^{δ}f̂_{d−2i}^{δ}.

Since d_{δ}! | [d/2]! for all δ, the proof is finished. □

4. Invariants and characteristic map

In the present section we investigate the relationships between the kernel of the characteristic map and the ideal of invariants. We refer to [2] for basic definitions and details.

4.1. Consider a crystallographic root system of Dynkin type 𝔇 with the weight lattice Λ. Let G be the associated split simple simply-connected linear algebraic group with a maximal torus T and a Borel subgroup B ⊃ T. Observe that Λ can be identified with the group of characters of T with a basis given by the fundamental weights ω_{1}, …, ω_{n}.

Consider the variety G/B of Borel subgroups of G (conjugate to B). To every character λ ∈ Λ we may associate the line bundle L(λ) over G/B. It induces the ring homomorphism from the symmetric algebra S^{∗}(Λ) to the Chow ring of G/B called the characteristic map [2, §8]

c_{a}: S^{∗}(Λ) → CH^{∗}(G/B), λ ↦ c_{1}(L(λ)).

4.2. The Weyl group W of G acts on the weight lattice Λ by means of simple reflections and, hence, on S^{∗}(Λ). Let I_{a}^{W} denote the ideal generated by non-constant W -invariants. According to [2, §4 Cor. 2] the kernel of the characteristic map ker c_{a} is generated by elements of S^{∗}(Λ) such that their multiples are in I_{a}^{W}. Moreover, for each homogeneous degree d there exists an integer b_{d} such that the prime decomposition of b_{d} consists only of torsion primes of G and

b_{d} · (ker c_{a})^{(d)} ⊆ (I_{a}^{W})^{(d)}.

The ideal (ker c_{a})⊗ℤ[\frac{1}{t_{G}}] = I_{a}^{W} ⊗ℤ[\frac{1}{t_{G}}], where t_{G} is the torsion index of G, is freely generated by the basic polynomial invariants which are homogeneous polynomials in fundamental weights.

The purpose of the present section is to get an upper bound for the integers b_{d}.

4.3. Example. For the type 𝔇 = A_{n} there are no torsion primes (the torsion index is 1). This implies that b_{d} = 1 for all d and the characteristic map c_{a} is surjective with the kernel ker c_{a} = I_{a}^{W}.

4.4. Example. According to [6] the basic polynomial invariants for the root system of type 𝔇 = B_{n} are given by even power sums

q_{2i} = ∑_{j=1}^{n} e_{j}^{2i}, 1 ≤ i ≤ n,

where e_{1} = ω_{1}, e_{j} = ω_{j} − ω_{j−1} for 2 ≤ j < n and e_{n} = 2ω_{n} − ω_{n−1}. Therefore,

q_{2i} = ω_{1}^{2i} + (ω_{2} − ω_{1})^{2i} + ... + (ω_{n−1} − ω_{n−2})^{2i} + (2ω_{n} − ω_{n−1})^{2i}.

The basic polynomial invariants for type D_{n} are given by q_{2i}, 1 ≤ i ≤ n − 1 and p_{n} = e_{1}e_{2} … e_{n}, where e_{1} = ω_{1}, e_{j} = ω_{j} − ω_{j−1} for 2 ≤ j ≤ n − 1 and e_{n−1} = ω_{n} − ω_{n−1}, e_{n} = ω_{n} + ω_{n−1} − ω_{n−2}.

From now on we assume that 𝔇 = B_{n}. In this case the torsion index of G is a power of 2.

4.5. Proposition. Let 𝔇 = B_{n} and let 1 ≤ d ≤ 2n + 1. Then 2^{d}[d/2]! · (ker c_{a})^{(d)} ⊂ I_{a}^{W}, i.e. b_{d} | 2^{d}[d/2]!.

Proof. Since (ker c_{a}) ⊗ ℤ[\frac{1}{2}] = I_{a}^{W} ⊗ ℤ[\frac{1}{2}] is generated by q_{2i}, i = 1, ..., n, given a polynomial f ∈ (ker c_{a})^{(d)}, we can write it as

2^{r}f = ∑_{i=1}^{[d/2]} f_{d−2i}q_{2i} ∈ I_{a}^{W} , for some f_{d−2i} ∈ ℤ[ω_{1}, ..., ω_{n}] and r ≥ 0.

