Equivariant formality in $ K $-theory

EQUIVARIANT FORMALITY IN K-THEORY

arXiv:1704.04796v2 [math.AT] 10 Jun 2017

CHI-KWONG FOK

Abstract. In this note we present an analogue of equivariant formality in K-theory and show that it is equivalent to equivariant formality ` a la Goresky-Kottwitz-MacPherson. We also apply this analogue to give alternative proofs of equivariant formality of conjugation action on compact Lie groups, left translation action on generalized flag manifolds, and compact Lie group actions with maximal rank isotropy subgroups.

1. Introduction Equivariant formality, first defined in [GKM], is a special property of group actions on topological spaces which allows for easy computation of their equivariant cohomology. A G-action on a space X is said to be equivariantly formal if the Leray-Serre spectral sequence of the fiber bundle X ֒→ X ×G EG → BG collapses on the E2 -page. The latter is also ∗ (X) ∼ H ∗ (pt) ⊗ H ∗ (X) as H ∗ (pt)-modules. There are various examples equivalent to HG = G G of interest which are known to be equivariantly formal, e.g. Hamiltonian group actions on compact symplectic manifolds and linear algebraic torus actions on smooth complex projective varieties (cf. [GKM, Section 1.2 and Theorem 14.1]). Though equivariant formality was first defined in terms of equivariant cohomology, in some situations working with analogous notions phrased in terms of other equivariant cohomology theories may come in handy. The notion of equivariant formality in K-theory was introduced and explored by Harada and Landweber in [HL], where they instead used the term ‘weak equivariant formality’ and exploited this notion to show equivariant formality of Hamiltonian actions on compact symplectic manifolds. Definition 1.1 (cf. [HL, Def. 4.1]). Let k be a ring, G a compact Lie group and X a ∗ (X; k)) to denote K ∗ (X) ⊗ k (resp. K ∗ (X) ⊗ k). We G-space. We use K ∗ (X; k) (resp. KG G also write R(G; k) = R(G) ⊗ k, and I(G; k) = I(G) ⊗ k, where I(G) is the augmentation ideal of R(G). Let ∗ fG : KG (X) → K ∗ (X) be the forgetful map. A G-action on a space X is k-weakly equivariantly formal if fG induces an isomorphism ∗ KG (X; k) ⊗R(G;k) k → K ∗ (X; k) We simply say the action is weakly equivariantly formal in the case k = Z. Harada and Landweber settled for Definition 1.1 as the K-theoretic analogue of equi∗ (X) ∼ K ∗ (pt) ⊗ K ∗ (X), variant formality, instead of the seemingly obvious candidate KG = G Date: June 10, 2017. 1

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citing the lack of the Leray-Serre spectral sequence for Atiyah-Segal’s equivariant Ktheory. The term ‘weak’ is in reference to the condition in Definition 1.1 being weaker ∗ (X) ∼ K ∗ (pt) ⊗ K ∗ (X) because of the possible presence of torsion. We would than KG = G like to define the following version of K-theoretic equivariant formality in exact analogy with another cohomological equivariant formality condition that the forgetful map ∗ (X) → H ∗ (X) be onto. HG Definition 1.2. We say that X is a rational K-theoretic equivariantly formal (RKEF for short) G-space if the forgetful map ∗ fG ⊗ IdQ : KG (X; Q) → K ∗ (X; Q) is onto. This condition admits a natural interpretation in terms of vector bundles: for every vector bundle V over X and its suspension ΣX, there are natural numbers m, n such that V ⊕m ⊕ Cn admits an equivariant G-structure. In this note, we will prove the following theorem, which asserts the equivalence of RKEF and equivariant formality in the classical sense. Theorem 1.3. Let G be a compact and connected Lie group which acts on a finite CWcomplex X. The following are equivalent. (1) X is a RKEF G-space. (2) X is an equivariantly formal G-space. (3) X is a Q-weakly equivariantly formal G-space. We will also give alternative proofs of equivariant formality of certain group actions which were proved in cohomological terms. These are conjugation action on compact Lie groups, left translation action on generalized flag manifolds, and compact Lie group actions with maximal rank isotropy subgroups. In the remainder of this note, the coefficient ring of any cohomology is always Q. Acknowledgment. We would like to thank Ian Agol for answering a question on the homotopy groups of classifying spaces of unitary groups. 2. The proof From now on, unless otherwise specified, X is a finite CW-complex equipped with an action by a torus T or more generally a compact connected Lie group G. The following K-theoretic abelianization result enables us to prove K-theoretic results in this Section in the T -equivariant case first and then generalize to the G-equivariant case. Theorem 2.1 (cf. [HLS, Theorem 4.9(ii)]). Let T be a maximal torus of G and W the ∗ (X; Q) → K ∗ (X; Q) restricting the G-action to the T -action Weyl group. The map r ∗ : KG T is an injective map onto KT∗ (X; Q)W . Here if w ∈ W and V is an equivariant T -vector bundle, w takes V to the same underlying vector bundle with T -action twisted by w, and this W -action on the set of isomorphism classes of equivariant T -vector bundles induces the W -action on KT∗ (X). We will also use the following result on equivariant Chern character on several occasions.


