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EQUIVARIANT COHOMOLOGY OF QUATERNIONIC FLAG MANIFOLDS

arXiv:math/0605539v1 [math.DG] 18 May 2006

AUGUSTIN-LIVIU MARE Abstract. The main result of the paper is a Borel type description of the Sp(1)n -equivariant cohomology ring of the manifold F ln (H) of all complete ?ags in Hn . To prove this, we obtain a Goresky-Kottwitz-MacPherson type description of that ring.

1. Introduction In this paper we study the quaternionic ?ag manifold F ln (H), which is the space of all sequences (V1 , . . . , Vn ) where Vν is a ν -dimensional quaternionic vector subspace (that is, left H-submodule) of Hn , for 1 ≤ ν ≤ n, such that V1 ? V2 ? . . . ? Vn . We can see that F ln (H) has a transitive action of the symplectic group Sp(n), with stabilizer K := Sp(1)n . We are interested in the K -equivariant cohomology1 of F ln (H). We describe this ring in terms of the canonical vector bundles Vν over F ln (H), where 0 ≤ ν ≤ n. More precisely, we prove the following theorem. Theorem 1.1. One has the ring isomorphism

? (F ln (H), Z) ? Z[x1 , . . . , xn , u1 , . . . , un ]/ (1 + x1 ) . . . (1 + xn ) = (1 + u1 ) . . . (1 + un ) , HK 4 where xν = eK (Vν /Vν ?1 ) ∈ HK (F ln (H), Z) is the K -equivariant Euler class of Vν /Vν ?1 and u1 , . . . , un are copies of the generator of H 4 (BSp(1), Z), so that

H ? (BK, Z) = H ? (BSp(1)n , Z) = Z[u1 , . . . , un ]. The strategy used is as follows. First we prove the Kirwan injectivity type result for the K ? ? action on F ln (H); that is, the restriction map ?? : HK (F ln (H)) → HK (F ln (H)K ) is injective, where F ln (H)K denotes the ?xed point set of the K action. Then we prove the GoreskyKottwitz-MacPherson (shortly GKM) type characterization of the image of ?? . Finally we compare what we have obtained with the GKM picture of the T -equivariant cohomology of F ln (C), where T = (S 1 )n is the maximal torus of U (n). We observe that there exists an abstract isomorphism between the two cohomology rings, which doubles the degrees. At the ? end, we use the known Borel-type description of HT (F ln (C)).

? Remarks. 1. The proof of the GKM presentation of HK (F ln (H)) we will give in section 3 uses the methods of Tolman and Weitsman [To-We] (see also Harada and Holm [Ha-Ho, section 2]).

Date : February 2, 2008. 1All cohomology rings in this paper will be with coe?cients in Z (unless otherwise speci?ed).

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AUGUSTIN-LIVIU MARE

2. A presentation of the usual cohomology ring of F ln (H) has been described by us in [Ma1, section 3] (the result was originally proved by Hsiang, Palais, and Terng [Hs-Pa-Te]). Theorem 1.1 gives the equivariant “deformation” of that presentation. Acknowledgements. I wanted to thank Megumi Harada and Tara Holm for reading a previous version of the paper and making some excellent suggestions. I also thank Jost Eschenburg for discussions about the topics of the paper. 2. The quaternionic flag manifold The goal of this section is to give a few alternative presentations of the manifold F ln (H), which will be used later. Let H = {a + bi + cj + dk | a, b, c, d ∈ R} be the skew ?eld of quaternions. The space Hn is a H module with respect to the scalar multiplication from the left. We equip it with the scalar product ( , ) given by