Suppose r is the smallest such integer. To finish the proof it suffices to show that r ≤ v_{2}([d/2]!) + d, where v_{2} denotes the 2-adic valuation.

Assume the contrary, i.e. that r ≥ v_{2}([d/2]!) + d + 1. Expressing ω_{j}’s in terms of e_{j}’s, we obtain f = \frac{1}{2^{d}} f̃ and f_{d−2i} = \frac{1}{2^{d−2i}} f̃_{d−2i} for some f̃, f̃_{d−2i} ∈ ℤ[e_{1}, ..., e_{n}]. So that

M = 2^{d+1}[d/2]! | 2^{r}s · f̃ = ∑_{i=1}^{[d/2]}(2^{2i}sf̃_{d−2i}) · q_{2i}, where s = \frac{[d/2]!}{2^{v_{2}([d/2]!)}} .

By Corollary 3.4, there exists h̃_{d−2i} ∈ ℤ[e_{1}, ..., e_{n}] divisible by \frac{M}{[d/2]!} = 2^{d+1} such

that 2^{r}s · f̃ = ∑_{i=1}^{[d/2]} h̃_{d−2i}q_{2i}. Expressing e_{j}’s in terms of ω_{j}’s back, we obtain 2^{d}2^{r}s · f = ∑_{i=1}^{[d/2]} h̃_{d−2i}q_{2i}, which implies

2^{r−1}f = ∑_{i=1}^{[d/2]} (\frac{1}{2^{d+1}} h̃_{d−2i} − \frac{s−1}{2}f_{d−2i}) · q_{2i}.

Since h̃_{d−2i} are divisible by 2^{d+1}, we have \frac{1}{2^{d+1}} h̃_{d−2i} ∈ ℤ[ω_{1}, ..., ω_{n}]. This contradicts to the minimality assumption on r. □

Assume now that 𝔇 = D_{n}. In this case we have an additional basic polynomial invarant p_{n} in degree n, however, this doesn’t change the situation much in view of the following slight modifications of Corollaries 3.3 and 3.4:

4.6. Lemma. Assume d ≥ n and n! | M. If M | ∑_{i=1}^{n} f_{d−i}q_{i} + g_{d−n}p_{n}, then there exists v_{d−i} and u_{d−n} such that ∑_{i=1}^{n} v_{d−i}q_{i} + u_{d−n}p_{n} = 0 and \frac{M}{n!} | g.c.d.(f_{d−i} + v_{d−i}, g_{d−n} + u_{d−n}).

If M | ∑_{i=1}^{n} f_{d−2i}q_{2i}+g_{d−n}p_{n}, then we can find v_{d−2i}, u_{d−n} such that ∑_{i=1}^{n} v_{d−2i}q_{2i}+ u_{d−n}p_{n} = 0 and \frac{M}{n!} | g.c.d.(f_{d−2i} + v_{d−2i}, g_{d−n} + u_{d−n}).

Proof. The proof of the first statement follows by the same arguments as the proof of Corollary 3.3. As for the second, multiplying by p_{n} we obtain

M | ∑_{i=1}^{n} f̃_{d+n−2i}q_{2i} + g_{d−n}p_{n}^{2}.

Following the proof of 3.4 we can rewrite this as

M | ∑_{δ} e^{δ}(∑_{i=1}^{n} f̃_{d+n−2i}^{δ}q_{2i} + g_{d−n}^{δ}p_{n}^{2})

and reduce it to the first statement (replacing e′_{j} = e_{j}^{2}). □

4.7. Proposition. Let 𝔇 = D_{n}. If d < n, then 2^{d}[d/2]!·(ker c_{a})^{(d)} ⊂ I_{a}^{W}. If d ≥ n, then

2^{d}n! · (ker c_{a})^{(d)} ⊂ I_{a}^{W} .

