ON THE ALGEBRAIC K-THEORY OF SOME HOMOGENEOUS VARIETIES.

ALEXEY ANANYEVSKIY

Abstract. In this paper the K-theory of the inner twisted forms of the homogeneous varieties G/H with the connected reductive algebraic groups H ⊂ G of the same rank is computed. We provide an explicit isomorphism with the K-theory of certain central simple algebras associated to the considered variety, as a consequence one has that K_{0}(G/H) is a free abelian group of rank [W (G) : W (H)]. The result is used for computing the K-theory of some affine homogeneous varieties including the octonionic projective plane and the quaternionic projective spaces.

1. Introduction.

It is known that K-theory of homogeneous projective varieties could be expressed by means of K-theory of central simple algebras. The most fundamental result concerns the case of projective space and states that there is an isomorphism

\xymatrix{K_{∗}(k)[t]/t^{n+1} \ar[r]^{∼} & K_{∗}(ℙ^{n}).}

The case of Severi-Brauer varieties was treated by Quillen [Q, �8], and one has

\xymatrix{⨁_{i=0}^{n} K_{∗}(A^{⊗i}) \ar[r]^{∼} & K_{∗}(ℙ_{γ}^{n}),}

where A is a central simple algebra defined by the cocycle γ. Swan [Sw] computed the K-theory of a smooth projective quadric and showed that there is an analogous isomorphism involving some Clifford algebras. It was shown by Panin [P1] that one has an analogous isomorphism for every homogeneous projective variety and one can express its K-theory in terms of the K-theory of the certain separable algebras. In the present paper we provide a unified approach to the K-theory of the homogeneous varieties and compute it for the inner forms of G/H with the connected reductive H ⊂ G of the same rank. The main result is the theorem 4 which claims that there is an isomorphism

\xymatrix{K_{∗}((G/H)_{γ}) \ar[r]^{∼} & ⨁_{i=1}^{r} K_{∗}(A(λ_{i})_{γ}),}

with r = [W (G) : W (H)] and the separable algebras A(λ_{i})_{γ} associated to (G/H)_{γ} in some canonical way. In the last section of the present paper it is shown that the known results concerning the projective varieties could be derived from this theorem, although it deals with the affine varieties.

An essential role in our computations plays the well-known equivalence between the category of the equivariant vector bundles over the homogeneous variety and the category of the finite dimensional representations of the stabilizer of a rational point. Another important ingredient is the spectral sequence contstructed by Merkurjev [M] that allows to pass from the equivariant Ktheory to the ordinary one. It turns out that when the groups have the same rank the spectral sequence degenerates and provides the very explicit answer. In order to show that the sequence degenerates we use the theorem proved by Steinberg [St] which states that in our case the representation ring R(H) is a free module over R(G). We give a new proof of the last theorem which provides us some good basis consisting of the irreducible represantations such that we can handle it in the twisted case.

Note that there is a decent classification of the connected reductive subgroups containing the maximal torus [BT, � 3]. They correspond to the quasiclosed (for char k = 0 one can say closed) symmetric subsets in the root system of the group G, so one can explicitly write down the varieties covered by theorem 4. For example we can compute by hand the K-theory for the variety G(E_{6})/G(A_{2} + A_{2} + A_{2}), the inclusion provided by 3A_{2} ⊂ E_{6}. The K_{0} in this case is a free abelian group of rank 240.

In the article everything is settled over the field k of an arbitrary characteristic. Algebraic groups are supposed to be the smooth algebraic varieties over the field k. The text is organized in the following way. In the second section we recall some well-known facts on the representation theory of the reductive groups, including the combinatorics concerning roots, weights and the Weyl group.

In the next section we introduce some useful combinatorics arising from the reductive subgroup of maximal rank. We define a linear order on the dominant weights and prove key lemmas providing the technical tool for the new approach to the Steinberg theorem.

In section 4 we show that with the given order one can choose some set resembling the Gröbner basis and could carry out the division relative to the chosen elements. Using the above idea we construct the basis for the representation ring in theorem 2 and show that there is a natural freedom in the choice of basis. The introduced division algorithm provides an explicit method for calculation of the multiplicative structure on the obtained free module.

The fifth section contains some examples, from the vivid two-dimensional case involving G_{2} to the non-obvious series of C_{n} root systems.

In section 6 we recall the basic notions from the equivariant K-theory and present the spectral sequence constructed by Merkurjev. The following section deals with the split case of the homogeneous varieties, the degeneration of the spectral sequence is demonstrated and the isomorphism for K-theory is constructed.

Section 8 deals with the twisted forms, separable algebras are introduced and the main result is proved by means of the splitting principle ??.

In the last section we use the developed technique towards concrete examples. First of all the relations with the known results are presented and K-theory for the twisted flags is computed. Then we turn to the case of characteristic zero and show that K-theory for any homogeneous variety with the stabilizer connected and having the maximal rank could be computed without assumption about reductiveness. Also some affine homogeneous examples are considered, including the octonionic projective plane and the quaternionic projective spaces.

I want to express my sincere gratitude to I.Panin for numerous discussions and useful suggestions concerning the subject of this paper.

2. Representations of reductive algebraic groups.

In this section we fix the notations and recall some well-known facts concerning the representation theory of split reductive algebraic groups. A comprehensive survey of this theme could be found in [Ja], the semisimple case is covered also in [H2].

Let G be a connected split reductive algebraic group and let T ⊂ G be a split maximal torus of G. Let W (G, T) = N_{G}(T)/Z_{G}(T) be the Weyl group of G. Since all split maximal tori are conjugate, W (G, T) does not depend on the choice of torus T so we will as usual denote it by W (G). Let

X^{∗}(T) = Hom(T, G_{m}) ≅ ℤ^{rk(T)}, Ch = Hom(Z(G), G_{m})

be the character groups of torus T and center Z(G) respectively. Recall that the Weyl group W (G) obviously acts on X^{∗}(T) and that there is a natural Weyl-equivariant Ch grading on X^{∗}(T).

Let Rep_{k}(G) be the category of finite dimensional k-rational representations of G and let R(G) = K_{0}(Rep_{k}(G)) be the representation ring of G. Recall that as an additive group the ring R(G) is a free abelian group generated by the isomorphism classes of irreducible representations. The following result is well-known (for example, see [Ja, Cor. 2.7]).

Theorem 1. Let G be a connected split reductive algebraic group and let T ⊂ G be a split maximal torus of G. Then there is a ring isomorphism R(T) ≅ ℤ[X^{∗}(T)] where the last one denotes the group ring. Moreover, the restriction of representetions induces R(G) ≅ ℤ[X^{∗}(T)]^{W (G)}.

We need some more combinatorial data on the connection between representations and characters group ring.

There is a root system Φ in X^{∗}(T), so there is a bilinear form on

V = X^{∗}(T) ⊗_{ℤ} ℝ ≅ ℝ^{rk(G)}

such that the Weyl group is generated by the reflections {w_{α}, α ∈ Φ}. The hyperplanes H_{α} orthogonal to the roots α ∈ Φ divide V into chambers which are the fundamental domains for the Weyl group action. The hyperplanes adjacent to the chamber are called walls of this chamber. Fix a set of simple roots Π ⊂ Φ and denote by

C(G) = {v ∈ V |(v, α) ≥ 0, α ∈ Π}

the fundamental Weyl chamber. The walls of the fundamental Weyl chamber C(G) coincide with the hyperplanes orthogonal to the simple roots. Let

Λ_{G}^{+} = C(G) ∩ X^{∗}(T)

be the cone of dominant weights. Note that in the semisimple case group is simply connected iff Λ_{G}^{+} ≅ ℕ_{0}^{+}.