n

(h, k ) =

ν =1 n

?ν , hν k

for all h = (h1 , . . . , hn ), k = (k1 , . . . , kn ) in H . Any linear transformation of Hn is described by a matrix A ∈ Matn×n (H) according to the formula Ah := h · A? , where h = (h1 , . . . , hn ) ∈ Hn . Here · denotes the matrix multiplication and the superscript ? indicates the transposed conjugate of a matrix. We denote by Sp(n) the group of linear transformations A of Hn with the property that (A.h, A.k ) = (h, k ), for all h, k ∈ Hn . Alternatively, Sp(n) consists of all n × n matrices A with entries in H with the property that A · A? = In . The ?ag manifold F ln (H) de?ned in the introduction can also be described as the space of all sequences (L1 , . . . , Ln ) of 1-dimensional H-submodules of Hn such that Lν is perpendicular to L? for all ?, ν ∈ {1, 2, . . . , n}, ? = ν . The group Sp(n) acts transitively on the latter space. Indeed, let us consider the canonical basis e1 = (1, 0, . . . , 0), . . . , en = (0, . . . , 0, 1) of Hn ; if h1 , . . . , hn is an orthonormal system in Hn , then the matrix A whose columns are (h1 )? , . . . , (hn )? is in Sp(n) and satis?es Aeν = hν , 1 ≤ ν ≤ n. The Sp(n) stabilizer of the ?ag (He1 , . . . , Hen ) consists of diagonal matrices, that is, it is equal to Sp(1)n . In this way we obtain the identi?cation (1) F ln (H) = Sp(n)/Sp(1)n .

Yet another presentation of the quaternionic ?ag manifold can be obtained by considering the conjugation action of Sp(n) on the space Hn := {X ∈ Matn×n (H) | X = X ? , Trace(X ) = 0}. We pick n distinct real numbers r1 , . . . , rn with r1 < r2 < . . . < rn and n ν =1 rν = 0, and consider the orbit of the diagonal matrix Diag(r1 , . . . , rn ). For any element X of the orbit

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there exist mutually orthogonal lines L1 , . . . , Ln such that X |Lν is the multiplication by rν , for all 1 ≤ ν ≤ n. This gives the identi?cation (2) F ln (H) = Sp(n).Diag(r1 , . . . , rn ).

Finally, we also mention that F ln (H) is an s-orbit or a real ?ag manifold. More precisely, it is a principal isotropy orbit of the symmetric space SU (2n)/Sp(n). It turns out that the isotropy representation of the latter space is just the conjugation action of Sp(n) on Hn mentioned above. Via this identi?cation, the metric on Hn turns out to be the standard one, given by X, Y := Re(Trace(XY )), X, Y ∈ Hn . Moreover, a maximal abelian subspace of Hn is the space d of all diagonal matrices with real entries and trace 0. For the details, we refer the reader to [Ma2, Example 5.4]. Let us consider an element A = Diag(a1 , . . . , an ) of d, where aν are real numbers such that a1 < a2 < . . . < an and n ν =1 aν = 0. The corresponding height function hA (X ) := A, X , X ∈ F ln (H), will be an important instrument in our paper. We will use the following result, a proof of which can be found in the last section of our paper. Proposition 2.1. group Sn via (3) (i) The critical set Crit(hA ) can be identi?ed with the symmetric

Sn ? w = Diag(rw(1) , . . . , rw(n) ) = (Hew(1) , . . . , Hew(n) ), w ∈ Sn (see also equation (2)). All critical points are non-degenerate. (ii) Take w ∈ Sn and v := spq w , for some 1 ≤ p < q ≤ n, where spq denotes the transposition of p and q in the symmetric group Sn . Assume that hA (w ) > hA (spq w ). Then the subspace

(4)

Sw,pq := Kpq .w,

of F ln (H) is a metric sphere of dimension four in (Hn , , ), for which w and spq w are the north, respectively south pole (with respect to the height function hA ). Here Kpq denotes the subgroup of Sp(n) consisting of matrices with all entries zero, except for those on the diagonal and on the positions pq and qp. The meridians of Sw,pq are gradient lines between w and spq w for the function hA : F ln (H) → R with respect to the submanifold metric induced by , . (iii) The negative space of the Hessian of hA at w is (p,q )∈I Tw Sw,pq , where by I we denote the set of all pairs (p, q ) with 1 ≤ p < q ≤ n such that hA (w ) > hA (wspq ). The sphere Sw,pq can also be described as the set of all ?ags (L1 , . . . , Ln ) with the property that Lν = Heν , for all ν ∈ / {p, q } and Lp and Lq are arbitrary (orthogonal) lines in Hew(p) ⊕ Hew(q) . It is obvious that Sw,pq can be identi?ed with the projective line HP 1 = P(Hew(p) ⊕ Hew(q) ), which is indeed a four-sphere.