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Proposition 2.2. Let G be a compact connected Lie group acting on a finite CW-complex + ∗ (pt). Denote the completion of H ∗ (X) at (pt) of HG X, and J the augmentation ideal HG G ∗∗ J by HG (X). Then the equivariant Chern character ∗ ∗∗ chG : KG (X; Q) → HG (X)

is injective, and ch−1 G (J) = I(G; Q) when X is a point. ∗ (X; Q) → K ∗ (X; Q)ˆ ∼ K ∗ (X × EG; Q) Proof. The map chG is the composition of ι : KG = G G ∗ ∗ (here KG (X; Q)ˆ means the completion of KG (X; Q) at I(G; Q), and the isomorphism is ∗∗ (X). asserted by [AS, Theorem 2.1]) and the Chern character ch : K ∗ (X ×G EG; Q) → HG The Chern character map is injective, and so is ι because the I(G; Q)-adic topology of the completion is Hausdorff if G is connected (cf. the Note immediately preceding [AH, Section 4.5]). Therefore chG is injective. Next, consider the commutative diagram

/ K ∗ (pt; Q)

R(G) chG

ch

∗∗ (pt) HG

/ H ∗ (pt)

where the two horizontal maps are forgetful maps. Since J is the kernel of the bottom map and both chG and ch are injective, ch−1 G (J) is the kernel of the top map, which is precisely I(G; Q). Under the condition of weak equivariant formality, [HL, Proposition 4.2] asserts that the ∗ (X). In fact, we also have kernel of f is I(G) · KG Lemma 2.3. Let X be a finite CW-complex which is acted on by a compact connected Lie group G equivariantly formally. Then the kernel of the forgetful map ∗ fG ⊗ IdQ : KG (X; Q) → K ∗ (X; Q) ∗ (X; Q). is I(G; Q) · KG

Proof. In the following diagram, (1)

fG ⊗IdQ

∗ (X; Q) KG

/ K ∗ (X; Q)

chG

ch

∗∗ (X) HG

geG ⊗IdQ

/ H ∗ (X)

∗∗ (X) is the completion of H ∗ (X) at the augmentation ideal J of H ∗ (pt). Since X is HG G G ∗ (X) is isomorphic to H ∗ (pt) ⊗ H ∗ (X) as a H ∗ (pt)an equivariantly formal G-space, HG G G module, and the forgetful map ∗ gG ⊗ IdQ : HG (X) → H ∗ (X) ∗ (X) as the kernel. Since H ∗ (X) is a finitely generated module over the Noetherian has J ·HG G ∗ (pt), a simple result on completions (cf. [Ma, Theorem 55]) implies that H ∗∗ (X) ∼ ring HG = G ∗∗ ∗∗ (X). By Proposition 2.2, the ∗ (X) ⊗ ∗ gG ⊗ IdQ is J · HG HG HG (pt) HG (pt). So the kernel of e