Let λ ∈ Λ_{G}^{+} be a dominant weight. Theorem 1 states that there is a bijection between such weights and irreducible G-modules, so we will denote by V_{G}(λ) the corresponding G-module.

At last, recall that there is a partial order on X^{∗}(T) which is defined by the set of simple roots Π: μ ⪯_{Π} λ if and only if λ − μ is the sum of positive roots. The interaction between this ordering and Weyl action is stated in the next lemma [H1, Lemma 13.2A].

Lemma 1. Let λ ∈ Λ_{G}^{+}, w ∈ W (G) then w(λ) ⪯_{Π} λ.

3. Subgroup combinatorics.

In this section we introduce the necessary combinatorics that we need in order to prove theorem 2. The main goal is to order dominant weights of the subgroup and show that there are several weights with good properties relative to the order.

Let G be a connected split semisimple simply connected group of rank r, let T ⊂ G be a split maximal torus of G and let T ⊂ H ⊂ G be a connected split reductive subgroup of maximal rank. Evidently, in this setting there is an inclusion of the Weyl groups W (H) ⊂ W (G). Hence we have the corresponding combinatorial data introduced in the previous section: lattice X^{∗}(T) ⊂ V in the euclidean space, root system Φ ⊂ X^{∗}(T) and actions of the Weyl groups W (H) ⊂ W (G) on the V. The following lemma shows that we could choose the compatible fundamental Weyl chambers and the corresponding cones of dominant weights

\xymatrix{C(G) \ar@{^{(}->}[r] & C(H) \\ Λ_{G}^{+} \ar@{^{(}->}[u] \ar@{^{(}->}[r] & Λ_{H}^{+} \ar@{^{(}->}[u]}

Let k = [W (G) : W (H)] be the Weyl group index.

Lemma 2. Any Weyl chamber of group H is the union of k Weyl chambers of group G.

Proof. It is clear that any wall for W (H) action is a wall for W (G) action, so in order to get chambers of group G we need to subdivide H chambers. The number of G subchambers is independent on H chamber. Since the number of chambers coincides with the order of Weyl group, the number of subchambers equals to the index k. □

So we choose some chambers C(G) ⊂ C(H) and elements

e = w_{1}, w_{2}, ..., w_{k} ∈ W (G),

such that

C(H) = ⋃_{1≤i≤k} w_{i}C(G), Λ_{H}^{+} = ⋃_{1≤i≤k} w_{i}Λ_{G}^{+}.

Let ω_{1}, ..., ω_{r} be the fundamental weights corresponding to C(G) and let Π, Π′ be the sets of simple roots for G, H respectively. By Φ^{+} denote the set of positive roots of G relative to Π.

Definition 1. Let μ ∈ X^{∗}(T) be some weight. Define

H(μ) = {H_{α}, α ∈ Φ^{+}|∃i : (μ, α) · (ω_{i}, α) < 0} = {H_{α}, α ∈ Φ^{+}|(μ, α) < 0}

to be the set of walls which separate μ from C(G). Let H(wC(G)) = H(μ) for some interior weight μ ∈ wC(G)^{o}.

Remark 1. The set H(μ) somehow measures the spherical distance from μ to C(G), the furthest weights are separated by the most hyperplanes. Also note that #H(wC(G)) = l(w), the usual length of an element of Weyl group, which is defined to be the number of simple reflections in the shortest word representing w.

Lemma 3. Let μ ∈ Λ_{H}^{+} be some dominant weight. Then there exists such i that μ ∈ w_{i}Λ_{G}^{+} and H(w_{i}C(G)) = H(μ).

Proof. If μ belongs to the interior of some chamber we of course should take that chamber. Otherwise we can choose an arbitrary ν ∈ C(G)^{o} and draw a segment connecting the points corresponding to μ and ν. Since ν is interior for C(G) this segment does not belong to hyperplanes H_{α} and we should take the chamber w_{i}C(G) which interior it crosses first, starting from the μ. There are no hyperplanes separating the choosen chamber from μ so H(w_{i}C(G)) = H(μ). □

Lemma 4. Let μ, λ ∈ w_{i}Λ_{G}^{+} for some i and w ∈ W (G). Suppose that there exists a hyperplane H_{α} such that (λ, α) · (wμ, α) < 0. Then (wμ, λ) < (μ, λ).

Proof. First of all we multiply weights by w_{i}^{−1} and consider

μ′ = w_{i}^{−1}, λ′ = w_{i}^{−1}λ, w′ = w_{i}^{−1}ww_{i}, α′ = w_{i}^{−1}α.

It follows that μ′, λ′ ∈ Λ_{G}^{+} and

(λ′, α′) · (w′μ′, α′) = (w_{i}^{−1}λ, w_{i}^{−1}α) · (w_{i}^{−1}wμ, w_{i}^{−1}α) = (λ, α) · (wμ, α),

and by the same vein

(w′μ′, λ′) = (wμ, λ), (μ′, λ′) = (μ, λ).

So from now on we suppose w_{i} = e.

Note that by lemma 1 μ − wμ equals to the sum of positive roots and λ is the sum of fundamental weights with nonnegative coefficients, so in general (wμ − μ, λ) ≤ 0, and we need to show that the difference wμ − μ is not orthogonal to λ. First of all wμ ∉ Λ_{G}^{+} (i.e. not equals μ) since there are no hyperplanes crossing C(G).

We can find a sequence α_{1}, α_{2}, ..., α_{n} ∈ Φ^{+} such that the following conditions hold, where s_{i} = w_{α_{i}}w_{α_{i−1}} … w_{α_{1}} and s_{0} = e.

(a) w = s_{n} = w_{α_{n}}w_{α_{n−1}} … w_{α_{1}}.

(b) For every 1 ≤ i < n the hyperplane H_{α_{i}} is a wall of s_{i−1}C(G) and

separates it and C(G) from s_{n}C(G) So this presentations divides w into the sequence of flips, and each flip drives the chamber further from the C(G).

We claim that the roots α_{i} should be the roots corresponding to hyperplanes in H(wC(G)) written in the appropriate order. Indeed, there exists some hyperplane H_{α_{1}} ∈ H(wC(G)) which is the wall of C(G), otherwise wC(G) = C(G) and w = e, contradicting μ ≠ wμ. Note that

H(s_{1}C(G)) = {H_{α_{1}}} ⊂ H(wC(G)),

and whenever s_{1} ≠ w we can find H_{α_{2}} ∈ H(wC(G)) \ H(s_{1}C(G)) satisfying the condition (b), i.e. it should be the wall of s_{1}C(G), the separating part is valid since we look at the separating hyperplanes. Now one has

H(s_{2}C(G)) = {H_{α_{1}}, H_{α_{2}}} ⊂ H(wC(G)).

If s_{2} ≠ w we can find α_{3} ∈ H(wC(G)) \ H(s_{2}C(G)) and so on.

For the above roots α_{i} one has

(wμ − μ, λ) = (∑_{i=1}^{n} s_{i}μ − s_{i−1}μ, λ) = ∑_{i=1}^{n} c_{i}(α_{i}, λ),

where c_{i} = −2\frac{(s_{i−1}μ,α_{i})}{(α_{i},α_{i})}. From the condition (b) it follows that H_{α_{i}} does not separate s_{i−i}C(G) from C(G), so (s_{i−1}μ, α_{i}) ≥ 0, hence c_{i} is nonpositive.