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AUGUSTIN-LIVIU MARE

? 3. The GKM type description of HK (F ln (H))

We consider the canonical embedding of U (1) = {a + bi | a2 + b2 = 1} into Sp(1) = {a + bi + cj + dk | a2 + b2 + c2 + d2 = 1}. This induces an embedding of T := U (1)n into K = Sp(1)n (as spaces of diagonal matrices). We are interested in the ?xed points of the action of the groups T and K on F ln (H). Lemma 3.1. The ?xed point sets F ln (H)T and F ln (H)K are both equal to Sn (see the identi?cation (3)). Proof. It is obvious that any ?ag of the form indicated in the lemma is ?xed by K . We prove that a ?ag ?xed by T has the form indicated in the theorem. Equivalently, we show that if the vector h = (h1 , . . . , hn ) ∈ Hn has the property that L := Hh is T -invariant, then L must be Heν , for some ν ∈ {1, . . . , n}. Indeed, for any A = Diag(z1 , . . . , zn ) ∈ T we have A.h ∈ L, that is, there exists λ ∈ H such that (5) h1 z ?1 = λh1 , . . . , hn z ?n = λhn .

If h? = 0 and hν = 0 for ν = ?, we pick z? = 1 and zν = ?1 and see that there exists no λ satisfying (5). This ?nishes the proof. We will prove the following analogue of the Kirwan injectivity theorem.

? ? Proposition 3.2. The map ?? : HK (F ln (H)) → HK (Sn ) induced by the inclusion ? : Sn → F ln (H) is injective.

Proof. By Proposition 2.1, hA is a Morse function, whose critical set is Sn . Let us order the latter set as w1 , . . . , wk such that hA (w1 ) < hA (w2 ) < . . . < hA (wk ). Take ? > 0, smaller than 1 the minimum of hA (w? ) ?hA (w??1 ), where 2 ≤ ? ≤ k . Denote M? := h? A (?∞, ha (w? )+ ?], and ? ? ? ? Sn = Sn ∩ M? . We prove by induction on ? ∈ {1, . . . , k } that the map ?? ? : HK (M? ) → HK (Sn ) is injective. For ? = 1, the result is obviously true, as M1 is equivariantly contractible to {w1 }. We assume the result is true for ? ? 1. In the same way as in [To-We, diagram 2.5], we have the following commutative diagram. ···

//

? HK (M? , M??1 ) ?

2

//

? HK (M? )

//

? HK (M??1 )

//

···

(6)

1

HK

??index(w? )

({ w ? } )

∪e?

//

? HK ({ w ? } )

? Here ? denotes the isomorphism obtained by composing the excision map HK (M? , M??1 ) ? ? HK (D, S ) (where D , S are the unit disk, respectively unit sphere in the negative normal ? bundle of the Hessian of hA at the point w? ), with the Thom isomorphism HK (D, S ) ? ??index(w? ) HK ({w? }); the map 1 is induced by the inclusion of {w? } in M? ; the class e? ∈ index(w? ) HK ({w? }) = H index(w? ) (BK ) is the K -equivariant Euler class of the negative space of the Hessian of hA at w? . According to Lemma 3.1, w? is an isolated ?xed point of the T action on F ln (H). By the Atiyah-Bott lemma (see appendix A of our paper) we deduce that

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the multiplication by e? is an injective map. We deduce that the long exact sequence of the pair (M? , M??1 ) splits into short exact sequences of the form

? ? ? 0 ?→ HK (M? , M??1 ) ?→ HK (M? ) ?→ HK (M??1 ) ?→ 0.

Let us consider now the following commutative diagram. 0

//

? HK (M? , M??1 )

//

? HK (M? )

//

? HK (M??1 )

//

0

3

?? ?

?? ? ?1

0

//

? HK ({ w ? } )

// //

? ? HK ( Sn )

? ??1 HK ( Sn )

//

0

? ? ??1 ? where we have identi?ed HK ( Sn , Sn ) = HK ({w? }). The map ?? ??1 is injective, by the induction hypothesis. The map 3 is the same as the composition of 1 and 2 (see diagram (6)), thus it is injective as well. By a diagram chase we deduce that ?? ? is injective.

The following result will be used later. Lemma 3.3. The group G := (H? )n = {Diag(γ1 , . . . , γn ) | γν ∈ H? } acts transitively on Sw,pq \ {w, wspq } (see Proposition 2.1 for the de?nition of Sw,pq ). There exists a point in the latter space whose stabilizer is the group Gpq = {(γ1 , . . . , γn ) ∈ G | γp = γq }. Proof. We use the description of Sw,pq given after Proposition 2.1. To prove the lemma, we only need to note that for any two lines Hv1 and Hv2 in H ⊕ H which are di?erent from the “coordinate axes” He1 and He2 we have Hv1 = Hv2 · γ1 0 0 γ2

for some γ1 , γ2 ∈ H? . Moreover, in the same set-up, the H? × H? stabilizer of the line H(e1 + e2 ) is the group {(γ, γ ) | γ ∈ H? }.