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preimage ch−1 G (J) is I(G; Q) and chG is injective. It follows that the kernel of fG ⊗ IdQ is ∗∗ (X)) = I(G; Q) · K ∗ (X; Q). (J · H ch−1 G G G Proof of Theorem 1.3, (1) ⇐⇒ (2). We first deal with the T -equivariant case, where T is a maximal torus of G. We claim that, if X is an equivariantly formal T -space, we have the following string of (in)equalities. dim K ∗ (X T ; Q) = rankR(T ;Q) KT∗ (X; Q) ≤ dim KT∗ (X; Q)/I(T ; Q)·KT∗ (X; Q) ≤ dim K ∗ (X; Q). The first equality follows from Segal’s localization theorem (cf. [Se, Proposition 4.1]) which, when applied to torus group action, says that the restriction map KT∗ (X; Q) → KT∗ (X T ; Q) becomes an isomorphism after localizing at the zero prime ideal, i.e. to the field of fraction of R(T ; Q). Next, we let n = dim KT∗ (X; Q)I(T ;Q) /I(T ; Q) · KT∗ (X; Q)I(T ;Q) and KT∗ (X; Q)I(T ; Q)/I(T ; Q) · KT∗ (X; Q)I(T ;Q) be spanned by x1 , · · · , xn as a vector space over R(T ; Q)I(T ;Q) /I(T ; Q) · R(T ; Q)I(T ;Q) ∼ = Q. Seeing that KT∗ (X; Q) is a finitely generated module over the Noetherian ring R(T ; Q), we invoke Nakayama lemma, and have that there exist lifts x b1 , · · · , x bn ∈ KT∗ (X; Q)I(T ;Q) that generate KT∗ (X; Q)I(T ;Q) as a R(T ; Q)I(T ;Q) -module. It follows, after further localization to the field of fraction of R(T ), that x b1 , · · · , x bn span KT∗ (X; Q)(0) as a R(T ; Q)(0) -vector space. Noting the isomorphism ∗ KT (X; Q)/I(T ; Q)·KT∗ (X; Q) ∼ = KT∗ (X; Q)I(T ;Q) /I(T ; Q)·KT∗ (X; Q)I(T ;Q) , we arrive at the first inequality. Finally, the last inequality follows from Lemma 2.3. If X is an equivariantly formal T -space, then dimH ∗ (X) = dimH ∗ (X T ). The Chern character isomorphism implies that dimK ∗ (X T ; Q) = dimK ∗ (X; Q) which, together with the (in)equalities in the above claim, yields dimKT∗ (X; Q)/I(T ; Q)·KT∗ (X; Q) = dimK ∗ (X; Q) or, equivalently, that X is RKEF. Assume on the other hand that X is RKEF. Consider the commutative diagram (1). Since fT ⊗ IdQ is onto and ch is an isomorphism, geT ⊗ IdQ is onto. By [Ma, Theorem 55], we have that HT∗∗ (X) ∼ = HT∗ (X) ⊗HT∗ (pt) HT∗∗ (pt). Applying geT ⊗ IdQ gives H ∗ (X) = Im(e gT ⊗ IdQ ) = Im(gT ⊗ IdQ ) ⊗Q Q = Im(gT ⊗ IdQ ). Hence X is T -equivariantly formal. For the G-equivariant case, it suffices to show that fT ⊗ IdQ is onto if and only if fG ⊗ IdQ is onto, because it is known that X is T -equivariantly formal if and only if it is G-equivariantly formal. One direction is easy: if fG ⊗ IdQ is onto, so is fT ⊗ IdQ because fG ⊗ IdQ = (fT ⊗ IdQ ) ◦ r ∗ . Conversely, suppose that fT ⊗ IdQ is onto. Then any x ∈ K ∗ (X; Q) admits a lift x e ∈ KT∗ (X; Q). Note that for any w ∈ W , (fT ⊗ IdQ )(w · x e) = x. It follows that the average 1 X w·x e x := |W | w∈W

∗ (X; Q) is also a lift of x. Moreover, by Theorem 2.1, x ∈ r ∗ KG (X; Q). So (r ∗ )−1 (x) ∈ KG is a lift of x and fG ⊗ IdQ is onto as well.