Proof. If d < n, then it is similar to the B_{n}-case. If d ≥ n, consider the equation

2^{r}f = ∑_{i=1}^{n} f_{d−2i}q_{2i} + g_{d−n}p_{n} ∈ I_{a}^{W} .

Following the proof of Proposition 4.5 and using Lemma 4.6 we show that the smallest 2^{r} satisfying the equation will divide 2^{d}n!. □

5. Exponents of types B_{n} and D_{n}

Following the notation of [1] let I_{m} := ker(ℤ[Λ] → ℤ) and I_{a} := ker(S^{∗}(Λ) → ℤ) be the augmentation ideals, where ℤ[Λ] → ℤ (respectively, S^{∗}(Λ) → ℤ) is the map sending e^{λ} to 1 (respectively, any element of positive degree to 0). For any d ≥ 0, we consider the ring homomorphism

ϕ^{(d)} : ℤ[Λ] → ℤ[Λ]/I_{m}^{d+1} → S^{∗}(Λ)/I_{a}^{d+1} → S^{d}(Λ),

where the first and the last maps are projections and the middle map sends e^{∑_{i=1}^{n} a_{i}ω_{i}} to ∏_{i=1}^{n}(1 − ω_{i})^{−a_{i}}. The dth-exponent of a root system (denoted by τ_{d}), as introduced in [1], is the gcd of all nonnegative integers N_{d} satisfying

N_{d} · (I_{a}^{W})^{(d)} ⊆ ϕ^{(d)}(I_{m}^{W}),

where I_{m}^{W} := ⟨ℤ[Λ]^{W} ∩ I_{m}⟩ (respectively, I_{a}^{W} := ⟨S^{∗}(Λ)^{W} ∩ I_{a}⟩) denotes the W - invariant augmentation ideal of ℤ[Λ] (respectively, S^{∗}(Λ)). For any d ≤ 4, it was shown that the dth-exponent divides the Dynkin index in [1].

In this section we show that all the remaining exponents of types B_{n} and D_{n} divide the Dynkin index 2.

5.1. For any λ ∈ Λ, we denote by W (λ) the W -orbit of λ. For any finite set S of weights, we denote −S the set of opposite weights. By the action of Weyl groups of types B_{n} and D_{n}, one has the following decomposition of W -orbits: if 𝔇 = B_{n} (respectively, 𝔇 = D_{n}), then for any 1 ≤ k ≤ n − 1 (respectively, 1 ≤ k ≤ n − 2)

(5.0.1)  W (ω_{k}) = W_{+}(ω_{k}) ∪ −W_{+}(ω_{k}),

where W_{+}(ω_{k}) = {e_{i_{1}} ± ⋯ ± e_{i_{k}}}_{i_{1}<···<i_{k}}. If n is even, then the W -orbits of the last two fundamental weights of D_{n} are given by

(5.0.2) W (ω_{n−1}) = W_{+}(ω_{n−1}) ∪ −W_{+}(ω_{n−1}) and W (ω_{n}) = W_{+}(ω_{n}) ∪ −W_{+}(ω_{n}),

where W_{+}(ω_{n−1}) (respectively, W_{+}(ω_{n})) is the subset of W (ω_{n−1}) (respectively, W (ω_{n})) containing elements of the positive sign of e_{1}.

For any λ = ∑_{i=1}^{n} a_{i}ω_{i} ∈ Λ and any integer m ≥ 0, we set λ(m) = ∑_{i=1}^{n} a_{i}ω_{i}^{m}. We shall need the following lemma:

5.2. Lemma. Let p be a positive integer and m_{1}, ⋯, m_{p} nonnegative integers.

(i) If 𝔇 = B_{n} (respectively, D_{n}), then for odd p and any 1 ≤ k ≤ n − 1 (respectively, any 1 ≤ k ≤ n − 2), we have

∑_{λ∈W (ω_{k})} λ(m_{1}) ⋯ λ(m_{p}) = 0.