In general (α_{i}, λ) ≥ 0 so it is sufficient to show that there exists some α_{i} such that (α_{i}, λ) ≠ 0 and (s_{i−1}μ, α_{i}) ≠ 0. The first condition is equivalent to λ ∉ H_{α_{i}} and the second means that s_{i}μ ≠ s_{i−1}μ. Now suppose that there is no such α_{i}, then we can get from μ to wμ by reflections w_{α_{i}} such that (s_{i−1}μ, α_{i}) = 0, i.e. through the hyperplanes which contain λ. Then wμ and λ lie in the same chamber and there are no hyperplanes separating them. So by cotradiction we can find such i that the correspodning term c_{i}(α_{i}, λ) < 0 and this finishes the proof.

□

Now we are ready to introduce a good order on Λ_{H}^{+} which uses W (G) action and hence somehow connects W (G) orbits with H weights.

Definition 2. Let μ_{1}, μ_{2} ∈ Λ_{H}^{+}, we say that μ_{1} ⪯′ μ_{2} if and only if one of the following conditions holds:

(1) μ_{1} = μ_{2}

(2) (μ_{1}, μ_{1}) < (μ_{2}, μ_{2})

(3) (μ_{1}, μ_{1}) = (μ_{2}, μ_{2}) and H(μ_{1}) ⊋ H(μ_{2})

Remark 2. The meaning of the above definition is that the dominant weight is smaller if the vector is shorter or the spherical distance to C(G) is greater.

Lemma 5.

(1) ⪯′ defines a partial order on Λ_{H}^{+}.

(2) For any μ ∈ Λ_{H}^{+} there are only finitely many μ′ such that μ′ ⪯̸′ μ.

(3) Let μ_{1}, μ_{2} ∈ Λ_{H}^{+} and μ_{1} ≺_{Π′} μ_{2}. Then μ_{1} ≺′ μ_{2}.

Proof.

(1) is checked by hand.

(2) Follows from the fact that there are finitely many weights μ′ such that (μ′, μ′) ≤ (μ, μ).

(3) There exists β ∈ X^{∗}(T) such that β equals a sum of positive roots and μ_{2} = μ_{1} +β. Then (μ_{2}, μ_{2}) = (μ_{1}, μ_{1})+(β, β)+2(μ_{1}, β). The last term is nonnegative since the scalar product of simple root and dominant weight is nonnegative and so is the scalar product of positive root and dominant weight. So μ_{1} ⪯′ μ_{2} follows from examining their lengths.

□

Definition 3. Let ⪯ be an arbitrary linear extension of order ⪯′, i.e. such linear order that from μ_{1} ⪯′ μ_{2} it follows that μ_{1} ⪯ μ_{2}.

Remark 3. Part (3) of the previous lemma is valid for ⪯ too and part (2) transforms into the property that there are only finitely many μ′ such that μ′ ⪯ μ.

The next lemma introduces basic and in some sense minimal and indecomposable elements λ_{i} ∈ Λ_{H}^{+}, one for each chamber w_{i}C(G).

Lemma 6. For every i there exists an element λ_{i} ∈ w_{i}Λ_{G}^{+} such that (1) H(λ_{i}) = H(w_{i}C(G)).

(2) For every μ ∈ w_{i}Λ_{G}^{+}, H(μ) = H(w_{i}C(G)) one has μ − λ_{i} ∈ w_{i}Λ_{G}^{+}.

(3) For every μ ∈ w_{i}Λ_{G}^{+} one has λ_{i} + w_{j}w_{i}^{−1}μ ⪯ λ_{i} + μ.

Proof. The set of weights {μ ∈ w_{i}Λ_{G}^{+}|H(μ) ≠ H(w_{i}C(G))} is just the intersection of w_{i}Λ_{G}^{+} with the union of chamber w_{i}C(G) walls which separate it from the C(G). Indeed, the only chance for the weight to have the lesser number of walls separating it from C(G) is to belong to such wall, and every weight lying on this wall has the lesser number of separating hyperplanes.

Now we use the fact that G is simply connected so w_{i}Λ_{G}^{+} ≅ ℕ_{0}^{r}. The walls of the chamber correspond to the hyperplanes where some coordinate equals 0, so the weights, which have the same H(μ) as the chamber, correspond to the points with certain coordinates, say 1, ..., l, strictly greater then 0. Let λ_{i} be the element corresponding to the point with first l coordinates equal 1 and others equal 0. From the above it follows that we get (1) and (2).

First of all note that all weights really lie in the Λ_{H}^{+}, so we can try to compare them. Examine their lengths:

(λ_{i} + w_{j}w_{i}^{−1}μ, λ_{i} + w_{j}w_{i}^{−1}μ) = (λ_{i}, λ_{i}) + 2(λ_{i}, w_{j}w_{i}^{−1}μ) + (w_{j}w_{i}^{−1}μ, w_{j}w_{i}^{−1}μ) =

= (λ_{i}, λ_{i}) + 2(λ_{i}, w_{j}w_{i}^{−1}μ) + (μ, μ)
(λ_{i} + μ, λ_{i} + μ) = (λ_{i}, λ_{i}) + 2(λ_{i}, μ) + (μ, μ)

From lemma 1 one has (λ_{i}, w_{j}w_{i}^{−1}μ) ≤ (λ_{i}, μ) and, consequently,

(λ_{i} + w_{j}w_{i}^{−1}μ, λ_{i} + w_{j}w_{i}^{−1}μ) ≤ (λ_{i} + μ, λ_{i} + μ).

Now look at H(λ_{i}). Observe that H(λ_{i}) = H(λ_{i} +μ). Indeed, λ_{i} +μ ∈ w_{i}Λ_{G}^{+} and from the first part of the lemma it follows that H(λ_{i}) ⊃ H(λ_{i} + μ). The opposite inclusion follows from the fact that since μ and λ_{i} lie in the same chamber there are no hyperplanes H_{α} separating them, i.e. one has (λ_{i}, α) · (μ, α) ≥ 0. For every H_{α} ∈ H(λ_{i}) one has (λ_{i}, α) < 0, so (μ, α) ≤ 0 and (λ_{i} + μ, α) < 0, then H_{α} ∈ H(λ_{i} + μ).

First suppose that there exists some H_{α} ∈ H(λ_{i}) such that (w_{j}w_{i}^{−1}μ, α) > 0. Since H_{α} ∈ H(λ_{i}) one has (λ_{i}, α) < 0 and

(λ_{i}, α) · (w_{j}w_{i}^{−1}μ, α) < 0.

Then we are in the setting of lemma 4 with a slight change of notation, so (λ_{i}, w_{j}w_{i}^{−1}μ) < (λ_{i}, μ) hence λ_{i} + w_{j}w_{i}^{−1}μ ⪯ λ_{i} + μ.

Otherwise for all H_{α} ∈ H(λ_{i}) one has (w_{j}w_{i}^{−1}μ, α) ≤ 0 and since (λ_{i}, α) < 0 one has (λ_{i} + w_{j}w_{i}^{−1}μ, α) < 0, so H(λ_{i}) ⊂ H(w_{j}w_{i}^{−1}μ + λ_{i}) and

H(λ_{i} + μ) ⊂ H(w_{j}w_{i}^{−1}μ + λ_{i}).

The last two sets coincide only if w_{j}w_{i}^{−1}μ + λ_{i} ∈ w_{i}Λ_{G}^{+} and the second part of the lemma in this case yields w_{j}w_{i}^{−1}μ ∈ w_{i}Λ_{G}^{+} that means μ = w_{j}w_{i}^{−1}μ. In any case one has λ_{i} + w_{j}w_{i}^{−1}μ ⪯ λ_{i} + μ.

□

Remark 4. Note that from the above construction one gets λ_{1} = 0 ∈ X^{∗}(T).

4. Restriction of representations.

In this section we study the representation restriction homomorphism on the representation rings and prove theorem 2.

Let T ⊂ H ⊂ G be the same groups as in the previous section. From theorem 1 we get the following commutative diagramm.