Proof of Theorem 1.3, (1) ⇐⇒ (3). That Q-weakly equivariant formality implies RKEF is immediate. On the other hand, if X is a RKEF G-space, then by Theorem 1.3, X is an equivariantly formal G-space. Using Lemma 2.3 and X being RKEF, the map ∗ KG (X; Q) ⊗R(G;Q) Q → K ∗ (X; Q)


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α ⊗ z 7→ fG (α)z is an isomorphism, and hence X is Q-weakly equivariantly formal G-space. This completes the proof. 3. Some applications In this Section, we shall demonstrate the utility of Theorem 1.3 by giving alternative proofs of some previous results. 3.1. Conjugation action on compact Lie groups. Let G be a compact connected Lie group with conjugation action by itself. It is well-known that this action is equivariantly formal. See, for example, [GS, Sect. 11.9, Item 6]) for a sketch of proof for the case G = U (n), and [J] for an explicit construction of equivariant extensions of the generators of H ∗ (G). We will show equivariant formality of conjugation action by proving that G is a RKEF G-space. By [Ho, II, Theorem 2.1], ^∗ K ∗ (G; Q) ∼ (R ⊗ Q), = Q

where R is the image of the map

δ : R(G) → K −1 (G) which sends ρ ∈ R(G) to the following complex of vector bundles1 0 −→ G × R × V −→ G × R × V −→ 0 (g, t, −tρ(g)v), if t ≥ 0, (g, t, v) 7→ (g, t, v), if t ≤ 0. ∗ (G) because G × R × V can be equipped For any ρ, δ(ρ) admits an equivariant lift in KG with the G-action given by

g0 · (g, t, v) = (g0 gg0−1 , t, ρ(g0 )v), with respect to which the middle map of the above complex of vector bundles is G∗ (G; Q) → K ∗ (G; Q) is onto, i.e., G is a RKEF G-space. equivariant. Thus fG ⊗ IdQ : KG 3.2. Left translation action on G/K where rank G = rank K. Let G be a compact connected Lie group and K a connected Lie subgroup of the same rank. The left translation action on G/K by G is well-known to be equivariantly formal, which can be proved by noting that G/K satisfies the sufficient condition for equivariant formality that its odd cohomology vanish (cf. [GHV, Chapter XI, Theorem VII]). Alternatively, by the rationalized version of [Sn, Theorem 4.2] and the remark following it, K ∗ (G/K; Q) ∼ = R(K; Q) ⊗R(G;Q) Q ∼ = R(K; Q)/r ∗ I(G; Q), where r ∗ : R(G; Q) → R(K; Q) is the restriction map. The forgetful map fG ⊗ IdQ : ∗ (G/K; Q) ∼ R(K; Q) → K ∗ (G/K; Q) is simply the projection map and hence surjective KG = (in fact the forgetful map sends any representation ρ ∈ R(K) to the K-theory class of the 1The map δ, which was defined in [BZ] and corrected in [F], is the same as the map β defined in [Ho].

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homogeneous vector bundle G ×K Vρ , where Vρ is the underlying complex vector space for ρ). Thus G/K is a RKEF G-space, and equivalently an equivariantly formal G-space. Remark 3.1. In the more general case where equality of ranks of G and K is not assumed, a representation theoretic characterization of equivariant formality of the left translation action of K on G/K is given by virtue of RKEF in [CF]. 3.3. Actions with connected maximal rank isotropy subgroups. Let G be a compact connected Lie group and X a finite G-CW complex. Suppose that the G-action on X has maximal rank connected isotropy subgroups. By [GR, Corollary 3.5]2, such a G-action on X is equivariantly formal. We would like to give a different proof of this result by using Theorem 1.3 and induction on the dimension of X. We shall remark that the group actions considered in Sections 3.1 and 3.3 are examples of group actions we discuss in this section. However, equivariant formality of left translation actions on generalized flag manifolds as in Section 3.3 is used in the following proof. Consider the n-skeleton Xn . It is obtained by gluing the cells G/Ki × Dn , 1 ≤ i ≤ k, to the (n − 1)-skeleton Xn−1 through some G-equivariant attaching maps fi : G/Ki × ∂Dn → Xn−1 . Let Fi : G/Ki × Dn → Xn be the inclusion of the cell G/Ki × Dn into Xn , and V be any given vector bundle over Xn . We shall show that, for some p and q, V ⊕p ⊕ Cq admits an equivariant structure, assuming that V0 , which is the restriction of V to Xn−1 , admits an equivariant structure by induction hypothesis. Note that V can be obtained by gluing V0 → Xn−1 and Vi → G/Ki × Dn , where Vi := Fi∗ V , through the clutching maps, i.e. vector bundle homomorphisms hi : Vi |G/Ki ×∂Dn → V0 which cover the maps fi and send fiber to fiber isomorphically. By the discussion in Section 3.2 and contractibility of Dn , Vi must be of the form (G × Dn ) ×Ki Wi for some Ki representation Wi . Hence Vi admits an equivariant structure. By the induction hypothesis, V0⊕p ⊕ Cq admits an equivariant structure for some p and q. Consider the clutching maps ⊕p ⊕p q q ⊕ Cq h⊕p i ⊕ Id : Vi |G/Ki ×∂Dn ⊕ C → V0