(ii) Let 𝔇 = D_{n}. Then we have

∑_{λ∈W (ω_{n})} λ(m_{1}) ⋯ λ(m_{p}) = ∑_{λ∈W (ω_{n−1})} λ(m_{1}) ⋯ λ(m_{p}) for odd n and even p,

∑_{λ∈W (ω_{n})} λ(m_{1}) ⋯ λ(m_{p}) = ∑_{λ∈W (ω_{n−1})} λ(m_{1}) ⋯ λ(m_{p}) = 0 for odd p < n.

Proof. (i) It follows from (5.0.1) that

∑_{λ∈W (ω_{k})} λ(m_{1}) ⋯ λ(m_{p}) = ∑_{λ∈W_{+}(ω_{k})} λ(m_{1}) ⋯ λ(m_{p}) + ∑_{λ∈−W_{+}(ω_{k})} λ(m_{1}) ⋯ λ(m_{p})

= ∑_{λ∈W_{+}(ω_{k})} λ(m_{1}) ⋯ λ(m_{p}) − ∑_{λ∈W_{+}(ω_{k})} λ(m_{1}) ⋯ λ(m_{p}) = 0.

(ii) If n is odd, then we have W (ω_{n}) = −W (ω_{n−1}). Hence, the result immediately follows from the assumption that p is even. If n is even, then the result follows from (5.0.2) by the same argument as in the proof of (i). □

Let p be an even integer and q ≥ 2 an integer. For any nonnegative integers m_{1}, ⋯, m_{p}, we define

Λ(p, q)(m_{1}, ⋯, m_{p}) := ∑ λ_{j_{1}}(m_{1}) ⋯ λ_{j_{p}}(m_{p}),

where the sum ranges over all different λ_{i_{1}}, ⋯, λ_{i_{q}} ∈ W_{+}(ω_{1}) and all λ_{i_{1}}, ⋯, λ_{i_{p}} ∈ {λ_{i_{1}}, ⋯, λ_{i_{q}}} such that the numbers of λ_{i_{1}}, ⋯, λ_{i_{q}} appearing in λ_{i_{1}}, ⋯, λ_{i_{p}} are all nonnegative even solutions of x_{1} + ⋯ + x_{q} = p. If p < 2q, then we set Λ(p, q)(m_{1}, ⋯, m_{p}) = 0. We simply write Λ(p, q) for Λ(p, q)(m_{1}, ⋯, m_{p}). For instance, Λ(4, 2) is the sum of λ_{j_{1}}(m_{1})λ_{j_{2}}(m_{2})λ_{j_{3}}(m_{3})λ_{j_{4}}(m_{4}) for all j_{1}, j_{2}, j_{3}, j_{4} ∈ {i, j} and all 1 ≤ i ≠ j ≤ n such that two i’s and two j’s appear in j_{1}, j_{2}, j_{3}, j_{4}.

5.3. Lemma. If 𝔇 = B_{n} (respectively, D_{n}), then for any 2 ≤ k ≤ n−1 (respectively, 2 ≤ k ≤ n − 2), any even p, and any nonnegative integers m_{1}, ⋯ m_{p} we have

∑_{W (ω_{k})} λ(m_{1}) ⋯ λ(m_{p}) = 2^{k−1}\binom{n − 1}{k − 1} ∑_{W (ω_{1})} λ(m_{1}) ⋯ λ(m_{p})+∑_{j=2}^{k} 2^{k}\binom{n − j}{k − j}Λ(p, j).

Proof. Let L be the LHS of the above equation. For any λ ∈ W (ω_{1}), there are 2^{k}\binom{n−1}{k−1} choices of the element containing λ in W (ω_{k}), thus we have the term 2^{k−1}\binom{n−1}{k−1} ∑_{W (ω_{1})} λ(m_{1}) ⋯ λ(m_{p}) in L.