\xymatrix{R(G) \ar@{^{(}->}[r] \ar[d]^{≃} & R(H) \ar@{^{(}->}[r] \ar[d]^{≃} & R(T) \ar[d]^{≃} \\ ℤ[X^{∗}(T)]^{W (G)} \ar@{^{(}->}[r] & ℤ[X^{∗}(T)]^{W (H)} \ar@{^{(}->}[r] & ℤ[X^{∗}(T)]}

Recall that Z(G) ⊂ T, hence all the rings above are Ch = X^{∗}(Z(G))- graded. We are interested in R(G)-module structure on R(H) and its connection with the grading.

We need the following easy lemma from commutative algebra.

Lemma 7. Let S ⊂ R be domains, let λ_{1}, ..., λ_{k} ∈ R generate R as S-module and let [Q(R) : Q(S)] = k. Then R is a free S-module with basis λ_{1}, ..., λ_{k}.

Proof. R is finitely generated as S-module hence it is integral over S. Then R ⊗_{S} Q(S) is integral over field S ⊗_{S} Q(S) = Q(S), hence itself is a field so R ⊗_{S} Q(S) ≅ Q(R).

We have the following short exact sequence induced by λ_{1}, ..., λ_{k}.

[figure: N; S^{k}; R] ,

hence

[figure: N ⊗_{S} Q(S); Q(S)^{k}; R ⊗_{S} Q(S)] .

The last term is isomorphic to Q(R) and comparing dimensions one can see that N ⊗_{S} Q(S) = 0 hence N = 0 and we get the claim of the lemma. □

We consider X^{∗}(T) as an additive group, so we will write the element of ℤ[X^{∗}(T)] corresponding to weight μ in such way: x^{μ}.

Definition 4. For μ ∈ X^{∗}(T) we will denote by (x^{μ})^{W (H)} the sum in the ℤ[X^{∗}(T)] of all elements corresponding to the weights in W (H)-orbit of μ and for any monomial ax^{μ} ∈ ℤ[X^{∗}(T)] by (ax^{μ})^{W (H)} = a(x^{μ})^{W (H)} we denote the similar orbit but with a coefficient.

With the above notation one has the unique decomposition of

f = ∑(a_{μ_{j}}x^{μ_{j}})^{W (H)} ∈ ℤ[X^{∗}(T)]^{W (H)}

into the sum of monomial orbits with distinct μ_{j} ∈ Λ_{H}^{+}. Recall that we have a linear order ⪯ on μ_{j} introduced in the previous section.

Definition 5. Let f = ∑(a_{μ_{j}}x^{μ_{j}})^{W (H)}, then define the degree deg(f) = max_{j} μ_{j} to be the maximal μ_{j} in the decomposition and the leading orbit

lo(f) = (a_{deg(f)}x^{deg(f)})^{W (H)} to be the orbit of the maximal monomial.

We will use the analogous notation for group G.

Theorem 2. Let G be a split semisimple simply connected group and let H be a connected split reductive subgroup of the maximal rank (i.e. H contains the split maximal torus T of G). Then R(H) is a free R(G)-module of rank [W (G) : W (H)] and there is a Ch-homogeneous basis.

Proof. First of all we will deal with the weight realization of the rings of representations, i.e. with the following sequence.

\xymatrix{ℤ[X^{∗}(T)]^{W (G)} \ar@{^{(}->}[r] & ℤ[X^{∗}(T)]^{W (H)} \ar@{^{(}->}[r] & ℤ[X^{∗}(T)]}

In the previous section in lemma 6 we have constructed some λ_{i} and we claim that the orbits (x^{λ_{i}})^{W (H)} form a homogeneous basis of ℤ[X^{∗}(T)]^{W (H)} over ℤ[X^{∗}(T)]^{W (G)}.

(a) Homogeneity. It is the easiest part since it follows at once from the equivarian e of W (H) action.

(b) (x^{λ_{i}})^{W (H)} generate ℤ[X^{∗}(T)]^{W (H)} as ℤ[X^{∗}(T)]^{W (G)}-module. We will show by induction on deg(f) that f ∈ ℤ[X^{∗}(T)]^{W (H)} could be expressed as linear combination of (x^{λ_{i}})^{W (H)} with ℤ[X^{∗}(T)]^{W (G)} coefficients. Note that λ_{1} = 0 ∈ X^{∗}(T) and

(x^{λ_{1}})^{W (H)} = 1 ∈ ℤ[X^{∗}(T)]^{W (H)},

so we have the constants. Now suppose that we can express as linear combinations all f ∈ ℤ[X^{∗}(T)]^{W (H)} such that deg(f) ≺ μ_{0} and we need to write such an expression for (x^{μ_{0}})^{W (H)}.

By lemma 3 we have some chamber w_{l}C(G) such that μ_{0} ∈ w_{l}Λ_{G}^{+} and H(μ_{0}) = H(w_{l}C(G)), hence, by lemma 6 ν = μ_{0} − λ_{l} ∈ w_{l}Λ_{G}^{+}. Choose the subset {w_{j}} of {w_{i}} such that one has all the distinct w_{j}w_{l}^{−1}ν. Then by subdividing W (G)-orbit into W (H)-orbits we have the following equality.

lo ((x^{ν})^{W (G)} (x^{λ_{l}})^{W (H)}) = lo ((∑_{j} (x^{w_{j}w_{l}^{−1}ν})^{W (H)}) (x^{λ_{l}})^{W (H)}) .

From lemma 1 and lemma 5 the last one equals to

lo ((∑_{j} x^{w_{j}w_{l}^{−1}ν}x^{λ_{l}})^{W (H)}) = lo ((∑_{j} x^{w_{j}w_{l}^{−1}ν+λ_{l}})^{W (H)}) ,

and finally, by lemma 6, one gets

lo ((∑_{j} x^{w_{j}w_{l}^{−1}ν+λ_{l}})^{W (H)}) = (x^{ν+λ_{l}})^{W (H)} = (x^{μ_{0}})^{W (H)} .

Hence

deg ((x^{μ_{0}})^{W (H)} − (x^{ν})^{W (G)} (x^{λ_{l}})^{W (H)}) ≺ μ_{0}

and we can use the induction.

(c) (x^{λ_{i}})^{W (H)} are linearly independent. From the sequence

\xymatrix{ℤ[X^{∗}(T)]^{W (G)} \ar@{^{(}->}[r] & ℤ[X^{∗}(T)]^{W (H)} \ar@{^{(}->}[r] & ℤ[X^{∗}(T)],}

one gets the sequence of fraction fields

[figure: Q(ℤ[X^{∗}(T)]^{W (G)}); Q(ℤ[X^{∗}(T)]^{W (H)}); Q(ℤ[X^{∗}(T)]); Q(ℤ[X^{∗}(T)])^{W (G)}; Q(ℤ[X^{∗}(T)])^{W (H)}; Q(ℤ[X^{∗}(T)])]

and the degree of field extension equals to [W (G) : W (H)]. Hence, by lemma 7 one gets the claim of the theorem.

□

Corollary 1. In the notation of theorem 2 one has a basis consisting of the irreducible representations of H.

Proof. One can take V_{H}(λ_{i}) and since their leading orbits coincide with the basis constructed in the theorem one gets the claim. □

Remark 5. We can choose various chambers C(G) and the different choices produce different basises. Also the proof of the theorem gives an explicit algorithm for calculating the coefficients of decomposition with respect to the chosen basis, so, for example, in every particular case one can write down the multiplication table for the basis, yet it seems that there is no elegant general formula.