for the vector bundles Vi⊕p ⊕ Cq and V0⊕p ⊕ Cq . The vector bundle V ⊕p ⊕ Cq admits an ⊕q is homotopy equivalent to another clutching map which equivariant structure if h⊕p i ⊕ Id ⊕p is G-equivariant. Note that hi ⊕ Id is the composition of ′

hi⊕p ⊕ Id : Vi⊕p |G/Ki ×∂Dn ⊕ C⊕q → fi∗ V0⊕p ⊕ Cq and

fei⊕p ⊕ Id : fi∗ V0⊕p ⊕ Cq → V0⊕p ⊕ Cq .

′

The latter map is obviously G-equivariant, so it suffices to show that hi⊕p ⊕ Id is homotopy ′ equivalent to a G-equivariant map. Note that h ⊕p ⊕Id is a vector bundle isomorphism, and 2The conditions of [GR, Corollary 3.5] are in general weaker because connectedness of the isotropy subgroups are not assumed. Though the space under consideration in [GR] is a G-manifold, the proof itself does not make essential use of this assumption and can be easily adapted to the more general case of G-CW complexes.


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again by the discussion in Section 3.2 and contractibility of Dn , both Vi⊕p |G/Ki ×∂Dn ⊕ Cq and fi∗ V0⊕p ⊕ Cq can be identified with (G × ∂Dn ) ×Ki Wi′ , where Wi′ is isomorphic to ′ Wi⊕p ⊕ Cqtriv as Ki -representations. Thus we may think of hi⊕p ⊕ Id as a vector bundle automorphism of (G×∂Dn )×Ki Wi′ or, more simply, G×Ki Wi′ (as Ki acts on ∂Dn trivially). After equipping G ×Ki Wi′ with a Hermitian structure and letting P be its unitary frame bundle (which is a principal U (m)-bundle, where m is dim Wi′ ), we may further identify ′ hi⊕p ⊕ Id with an element in the gauge group G(P ) of P . In fact, if q is sufficiently large, ′ then hi⊕p ⊕ Id is homotopy equivalent to the identity automorphism which is obviously G-equivariant because Proposition 3.2. The gauge group G(P ) is connected for q sufficiently large. Proof. We will follow the discussion in [AB, pp. 540-542] to prove connectedness of G(P ). Since π1 (BG(P )) = π0 (G(P )), and BG(P ) is homotopy equivalent to the component of the space of maps Map(G/Ki , BU (m)) containing a map of G/Ki into BU (m) which induces P ([AB, Proposition 2.4]), it remains to show that π1 (Map(G/Ki , BU (m))) is trivial. It is well-known that G/Ki is a CW-complex consisting of only even dimensional cells. Let Yk be its k-skeleton. Then the skeletal filtration Y0 ⊂ Y2 ⊂ · · · ⊂ Y2r = G/Ki induces a series of cofibrations _ Y2r−2 −→ G/Ki −→ S2r _ Y2r−4 −→ Y2r−2 −→ S2r−2 .. .

Y0 −→ Y2 −→

_

S2

which gives rise to a series of fiberings _ Map∗ ( S2r , BU (m)) −→ Map∗ (G/Ki , BU (m)) −→ Map∗ (Y2r−2 , BU (m)) _ Map∗ ( S2r−2 , BU (m)) −→ Map∗ (Y2r−2 , BU (m)) −→ Map∗ (Y2r−4 , BU (m)) .. .