Set N := Sw,pq , where the union runs over all w, p, q such that w ∈ Sn , 1 ≤ p < q ≤ n, and hA (w ) > hA (wspq ). We recall that since K = Sp(1)n , the K -equivariant cohomology of a point pt. is given by

? HK (pt.) = H ? (BK ) = H ? (BSp(1)n ) = Z[u1 , . . . , un ].

Lemma 3.4. (i) Fix p ∈ {1, . . . , n} and consider the action of K on the vector bundle H over a point pt. given by γ.h := γp h, for any γ = (γ1 , . . . , γn ) ∈ K and any h ∈ H. The K -equivariant Euler class of the bundle is up . (ii) Fix p, q ∈ {1, . . . , n} and consider the action of K on the vector bundle H over a point pt. given by γ.h := γp hγ ?q , for any γ = (γ1 , . . . , γn ) ∈ K and any h ∈ H. The K -equivariant Euler class of the bundle is up ? uq .

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AUGUSTIN-LIVIU MARE

Proof. (i) Let E = EK = E (Sp(1))n be the total space of the classifying bundle of K . By de?nition (see for instance section 6.2 of [Gu-Gi-Ka]), the equivariant Euler class is eK (H) = e(E ×K H), where E ×K H is a vector bundle with ?ber H over E/K = BK = B (Sp(1))n . Thus eK (H) is an element of degree 4 in H ? (BK ) = R[u1 , . . . , un ], hence a degree one polynomial in ∞ ∞ ∞ u1 , . . . , un . We denote BK = B (Sp(1))n = HP1 × . . . HPn , where HPν , 1 ≤ ν ≤ n, are ∞ copies of HP (see e.g. [Hu, chapter 8, Theorem 6.1]). The latter is the space of all H lines (that is, one dimensional H-submodules) in H∞ , where the scalar multiplication is from the ? . It is right. By this we mean that if e ∈ H∞ and h ∈ H, then, by de?nition, h.e := eh ∞ su?cient to show that if ?ν is the inclusion of HPν into BK , then we have ?? ν (e(E ×K H)) = 0, if ν = p x, if ν = p.

Here x ∈ H 4 (HP ∞ ) denotes the Euler class of the tautological vector bundle over HP ∞ . To ∞ show this, we note ?rst that if ν = p, then the restriction of the bundle E ×K H to HPν is ∞ the trivial bundle. Also, the restriction of E ×K H to HPp is the bundle ESp(1) ×Sp(1) H over ESp(1)/Sp(1) = HP ∞ (here the action of Sp(1) on H is by left multiplication). The latter bundle is actually the tautological bundle τ = {(v, L) | L is a H line in H∞ , v ∈ L} over HP ∞ . Indeed, we take into account that ESp(1) is the unit sphere S ∞ in H∞ (again by [Hu, chapter 8, Theorem 6.1]), thus we can identify ESp(1) ×Sp(1) H with τ via [e, h] → (eh, eH), for all e ∈ ESp(1) and h ∈ H. We conclude that ?? p (e(E ×K H)) = e(τ ) = x. (ii) This time we have to check that ? ? ?0, if ν = p or q ? ?ν (e(E ×K H)) = x, if ν = p ? ??x, if ν = q.

The cases ν = q have been discussed before, at point (i). To analyze the case ν = q , we note that the restriction of E ×K H to HPq∞ is ESp(1) ×Sp(1) H, where Sp(1) acts on H from the right. The map ? eH) ESp(1) ×Sp(1) H ? [e, h] → (eh, is an isomorphism between ESp(1) ×Sp(1) H and the rank four vector bundle over HP ∞ , call ? := {v it τ ?, whose ?ber over L is L ? | v ∈ L}. Because the linear automorphism of R4 = H given by (x1 , x2 , x3 , x4 ) = x1 + x2 i + x3 j + x4 k → x1 ? x2 i ? x3 j ? x4 k = (x1 , ?x2 , ?x3 , ?x4 ) is orientation changing (its determinant is equal to ?1), we deduce that ?? τ ) = ?e(τ ) = ?x. q (e(E ×K H)) = e(? The claim, and also the lemma, are completely proved.