5. Examples: A_{1} + A_{1} ⊂ G_{2}, B_{4} ⊂ F_{4} and C_{1} + C_{n−1} ⊂ C_{n}.

In this section we compute some examples of basises. Every reductive subgroup containing the maximal torus is defined by some quasi-closed root subset [BT, � 3], so we use the root system notation. Every maximal root subsystem of full rank corresponds to some node in the Dynkin diagram and the subsystem diagram is just the extended Dynkin diagram with the chosen node removed. We label simple roots in a way of [Bo].

5.1. A_{1} + A_{1} ⊂ G_{2}. In this example we take the the subsystem in G_{2} defined by the short simple root and the maximal one. The corresponding Dynkin diagram is the next one, with the white node removed.

[figure: α_{1}; α_{2}; −α_{max}; ⟨]

We label the roots in a way shown above and α_{max} = 3α_{1} + 2α_{2}. The fundamental chamber C(G_{2}) is the chamber spanned by the fundamental weights

ω_{1} = α_{max} = 3α_{1} + 2α_{2}, ω_{2} = 2α_{1} + α_{2}.

The fundamental chamber C(A_{1}+A_{1}) should contain C(G_{2}) so it is the quarter of the plane bounded by α_{1} and α_{max}. Note, by the way, that the considered group G(A_{1} + A_{1}) is not simply connected since there are no weights \frac{1}{2}α_{1} and \frac{1}{2}α_{max} in our lattice and one can see that

G(A_{1} + A_{1}) = SL_{2} ⊗ SL_{2} = (SL_{2} × SL_{2})/μ_{2}

with μ_{2} embedded diagonally.

The chamber C(A_{1} + A_{1}) subdivides into the G_{2} chambers in the following way:

C(A_{1} + A_{1}) = C(G_{2}) ∪ w_{α_{2}}C(G_{2}) ∪ w_{α_{1}+α_{2}}w_{α_{2}}C(G_{2})

So the theorem 2 tells us that we should take in each subchamber the shortest of the furthest by spherical distance weights, i.e. the generator for the furthest wall, hence one has

0 ∈ C(G), 3α_{1} + α_{2} ∈ w_{α_{2}}C(G), α_{1} ∈ w_{α_{1}+α_{2}}w_{α_{2}}C(G_{2})

and the corresponding sums over W (A_{1} +A_{1}) would form the basis. The basis from the theorem is the follows:

1, x^{3α_{1}+α_{2}} + x^{α_{2}} + x^{−α_{2}} + x^{−3α_{1}−α_{2}}, x^{α_{1}} + x^{−α_{1}}.

One could compute the corresponding basis consisting of irreducible modules from corollary 1 having the following weight subspaces:

V (0)  = 1
V (3α_{1} + α_{2}) = x^{2α_{1}+α_{2}} + x^{α_{1}+α_{2}} + x^{−α_{1}−α_{2}} + x^{−2α_{1}−α_{2}}+
+x^{3α_{1}+α_{2}} + x^{α_{2}} + x^{−α_{2}} + x^{−3α_{1}−α_{2}}
V (α_{1})  = 1 + x^{α_{1}} + x^{−α_{1}}

In fact after identifying G(A_{1} + A_{1}) = SL_{2} ⊗ SL_{2} one can write the above representations in more natural way, denoting by W_{1}, W_{2} the regular representations of the factors one has

V (0) = S^{0}W_{1} ⊗S^{0}W_{2}, V (3α_{1} +α_{2}) = S^{3}W_{1} ⊗W_{2}, V (α_{1}) = S^{2}W_{1} ⊗S^{0}W_{2}.

5.2. B_{4} ⊂ F_{4}. In this case we remove the α_{4} node from the extended Dynkin diagram of type F_{4}. One can show that it corresponds to Spin_{9} ⊂ G(F_{4}).

[figure: −α_{max}; α_{1}; α_{2}; α_{3}; α_{4}; ⟩]

With the above labeling one has

α_{max} = 2α_{1} + 3α_{2} + 4α_{3} + 2α_{4}.

We will shorten the above notation to (2, 3, 4, 2). The fundamental weights defining C(F_{4}) are

ω_{1} = (2, 3, 4, 2), ω_{2} = (3, 6, 8, 4), ω_{3} = (2, 4, 6, 3), ω_{4} = (1, 2, 3, 2).

Choose the simple roots for B_{4} in the following way:

α′_{1} = w_{α_{4}}(−α_{max}) = (0, 1, 2, 2), α′_{2} = α_{1}, α′_{3} = α_{2}, α′_{4} = α_{3},

hence we have the fundamental weights defining C(B_{4}) ∼ {ω′_{1}, ω′_{2}, ω′_{3}, ω′_{4}}:

ω′_{1} = (1, 2, 3, 2) = ω_{4}, ω′_{2} = (2, 3, 4, 2) = ω_{1},
ω′_{3} = (2, 4, 5, 2), ω′_{4} = (1, 2, 3, 1).

Since these weights belong to the considered lattice the chosen G(B_{4}) is simply connected, so it really is Spin_{9}. Now we compute the subdividing of C(B_{4}):

C(B_{4}) = C(F_{4}) ∪ w_{α_{4}}C(F_{4}) ∪ w_{α_{3}+α_{4}}w_{α_{4}}C(F_{4}),
w_{α_{4}}C(B_{4}) ∼ {ω_{1}, ω_{2}, ω_{3}, ω′_{4}}, w_{α_{3}+α_{4}}w_{α_{4}}C(B_{4}) ∼ {ω_{1}, ω_{2}, ω′_{3}, ω′_{4}}.

Theorem 2 suggests to look at the elements appeared after flips, since they are the spherically furthest, so the basis would consist of W (B_{4}) orbits of 0, ω′_{4}, ω′_{3}.

Another basis comes from the corollary 1 that claims V (0), V (ω′_{3}), V (ω′_{4}) to be a basis. These representations are just the trivial one, the Λ^{3}W for the regular W and the spin one.

5.3. C_{1} + C_{n−1} ⊂ C_{n}. In this case we remove the α_{1} node from the extended Dynkin diagram of type C_{n} and it corresponds to (Sp_{2} ×Sp_{2n−2}) ⊂ Sp_{2n} with the quotient variety HP^{n−1} = Sp_{2n}/(Sp_{2} × Sp_{2n−2}) being the quaternionic projective space in notation of section 9.3.

[figure: −α_{max}; α_{1}; α_{2}; α_{n−2}; α_{n−1}; α_{n}; …; ⟩; ⟨]

One has

α_{max} = 2α_{1} + 2α_{2} + ⋯ + 2α_{n−1} + α_{n} = (2, 2, …, 2, 1).

The fundamental weights for C(C_{n}) are

ω_{i} = (1, 2, …, i − 1, i, i, … i, \frac{i}{2}).

Choosing α_{2}, α_{3}, …, α_{n}, α_{max} to be the simple roots of C_{1} +C_{n−1} one gets the following C(C_{1} + C_{n−1}):

ω′_{i} = (0, 1, …, i − 1, i, i, … i, \frac{i}{2}), ω′_{n} = (1, 1, …, 1, \frac{1}{2}).

The subdividing of C(C_{1} + C_{n−1}) is straightforward:

C(C_{1} + C_{n−1}) = C(C_{n}) ∪ w_{α_{1}}C(C_{n}) ∪ w_{α_{1}+α_{2}}w_{α_{1}}C(C_{n}) ∪ … ⋯ ∪ (w_{α_{1}+···+α_{n−1}} … w_{α_{1}+α_{2}}w_{α_{1}})C(C_{n}),

(w_{α_{1}+···+α_{i}} … w_{α_{1}+α_{2}}w_{α_{1}})C(C_{n}) ∼ {ω′_{1}, ω′_{2}, …, ω′_{i}, ω_{i+1}, … ω_{n}}.