If an element λ ∈ W (ω_{1}) appears odd times in a term λ_{i_{1}}(m_{1}) ⋯ λ_{i_{p}}(m_{p}) of L, where λ_{i_{1}}, ⋯, λ_{i_{p}} ∈ W (ω_{1}), then by the action of Weyl group this term vanishes in L. Hence, the remaining terms in L are linear combinations of Λ(p, j) for all 2 ≤ j ≤ k such that p ≥ 2k. As each term Λ(p, j) appears 2^{k}\binom{n−j}{k−j} times in ∑_{W (ω_{k})} λ(m_{1}) ⋯ λ(m_{p}), the result follows. □

For any λ ∈ Λ, we denote by ρ(λ) the sum of all elements e^{μ} ∈ ℤ[Λ] over all elements μ of W (λ). By the recursive formulas in [1, Section 1], we can let d! · ϕ^{(d)}(e^{λ}) = λ^{d} + Q_{d} for any d ≥ 1, where Q_{d} is the sum of remaining terms in d! · ϕ^{(d)}(e^{λ}). Hence, for any fundamental weight ω_{k} we have

(5.0.3)  d! · ϕ^{(d)}(ρ(ω_{k})) = ∑_{W (ω_{k})} λ^{d} + ∑_{W (ω_{k})} Q_{d}.

We view d! · ϕ^{(d)}(e^{λ}) as a polynomial in variables λ, λ(m_{1}), ⋯, λ(m_{j}) for some nonnegative integers m_{1}, ⋯, m_{j}. Let T_{d} be the sum of monomials in Q_{d} whose degrees are even.

If 𝔇 = B_{n} (respectively, D_{n}), then by Lemma 5.2(i) the equation (5.0.3) reduces to

(5.0.4)  d! · ϕ^{(d)}(ρ(ω_{k})) = ∑_{W (ω_{k})} λ^{d} + ∑_{W (ω_{k})} T_{d}

for any 1 ≤ k ≤ n − 1 (respectively 1 ≤ k ≤ n − 2).

Given p and q, we define Ω(p, q) := ∑ Λ(p, q)(m_{1}, ⋯, m_{p}), where the sum ranges over all m_{1}, ⋯, m_{p} which appear in all monomials of T_{d}.

5.4. Example. If 𝔇 = B_{n} (n ≥ 4) or D_{n} (n ≥ 5) and d = 6, then by (5.0.4) and Lemma 5.3 we have

6!ϕ^{(6)}(ρ(ω_{1})) = ∑_{W (ω_{1})} λ^{6} + ∑_{W (ω_{1})} T_{6},
6!ϕ^{(6)}(ρ(ω_{2})) = ∑_{W (ω_{2})} λ^{6} + 2(n − 1) ∑_{W (ω_{1})} T_{6} + 4Ω(4, 2),

6!ϕ^{(6)}(ρ(ω_{3})) = ∑_{W (ω_{3})} λ^{6} + 4\binom{n − 1}{2} ∑_{W (ω_{1})} T_{6} + 8(n − 2)Ω(4, 2),

which implies that

ϕ^{(6)}(ρ(ω_{3})) − 2(n − 2)ϕ^{(6)}(ρ(ω_{2})) + 2(n − 1)(n − 2)ϕ^{(6)}(ρ(ω_{1})) = ∑_{i<j<k} e_{i}^{2}e_{j}^{2}e_{k}^{2}.

5.5. Lemma. Let 𝔇 = D_{n} and let 1 ≤ p ≤ n − 1 and m_{1}, ⋯, m_{p} be nonnegative integers. Then we have

∑_{W (ω_{n})} λ^{n} − ∑_{W (ω_{n−1})} λ^{n} = n!e_{1} ⋯ e_{n} and

∑_{W (ω_{n})} λ(m_{1}) ⋯ λ(m_{p}) = ∑_{W (ω_{n−1})} λ(m_{1}) ⋯ λ(m_{p}).

Proof. Let L be the LHS of the upper equation. First, assume that n ≥ 4 is even. We show that

∑_{W_{+}(ω_{n})} λ^{n} − ∑_{W_{+}(ω_{n−1})} λ^{n} = (n!/2)e_{1} ⋯ e_{n}.