QUATERNIONIC FLAG MANIFOLDS

7

The following result will also be needed later. Lemma 3.5. Fix w ∈ Sn , ? > 0 strictly smaller than |hA (w ) ? hA (v )|, for any v ∈ Sn , 1 ?1 ? v = w , and set N+ := N ∩ h? A (?∞, hA (w ) + ?], Sn = Sn ∩ hA (?∞, hA (w ) ? ?]. Let η be ? ? an element of HK (N + ) which vanishes when restricted to Sn . Then the restriction η |w is a multiple of ew , where ew is the K -equivariant Euler class of the negative space of the Hessian of hA at w . Proof. By Proposition 2.1, (iii), the negative space of the Hessian of hA at w is (p,q)∈I Tw Sw,pq . ? (Sw,pq ) vanishes when restricted to wspq , For each (p, q ) ∈ I , the class η1 := η |Sw,pq ∈ HK 1 which is the South pole of Sw,pq . Consequently, η1 vanishes when restricted to h? A (?∞, hA (w )? ?] ∩ Sw,pq , since the latter can be equivariantly retracted onto wspq . The long exact sequence 1 of the pair (Sw,pq , h? A (?∞, hA (w ) ? ?] ∩ Sw,pq ) is

? 1 ? ? ?1 . . . → HK (Sw,pq , h? A (?∞, hA (w )??]∩Sw,pq ) → HK (Sw,pq ) → HK (hA (?∞, hA (w )??]∩Sw,pq ) → . . . . 1 ? ? Here we can replace HK (Sw,pq , h? A (?∞, hA (w ) ? ?] ∩ Sw,pq ) with HK (D, S ) where D, S are ? a disk, respectively sphere, around w (the north pole of Sw,pq ). In turn, HK (D, S ) ? ??4 HK ({w }). Like in the proof of Proposition 3.2 (see diagram (6)), we deduce that η1 |w , which is the same as η |w , is a multiple of eK (Tw (Sw,pq )), the K -equivariant Euler class of the tangent space Tw (Sw,pq ). Now we recall that w is a ?xed point of the K action and Sw,pq = Kpq .w ; consequently, the tangent space at w to Sw,pq is

Tw (Sw,pq ) = k0 pq .w where the dot indicates the in?nitesimal action and ? k0 pq = {hEpq + hEqp | h ∈ H}. Here Epq denotes the n × n matrix whose entries are all 0, except for the one on position pq , which is equal to 1 (and the same for Eqp ). This implies that Tw (Sw,pq ) is K -equivariantly isomorphic to H, with the K action given by (γ1 , . . . , γn ).h := γp hγq ?1 , for any (γ1 , . . . , γn ) ∈ K = (Sp(1))n and any h ∈ H. By Lemma 3.4, we have eK (Tw (Sw,pq )) = up ? uq . Thus the polynomial η |w is divisible by up ? uq , for all (p, q ) ∈ I . Because the latter polynomials are relatively prime with each other, we deduce that η |w is actually divisible by their product, which is just ew . Lemma 3.6. If ?, ? are the inclusion maps of Sn into F ln (H), respectively N , then the ? ? ? ? images of ?? : HK (F ln (H)) → HK (Sn ) and ?? : HK (N ) → H K (Sn ) are the same. Proof. Like in the proof of Proposition 3.2, we order Sn = {w1 , . . . , wk } such that hA (w1 ) < hA (w2 ) < . . . < hA (wk ). We choose ? > 0 smaller than the minimum of hA (w? ) ? hA (w??1 ), 1 ? where ? = 2, . . . , k . We denote M? = h? A (?∞, hA (w? ) + ?]), N? := N ∩ M? , Sn = Sn ∩ M? ? ? ? and will prove by induction on ? that the images of the maps ?? ? : HK (M? ) → HK (Sn ) and ? ? ? ? ?? : HK (N? ) → HK (Sn ) are the same. For ? = 1, the assertion is clear, because both M1 and its subset N1 can be retracted equivariantly onto w1 . Now we assume that the assertion is

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AUGUSTIN-LIVIU MARE

true for ? ? 1 and we prove that it is true for ?. Let us consider the following commutative diagram.