The corollary 1 claims V (0), V (ω′_{1}), … V (ω′_{n−1}) to be the basis and this representations are just Λ^{i}W for regular representation W of group Sp_{2n−2}.

6. Representations, vector bundles and equivariant K-theory.

In this section we recall some results on the equivariant K-theory. An extensive exposition and further references could be found in [M].

Let G be an algebraic group, let H ⊂ G be a closed subgroup and let X = G/H be the corresponding smooth homogeneous G-variety. There is a well-known tensor equivalence [M, Example 2]

\xymatrix{Rep_{k}(H) \ar[r]^{∼} & V ect^{G}(X)}

between the categories Rep_{k}(H) of finite dimensional k-rational representations of H and V ect^{G}(X) of G-equivariant vector bundles over X. The inverse for the above equivalence is given by the fiber over the extinguished point eH of X. Further we will use the following notation.

Definition 6. Let V_{H}(λ) be the irreducible representation of H with the highest weight λ ∈ Λ_{H}^{+}, then denote by V_{H}(λ) the corresponding vector bundle over G/H. For an irreducible representation V_{G}(μ) of group G with the highest weight μ ∈ Λ_{G}^{+} one can use the restriction of representations, get the representation of H (not necessary irreducible) and then take the corresponding vector bundle V_{G}(μ). Occasionally we will write V_{G}(λ) and V_{G}(λ) for a λ ∈ Λ_{H}^{+} and it means that one should find μ ∈ Λ_{G}^{+} from W (G)-orbit of λ and then take the corresponding V_{G}(μ) and V_{G}(μ).

Remark 6. Note that after forgetting about the G-action the last bundle becomes trivial, i.e. the composition

\xymatrix{Rep_{k}(G) \ar[r]^{Res} & Rep_{k}(H) \ar[r]^{∼} & V ect^{G}(X) \ar[r] & V ect(X)}

takes G representations to trivial bundles.

Set

K_{n}(G; X) = K_{n}(V ect^{G}(X)).

The above equvalence yields

K_{n}(G; X) ≅ K_{n}(Rep(H)),

in particular

K_{0}(G; X) ≅ R(H).

Note that R(H) is a R(G)-module, hence every K_{n}(G, X) also is. The following proposition, being a straightforward consequence of [M, Theorem 10], compares K_{n}(G; X) with K_{n}(X).

Proposition 1. Let G be a split simply connected semisimple group. Then there is a spectral sequence

E_{p,q}^{2} = Tor_{p}^{R(G)}(ℤ, K_{q}(G; X)) ⟹ K_{p+q}(X).

7. K-theory of a homogeneous variety.

In this section we calculate K-theory of a homogeneous variety X = G/H for connected split reductive algebraic groups H ⊂ G of the same rank.

Lemma 8. K_{n}(Rep_{k}(H)) ≅ R(H) ⊗_{ℤ} K_{n}(k).

Proof. Note that char k not necessary equals 0, so the reductive group H not necessary geometrically reductive, i.e. the category Rep_{k}(H) may be not semisimple. But, nevertheless, all objects of Rep_{k}(H) have finite length and, thanks to Devissage property of K-theory, one has

K_{n}(Rep_{k}(H)) ≅ K_{n}(Rep_{k}(H)_{ss}),

where Rep_{k}(H)_{ss} stands for the subcategory of semisimple representations. By Shur’s Lemma we can pass to the endomorphisms of irreducible represenations

K_{n}(Rep_{k}(H)_{ss}) ≅ ⨁_{i} K_{n}(End(V_{i})),

and, since End(V_{i}) = k, the last one equals ⨁ K_{n}(k) with the sum over isomorphism classes of irreducible representations, which could be identified with R(H) ⊗_{ℤ} K_{n}(k). □

Proposition 2. Let G be a connected split simply connected semisimple algebraic group and let H ⊂ G be a connected split reductive subgroup of the same rank. Then the spectral sequence in proposition 1 degenerates, i.e.

Tor_{p}^{R(G)}(ℤ, K_{n}(G; X)) = \begin{cases}K_{n}(X), & if p = 0; \\ 0, & if p > 0. \end{cases}

Proof. Due to lemma 8 it is sufficient to show that for p ≥ 1 one has

Tor_{p}^{R(G)}(ℤ, R(H) ⊗_{ℤ} K_{n}(k)) = 0.

Replace K_{n}(k) with an arbitrary abelian group M. Since Tor commutes with the direct limits we can reduce the problem to the finitely generated abelian groups, and, moreover, to M = ℤ or M = ℤ/mℤ.

In the first case we at once get the claim from theorem 2,

Tor_{p}^{R(G)}(ℤ, R(H)) = 0,

since R(H) is a free R(G)-module.

In the second case we can write the resolution

\xymatrix{0 \ar[r] & R(H) \ar[r]^{m} & R(H) \ar[r] & R(H) ⊗_{ℤ} ℤ/mℤ \ar[r] & 0} ,

which is exact since R(H) is a domain of a zero characteristic. Denoting the rank of R(H) over R(G) by r, after tensoring with ℤ one still gets an exact sequence sequence

\xymatrix{0 \ar[r] & ℤ^{r} \ar[r]^{m} & ℤ^{r} \ar[r] & (ℤ/mℤ)^{r} \ar[r] & 0} .

So, we conclude

Tor_{p}^{R(G)}(ℤ, R(H) ⊗ ℤ/mℤ) = 0,

finishing the proof. □

In order to remove the annoying restriction that G should be simply connected we need the following lemma.

Lemma 9. Let H ⊂ G be a pair of connected split reductive groups of the same rank. Then there exists a connected split simply connected semisimple group G̃ and a connected split reductive subgroup H̃ ⊂ G̃ of the same rank such that G̃/H̃ ≅ G/H.

Proof. Let G̃ be the simply connected covering of the derived group G′. There exists a covering

[figure: Z; (G_{m})^{l} × G̃; G]

with finite kernel Z. Since H contains the maximal torus the preimage of H under this projection contains the factor (G_{m})^{l}, so we have the following diagram.

\xymatrix{(G_{m})^{l} × G̃ \ar@{->>}[r] & G \\ (G_{m})^{l} × H̃ \ar@{^{(}->}[u] \ar@{->>}[r] & H \ar@{^{(}->}[u]}

The above consideration yields

G̃/H̃ ≅ (G_{m})^{l} × G̃/(G_{m})^{l} × H̃ ≅ G/H,

so we need to show that H̃ is connected. One has

[figure: Z; (G_{m})^{l} × H̃; H]

with finite central Z and connected H. The identity component of (G_{m})^{l} × H̃ contains the maximal torus, hence it contains Z and the connectedness of H yields that the identity component coincides with the whole group. The group H̃ is connected as the quotient of the connected group. □

Theorem 3. Let H ⊂ G be a pair of connected split reductive groups of the same rank. Denote r = [W (G) : W (H)]. Then there exist V_{1}, ..., V_{r} ∈ V ect(G/H) such that

K_{∗}(G/H) = ⨁_{i=1}^{r} K_{∗}(k)[V_{i}].

Proof. By lemma 9 we can pass to a simply connected semisimple group G and from the proof one has that [W (G) : W (H)] remains the same. Due to proposition 2, the spectral sequence in proposition 1 degenerates, so, using lemma 8, one has

K_{n}(X) ≅ T or_{0}^{R(G)}(ℤ, R(H) ⊗_{ℤ} K_{n}(k)) = = ℤ ⊗_{R(G)} R(H) ⊗_{ℤ} K_{n}(k) = ℤ^{r} ⊗_{ℤ} K_{n}(k).