As |W_{+}(ω_{n})| = |W_{+}(ω_{n−1})| = 2^{n−2}, we have

(n!/2^{n})2^{n−2}e_{1} ⋯ e_{n} − (−(n!/2^{n})2^{n−2}e_{1} ⋯ e_{n}) = (n!/2)e_{1} ⋯ e_{n}

in L. If one of the exponents i_{1}, ⋯, i_{n} in e_{1}^{i_{1}} ⋯ e_{n}^{i_{n}} (except the case i_{1} = ⋯ = i_{n} = 1) is odd, then from the orbits W_{+}(ω_{n}) and W_{+}(ω_{n−1}) this monomial vanishes in each sum of ∑_{W_{+}(ω_{n})} λ^{n} − ∑_{W_{+}(ω_{n−1})} λ^{n}. Otherwise, the terms 2^{n−2} ∑_{j=1}^{n} e_{j}^{n}, Λ(n, 2) ⋯, Λ(n, n/2) with m_{1} = ⋯ = m_{n} = 1 are in both ∑_{W_{+}(ω_{n})} λ^{n} and ∑_{W_{+}(ω_{n−1})} λ^{n}.

Now, we assume that n ≥ 4 is odd. As |W (ω_{n})| = |W (ω_{n−1})| = 2^{n−1}, we have

(n!/2^{n})2^{n−1}e_{1} ⋯ e_{n} − (−(n!/2^{n})2^{n−1}e_{1} ⋯ e_{n}) = n!e_{1} ⋯ e_{n}

in ∑_{W (ω_{n})} λ^{n} − ∑_{W (ω_{n−1})} λ^{n}. By the same argument, if one of the exponents i_{1}, ⋯, i_{n} in e_{1}^{i_{1}} ⋯ e_{n}^{i_{n}} (except the case i_{1} = ⋯ = i_{n} = 1) is odd, then this monomial vanishes in each sum of L. This completes the proof of the first equation.

By Lemma 5.2(ii), it is enough to consider the case where both n and p are even. Let A = ∑_{W_{+}(ω_{n})} λ(m_{1}) ⋯ λ(m_{p}) and B = ∑_{W_{+}(ω_{n−1})} λ(m_{1}) ⋯ λ(m_{p}). For any p and any n ≥ p + 2, we have C := 2^{n−2}(∑_{W_{+}(ω_{1})} λ(m_{1}) ⋯ λ(m_{p})) in both A and B. By the action of Weyl group, any term λ_{i_{1}}(m_{1}) ⋯ λ_{i_{p}}(m_{p}) with λ_{i_{j}} ∈ W (ω_{1}) appearing odd times in either A − C or B − C vanishes. As each term of Λ(p, 2), ⋯, Λ(p, p/2) appears in both A and B, this completes the proof. □

5.6. Proposition. If 𝔇 = B_{n} (respectively, D_{n}), then for any d ≥ 3 and any n ≥ [d/2] + 1 (respectively, n ≥ [d/2] + 2) the exponent τ_{d} divides the Dynkin index τ_{2} = 2.

Proof. As B_{2} = C_{2} and D_{3} = A_{3}, we have 1 = τ_{3} | 2 by [1, Theorem 5.4]. If 𝔇 = D_{n} for any n ≥ 4, then by Lemma 5.5 we have

p_{n} = ϕ^{(n)}(ρ(ω_{n})) − ϕ^{(n)}(ρ(ω_{n−1})),

which implies that the invariant p_{n} is in the ideal generated by the image of ϕ^{(n)}. As there are no invariants of odd degree except p_{n}, we have

τ_{2d+1} | τ_{2d}

for all d ≥ 1. Therefore, it suffices to show that τ_{2d} | τ_{2} for any d ≥ 2.

By Lemma 5.3 together with the same argument as in Example 5.4 we have

(5.0.5)  ϕ^{(2d)}(ρ(ω_{d})) + ∑_{j=1}^{d−1} a_{j}ϕ^{(2d)}(ρ(ω_{d−j})) = ∑_{j_{1}<···<j_{d}} e_{j_{1}}^{2} ⋯ e_{j_{d}}^{2},

where the integers a_{1}, ⋯, a_{d−1} satisfy

( ∑_{j=k}^{d−2} 2^{j+1}\binom{n − 1 − k}{j − k}a_{j+1}) + 2^{d}\binom{n − 1 − k}{d − 1 − k} = 0,

for 0 ≤ k ≤ d − 2. Let r_{d} be the RHS of (5.0.5). Then this equation implies that r_{d} is in the image of ϕ^{(2d)}.