_ Map∗ ( S2 , BU (m)) −→ Map∗ (Y2 , BU (m)) −→ Map∗ (Y0 , BU (m))

where Map∗ means the space of base-point preserving maps. Together with the fibration Map∗ (G/Ki , BU (m)) −→ Map(G/Ki , BU (m)) −→ BU (m),Wwe see that Map(G/K Q i , BU (m)) is a series of fibrations constructed out of BU (m) and Map∗ ( S2k , BU (m)) = Ω2k BU (m) for 1 ≤ k ≤ r. It suffices to show that all these spaces are simply-connected. Note that π1 (BU (m)) = 0, and π1 (Ω2k BU (m)) = π2k+1 (BU (m)) = π2k (U (m)) which, when q is sufficiently large (and so is m), stabilizes and equals π2k (U (∞)). The latter is, by Bott periodicity, π0 (U (∞)) = 0. This finishes the proof of the Proposition.

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We have shown that, by induction on the dimension of X, for any given vector bundle V → X, V ⊕p ⊕ Cq admits an equivariant structure for some p and q. The same is true for the suspension ΣX by analogous arguments. It follows that the G-action on X is equivariantly formal by Theorem 1.3. References [AB]

M. F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond. A, Vol. 308, pp. 523-615, 1982. [AH] M. F. Atiyah, F. Hirzebruch, Vector bundles and homogeneous spaces, AMS Symposium in Pure Math. III (1960). [AM] M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969. [AS] M. F. Atiyah, G. Segal, Equivariant K-theory and completion, J. Differential Geometry 3, 1-18, 1969. [BZ] J.-L. Brylinski, B. Zhang, Equivariant K-theory of compact connected Lie groups, K-theory, Vol. 20, pp. 23-36, 2000. [CF] J. D. Carlson, C.-K. Fok, Equivariant formality of homogeneous spaces, preprint available at https://arxiv.org/abs/1511.06228 [F] C.-K. Fok, The Real K-theory of compact Lie groups, Symmetry, Integrability and Geometry: Methods and Applications, Vol. 10, 022, 26 pages, 2014. [GR] O. Goertsches, S Rollenske, Torsion in equivariant cohomology and Cohen-Macaulay G-actions, Transformation Groups, Vol. 16, No. 4, pp. 1063-1080, 2011. [GKM] M. Goresky, R. Kottwitz, R. Macpherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131, no. 1, 25-83, 1998. [GHV] W. Greub, S. Halperin, R. Vanstone, Connections, curvature, and cohomology, Vol. 3: Cohomology of principal bundles and homogeneous spaces, Academic Press, 1976. [GS] V. W. Guillemin, S. Sternberg, Supersymmetry and Equivariant de Rham Theory, Mathematics Past and Present, Springer-Verlag, Berlin, 1999. [HL] M. Harada, G. Landweber, Surjectivity for Hamiltonian G-spaces in K-theory, Trans. Amer. Math. Soc., Vol. 359, No. 12, pp. 6001-6025, Dec. 2007. [Har] R. Hartshorne, Algebraic geometry, Graduate Texts in Math. 52, Springer, 1977. [HLS] M. Harada, G. Landweber, R. Sjamaar, Divided differences and the Weyl character formula in equivariant K-theory, Math. Res. Lett. 17, no. 3, 507-527, 2010. [Ho] L. Hodgkin, On the K-theory of Lie groups, Topology, Vol. 6, pp. 1-36, 1967. [J] L. Jeffrey, Group cohomology construction of the cohomology of moduli spaces of flat connections on 2-manifolds, Duke Math. J., Vol. 77, No. 2, pp. 407-429, 1995. [Ma] H. Matsumura, Commutative Algebra, Benjamin/Cummings, 1980. ´ [Se] G. Segal, Equivariant K-theory, Inst. Hautes Etudes Sci. Publ. Math., No. 34, 129-151, 1968. [Sn] V. P. Snaith, On the K-theory of homogeneous spaces and conjugate bundles of Lie groups, Proc. London. Math. Soc., Vol. 3, No. 3, pp. 562-584, 1971. National Center for Theoretical Sciences, Mathematics Division, National Taiwan University, Taipei 10617, Taiwan E-mail: [email protected] URL: https://sites.google.com/site/alexckfok