The above isomorphism is induced by the elements of basis R(H) over R(G) constructed in theorem 2, so we can take as V_{i} the corresponding elements of V ect(G/H). □

Remark 7. In remark 5 we noted that there is an expicit algorithm to write down the multiplication for the basis elements and now it describes the ring structure on the K_{0}(G/H). Also one can drop the assumption of the connectedness of G (but not the H), since G/H is just the disjoint union of [G : G_{0}] varieties isomorphic to G_{0}/H and so be the K-theory.

8. K-theory of twisted forms.

In this section we deal with certain twisted forms of homogeneous varieties X = G/H with connected split reductive groups H ⊂ G of the same rank. From now on we suppose G to be simply connected semisimple, lemma 9 shows that in fact it is not a restriction. Denote as before r = [W (G) : W (H)].

One has an obvious left action of G on G/H and since H contains the maximal torus hence the center, this action extends to the action of

\overline{G} = G/Z(G).

Now fix a 1-cocycle γ : Gal(k^{sep}/k) → \overline{G}(k^{sep}). Twisting the variety with this cocycle we obtain

X_{γ} = (G/H)_{γ}.

The following lemma provides a splitting variety for such cocycle.

Lemma 10. For the above cocycle γ there exists a variety Y such that the following conditions hold:

(1) Y is a smooth projective variety.

(2) The Euler characteristic χ(Y) equals to 1.

(3) For every point (not necessary closed) y ∈ Y the cocycle γ_{k(y)} is a coboundary.

Proof. Note that \overline{G} is split semisimple group and we can twist it with γ as well. The last condition is equivalent to the condition that for every point y ∈ Y the group (\overline{G}_{γ})_{k(y)} is split. Consider

Y = (G/B)_{γ}.

We claim that for the Borel subgroup B ⊂ \overline{G} the variety

Y_{F} = ((G/B)_{γ})_{F}

has rational point if and only if G_{γ} splits over F. The existence of a rational point on this variety is equivalent to the existence of a Borel subgroup defined over F, which is the stabilizer ot this point, in (\overline{G}_{γ})_{F}. The existence of Borel subgroup means that group is quasi-split that in our case is equivalent to be split, since we work with inner form.

Y = (G/B)_{γ} is clearly a smooth projective variety. In order to compute the Euler characteristic χ(G/B) it can be shown [Ja, Proposition 4.5] that

h^{i}(O_{G/B}) = \begin{cases}0, & if i > 0; \\ 1, & if i = 0. \end{cases}

so we get the claim. □

The idea lying behind the calculation of K-theory of twisted form is quite simple: one needs to construct some candidate for the K-theory and a morphism such that they will produce the correct answer and an isomorphism in the split case. The isomorphism in the split case is written by means of some vector bundles V_{i}, so in general we want to twist them. And here is the problem – there is no action of \overline{G} on them since the center could act non-trivially. In order to get over that we should tensor V_{i} with some bundles to trivialize the center action, and then for cancellation of this tensoring we should look at modules over the endomorphisms of the excessive factors. Twisting these endomorphisms algebras we get the separable algebras which produce the answer.

Definition 7. Let V_{G}(λ) be a representation of G then we denote

A(λ) = End_{k}(V_{G}(λ)) = V_{G}(λ) ⊗ V_{G}^{∗}(λ)

the endomorphism algebra of the underlying vector space. There is an obvious G action on A(λ) which extends to the \overline{G} action, so we can twist this algebra and get the separable algebra A(λ)_{γ}. Also one can pass to the corresponding trivial sheaf of algebras A(λ)_{γ} over X_{γ}.

Now we fix λ_{i} from theorem 2 and the corresponding A(λ_{i}). Denote

W (λ_{i}) = V_{H}(λ_{i}) ⊗ V_{G}(λ_{i}),

and the corresponding vector bundle W(λ_{i}). Note that W (λ_{i}) is a right module over A(λ_{i}) through the second factor and so W(λ_{i}) is. The center acts trivially on W (λ_{i}) and W(λ_{i}) so one can obtain the twisted form W(λ_{i})_{γ}.

All the considered above structures are agreed, so now we have trivial sheafs of separable algebras A(λ_{i})_{γ} and vector bundles W(λ_{i})_{γ} that are right A(λ_{i})_{γ}- modules.

Definition 8. For a variety Z and a separable algebra A let P(Z, A) be the category of coherent O_{Z} ⊗A-modules which are locally free O_{Z}-modules. Then we denote

K_{∗}(Z, A) = K_{∗}(P(Z, A)).

There is a corresponding notion of K′_{∗}(Z, A) and it satisfies all the usual properties of K-theory [M].

Proposition 3. Let Z be a variety such that every point z ∈ Z (not necessary closed) splits γ, i.e. γ_{k(z)} is a coboundary. Then in the above notation one has an isomorphism

\xymatrix{∑_{i=1}^{r} ϕ_{i} : ⨁_{i=1}^{r} K′_{∗}(Z, A(λ_{i})_{γ}) \ar[r] & K′_{∗}(X_{γ} × Z),}

where

ϕ_{i}(U) = p_{X}^{∗}(W(λ_{i})_{γ}) ⊗_{A(λ_{i})_{γ}} p_{Z}^{∗}(U).

Proof. This is proved by induction on the variety dimension.

Suppose first that dim Z = 0, i.e. Z is a point, then we are in fact in the split case. Let F = k(Z), so we have

X_{γ} × Z = X_{F} , (A(λ_{i})_{γ})_{F} = A(λ_{i})_{F} = End_{F} (V_{G}(λ_{i}) ⊗ F),

(W(λ_{i})_{γ})_{F} = W(λ_{i})_{F} = V_{H}(λ_{i}) ⊗ V_{G}^{∗}(λ_{i}) ⊗ F.

Since every module over End_{F} (V_{G}(λ_{i}) ⊗ F) is isomorphic to V_{G}(λ_{i}) ⊗ F^{n} and

V_{G}^{∗}(λ_{i}) ⊗ F ⊗_{End_{F} (V_{G}(λ_{i})⊗F)} V_{G}(λ_{i}) ⊗ F ≅ F,

one has

ϕ_{i}(V_{G}(λ_{i}) ⊗ F^{n}) = V_{H}(λ_{i}) ⊗ V_{G}^{∗}(λ_{i}) ⊗ F ⊗_{End_{F} (V_{G}(λ_{i})⊗F)} V_{G}(λ_{i}) ⊗ F^{n} = = V_{H}(λ_{i}) ⊗ F^{n}.

The above considerations show that we are in the setting of theorem 3 claiming ∑ ϕ_{i} to be an isomorphism.

For the dimension greater then 0 we can write the localization sequence for all subvarieties Z′ ⊂ Z of codimension one, so for F = k(Z) one has

\xymatrix{\underrightarrow{lim}_{Z′⊂Z} ⨁_{i=1}^{r} K′_{∗}(Z′, A(λ_{i})_{γ}) \ar[r] \ar[d] & ⨁_{i=1}^{r} K′_{∗}(Z, A(λ_{i})_{γ}) \ar[r] \ar[d]^{∑ ϕ_{i}} & ⨁_{i=1}^{r} K′_{∗}(Spec F, A(λ_{i})_{γ}) \ar[d] \\ \underrightarrow{lim}_{Z′⊂Z} K′_{∗}(X_{γ} × Z′) \ar[r] & K′_{∗}(X_{γ} × Z) \ar[r] & K′_{∗}(X_{γ} × Spec F)}

This sequence extends to the right and to the left with the shifts in K-theory, and both the side vertical morpisms in each triple are isomorphisms by induction, so using the five lemma one concludes that the middle one is an isomorphism. □

Corollary 2. In the notation of proposition 3 for the smooth Z one has

\xymatrix{∑_{i=1}^{r} ϕ_{i} : ⨁_{i=1}^{r} K_{∗}(Z, A(λ_{i})_{γ}) \ar[r] & K_{∗}(X_{γ} × Z).}

Proof. One has K_{∗}(Z, A) = K′_{∗}(Z, A) and K_{∗}(X_{γ} × Z) = K′_{∗}(X_{γ} × Z). □

Theorem 4. In the above notation there is an isomorphism

\xymatrix{∑_{i=1}^{r} ψ_{i} : ⨁_{i=1}^{r} K_{∗}(A(λ_{i})_{γ}) \ar[r] & K_{∗}(X_{γ}),}

where

ψ_{i}(U) = W(λ_{i})_{γ} ⊗_{A(λ_{i})_{γ}} U.