We show that the invariant q_{2d} is in the ideal ϕ^{(2d)}(I_{m}^{W}) for any d ≥ 2. We proceed by induction on d. By [1, Lemma 5.3], the case d = 2 is obvious. By Newton’s identities we have

(5.0.6)  (−1)^{d−1}q_{2d} = dr_{d} − ∑_{j=1}^{d−1}(−1)^{j−1}r_{d−j}q_{2j}.

By the induction hypothesis, the sum of (5.0.6) is in ϕ^{(2d)}(I_{m}^{W}). Hence, q_{2d} is in ϕ^{(2d)}(I_{m}^{W}). □

6. Annihilators of torsion of twisted spin-flags

In the present section we apply Propositions 4.5, 4.7, and 5.6 to prove the main result of the paper:

6.1. Theorem. Let G be a split simple simply-connected linear algebraic group of Dynkin type B_{n} (n ≥ 3) or D_{n} (n ≥ 4) over a field k of characteristic different from 2, i.e. a Spin group. Let X = _{ξ}G/B be a twisted form of the variety of Borel subgroups of G by means of a cocycle ξ ∈ Z^{1}(k, G).

If G is of type B_{n} (resp. of type D_{n}), then for all 2 ≤ d ≤ 2n − 1 (resp. 2 ≤ d ≤ n − 1), the integer M_{d} = (d − 1)! ∏_{i=2}^{d}(i − 1)! · [i/2]! · 2^{i+1} annihilates the torsion part of CH^{d}(X), i.e.

M_{d} · T ors CH^{d}(X) = 0.

6.2. Remark. Observe that since 2 is the only torsion prime of G we can replace the integer M_{d} by its 2-primary part. Note also that the integer M_{d} depends only on the codimension d but not on the rank n of G.

Proof. We follow the arguments of section 6 of [1]. Following the proof of [1, Thm. 6.5] and using Propositions 4.5, 4.7, and 5.6, we obtain that the integer m_{d} = (d − 1)! · [d/2]! · 2^{d+1} annihilates the torsion of γ^{d}(G/B)/γ^{d+1}(G/B).

By [13, Thm.2.2.(2)] the restriction map K_{0}(X) → K_{0}(G/B) on Grothendieck’s K_{0} is an isomorphism (here we identify K_{0}(G/B) with the K_{0}(X ×_{k} k̄) over the algebraic closure k̄). Since the characteristic classes commute with restrictions, this induces an isomorphism between the respective γ-filtrations on X and on G/B and, hence, between the respective quotients (see [1, p.12])

γ^{d}(X)/γ^{d+1}(X) ≃ γ^{d}(G/B)/γ^{d+1}(G/B) for every d ≥ 0.

This implies that m_{d} annihilates the torsion of γ^{d}(X)/γ^{d+1}(X). Finally, by the proof of [1, Cor.6.8] we obtain that the torsion of CH^{d}(X) is annihilated by the product M_{d} = (d − 1)! ∏_{i=2}^{d} m_{i}. □

6.3. Remark. Using the motivic decomposition of [14] one immediately extends this result to any generically split twisted form X = _{ξ}G/P, where P is a parabolic subgroup of G. In particular, it holds for any maximal orthogonal Grassmannian of a quadratic form with trivial discriminant and trivial Clifford invariant.

Acknowledgments. The first and the third authors have been partially supported from the Fields Institute and the NSERC DAS grant of the second author. The second author has been supported by the NSERC Discovery grant 385795-2010, NSERC Accelerator Supplement grant 396100-2010 and the Early Researcher Award (Ontario).

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Sanghoon Baek, Department of Mathematics and Statistics, University of Ottawa

E-mail address: baek.sanghoon@gmail.com

Kirill Zainoulline, Department of Mathematics and Statistics, University of Ottawa

E-mail address: kirill@uottawa.ca

Changlong Zhong, Department of Mathematics and Statistics, University of Ottawa

E-mail address: zhongusc@gmail.com