Proof. We can insert two copies of our morphism into the following diagram with the middle arrow from corollary 2 and Y being the splitting variety constructed in lemma 10.

\xymatrix{⨁_{i=1}^{r} K_{∗}(A(λ_{i})_{γ}) \ar@{-->}[dr]^{p^{∗}} \ar[dd]^{χ(Y)·} \ar[rrr]^{∑ ψ_{i}} &  &  & K_{∗}(X_{γ}) \ar[dl]_{p^{∗}} \ar[dd]^{χ(Y)·} \\  & ⨁_{i=1}^{r} K_{∗}(Y, A(λ_{i})_{γ}) \ar[r]^{∑ ϕ_{i}} \ar@{-->}[dl]_{p_{∗}} & K_{∗}(X_{γ} × Y) \ar[dr]^{p_{∗}} \\ ⨁_{i=1}^{r} K_{∗}(A(λ_{i})_{γ}) \ar[rrr]^{∑ ψ_{i}} &  &  & K_{∗}(X_{γ})}

A direct verification shows that the diagram is commutative. The vertical morphisms are just multiplications by χ(Y) = 1 since they are equal to the composition p_{∗}p^{∗} with p being a projection from Y to a point. The above yields that our morphism ∑ ψ_{i} is a retraction of an isomorphism ∑ ϕ_{i} hence is an isomorphism itself. □

Remark 8. It can be shown [P1] that K_{∗}(A(λ_{i})) depends only on the Z(G) action on V_{G}(λ_{i}). The explicit description of the arising algebras could be found in [Ti].

9. Examples.

9.1. Twisted flag variety. K-theory of twisted flag varieties was computed in [P1] and our computation gives the same description for the inner forms. Flag variety is a homogeneus variety G/P with split semisimple G and parabolic P ⊂ G, and this definition includes projective spaces, flag varieties in usual sense (for G = SL_{n}), split projective quadrics, etc.

There is a decomposition P = LU into the semidirect product of Levi subgroup and unipotent radical of P hence there is a morphism

\xymatrix{G/L \ar[r] & G/P}

with the fiber U. Over the extinguished point P acts on U by

l_{p}u_{p} · u = l_{p}u_{p}ul_{p}^{−1},

so one has the representation of P on the Lie algebra of U, and G/L is the corresponding vector bundle.

Levi subgroup is a reductive subgroup of maximal rank hence theorem 4 gives an explicit answer for the K_{∗}(G/L) and by the homotopy invariance of K-theory this is an answer for K_{∗}(G/P). Note that the center Z(G) acts trivially on the above bundle so one can twist it and get a new vector bundle

\xymatrix{(G/L)_{γ} \ar[r] & (G/P)_{γ},}

hence K-theory for the inner form of a flag variety could be computed by our method as well.

9.2. Even dimensional affine quadric. This case corresponds to the inclusion SO_{2n} ⊂ SO_{2n+1} and for the root systems it is D_{n} ⊂ B_{n}.

First of all we pass to the simply connected group,

Spin_{2n+1}/Spin_{2n} = SO_{2n+1}/SO_{2n}.

One has [W (B_{n}) : W (D_{n})] = 2 so there are two elements in the basis. First element V (λ_{1}) as usual corresponds to the trivial representation of Spin_{2n} and as the second we can take one of the half-spin representations V (ω_{n−1}), V (ω_{n}), since the algorithm from theorem 2 suggestes one of the fundamental weights having the orbit consisting of two points.

After twisting with γ we get a quadric X(q) defined by a quadratic form q, then the algebra A(λ_{2})_{γ} = C_{0}(q) is the even Clifford algebra for the form q [Ti]. Hence one has

K_{∗}(X(q)) = K_{∗}(k) ⊕ K_{∗}(C_{0}(q)).

This answer coincides with the one obtained in [Sw].

9.3. Quaternionic projective space. We consider

HP^{n} = Sp_{2n+2}/(Sp_{2} × Sp_{2n})

as an algebraic model for the quaternionic projective space. The motivation comes from the fact that HP^{n}(ℂ) is homotopy equivalent to the usual quaternionic projective space ℍℙ^{n}. An extensive treatment of the quaternionic flag varieties including the simplest case of projective spaces one can find in [PW].

The root systems in this case are C_{1} + C_{n} ⊂ C_{n+1}, so basis consists of

[W (C_{n+1}) : (W (C_{1}) × W (C_{n}))] = n + 1

elements. We have dealt with this case in section 5.3 and the basis consists of Λ^{i}W for regular representation W of Sp_{2n}. The center acts trivially on the even degrees and nontrivially on the odd ones, so one has

K_{∗}(HP_{γ}^{n}) = K_{∗}(k)^{⌈ \frac{n+1}{2} ⌉} ⊕ K_{∗}(A(λ_{1})_{γ})^{⌊ \frac{n+1}{2} ⌋}.

In the split case it reduces to K_{∗}(HP^{n}) = K_{∗}(k)^{n+1}, and it agrees with the result obtained in [PW].

9.4. Zero characteristic. In this case we can treat non-reductive groups. When char k = 0 one has the Levi decomposition G = L_{G}U_{G} of group G into the semidirect product of some reductive subgroup and the unipotent radical [Mc], which in general fails in the positive characteristic. Also in this case the unipotent radical U_{G} splits, i.e. it has a filtration with vector factors [KMT] so the underlying variety is 𝔸^{n}, which also can fail over nonperfect fields. Hence for the connected split groups of the same rank H ⊂ G one has the following triangle.

\xymatrix{G/L_{H} \ar[r]^{p_{1}} \ar[d]^{p_{2}} & G/H \\ L_{G}/L_{H}}

The fibres of p_{1} and p_{2} are isomorphic to U_{G} and U_{H} respectively and both are affine spaces. One can show that both the projections define some vector bundles with the trivial action of the center Z(L_{G}), so one can twist with L_{G}/Z(L_{G})-cocycle γ and from the homotopy invariance obtain that

K_{∗}((G/H)_{γ}) = K_{∗}((L_{G}/L_{H})_{γ}).

The last one could be computed using the methods introduced in this paper.

9.5. Octonionic projective plain. It could be shown [Ba] that

𝕆ℙ^{2} ≅ G(F_{4})/Spin(9),

where G(F_{4}) stands for the compact form of the simple algebraic group with the root system F_{4}. We consider as an algebraic model

OP^{2} = G(F_{4})/Spin_{9}

with split G(F_{4}). It corresponds to the root systems B_{4} ⊂ F_{4} treated in section 5.2. One has

[W (F_{4}) : W (B_{4})] = 3,

and the corresponding representations for Spin_{9} are k, V (ω_{3}), V (ω_{4}), i.e. the trivial one, 84-dimensional Λ^{3}W and 16-dimensional spinor representation. Since the center of G(F_{4}) is trivial the twisting does not produce interesting algebras, though it changes the variety. Hence one has

K_{∗}(OP_{γ}^{2}) = K_{∗}(k) ⊕ K_{∗}(k) ⊕ K_{∗}(k).

References

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