? HK (N? )

//

? HK (N??1 ) ?? ? ?1

(7)

?? ?

r ? ? HK ( Sn )

//

? ??1 HK ( Sn )

? where r is the restriction map. We deduce that r maps im(?? ? ) to im(???1 ). From now on, by r we will denote the induced map ? r : im(?? ? ) → im(???1 ).

Let us consider another commutative diagram, namely 0

//

? HK (M? , M??1 )

//

? HK (M? )

//

? HK (M??1 )

//

0

h

?? ?

?? ? ?1

0

//

ker r

g

//

im(?? ?)

r

//

im(?? ??1 )

//

0

? ? ??1 In this diagram we have made two identi?cations: ?rst, HK ( Sn , Sn ) is identi?ed with ? ? ? ? HK ({w? }); then HK ({w? }) is canonically embedded in HK (Sn ). In this way, h goes from ? ? ? ??1 ? HK (M? , M??1 ) to HK ( Sn , Sn ) = HK ({w? }); but one can easily see from the diagram that h actually takes values in the subspace ker r of the latter space. The map g is just the inclusion ? map. We will prove that the image of h is the whole ker r . To this end, we take η ∈ HK (N? ) ? ? such that r (?? (η )) = 0 and we prove that ?? (η ) is in the image of g ? h, or, equivalently, that η |w? is in the image of h. From the commutative diagram (7) we deduce that the restriction ??1 of η to Sn is equal to 0. From Lemma 3.5 we deduce that η |w? is a multiple of the Euler class ew? . Now let us consider again the diagram (6). We deduce that η |w? is in the image of 1 ? 2 , which is the same as h. In conclusion, we can use that h and ?? ??1 are surjective and, by a chase diagram, deduce that ?? is surjective as well. The proposition is proved. ?

Now we are ready to characterize the image of ?? .

? ? Proposition 3.7. The image of ?? : HK (F ln (H)) → HK ( Sn ) = w ∈Sn

R[u1 , . . . , un ] is

(8)

{(fw )w∈Sn | up ? uq divides fw ? fwspq , ?w ∈ Sn , ?1 ≤ p < q ≤ n}.

Proof. Denote by im the space described by (8). By Lemma 3.6, it is su?cient to show that ? ? the image of ?? : HK (N ) → H K (Sn ) is equal to im. First, let (fw )w∈Sn be in the image of ?? . Pick w ∈ Sn and p, q integers such that 1 ≤ p < q ≤ n and hA (w ) > hA (wspq ). The pair ? ? (Sw,pq ) → HK ({w, wspq }). (fw , fwspq ) is in the image of the restriction map HK Claim. The polynomial fw ? fwspq is divisible by up ? uq . Put U1 := Sw,pq \ {w }, U2 := Sw,pq \ {wspq }. We use the Mayer-Vietoris sequence in equivariant cohomology for the pair (U1 , U2 ). We note that U1 ∩ U2 = Sw,pq \ {w, wspq }. By Lemma 3.3, we can identify Sw,pq \ {w, wspq } with the homogeneous space G/Gpq . Now, because G/K is contractible, we have that

? ? HK (U1 ∩ U2 ) = HG (G/Gpq ).

QUATERNIONIC FLAG MANIFOLDS

9

The ring in the right hand side consists of all polynomials in u1 , . . . , un which are divisible by up ? uq , that is ? HK (U1 ∩ U2 ) = Z[u1 , . . . , un ]/ up ? uq . ? ? ? Also note that since U1 and U2 are contractible, we have HK (U1 ) = HK ({w }) and HK (U2 ) = ? HK ({wspq }), which are both isomorphic to H ? (BK ) = R[u1 , . . . , un ]. Moreover, the restric? ? ? ? tion maps HK (U1 ) → HK (U1 ∩ U2 ) and HK (U2 ) → HK (U1 ∩ U2 ) are both the canonical projection Z[u1 , . . . , un ] → Z[u1 , . . . , un ]/ up ? uq . The Mayer-Vietoris sequence is (9)

? ? ? ? . . . → HK (Sw,pq ) → HK (U1 ) ⊕ HK (U2 ) → HK (U1 ∩ U2 ) → . . . . g

The claim follows from the fact that g (fw , fwspq ) = fw ? fwspq mod up ? uq is equal to 0. Now we prove that im is contained in the image of ?? . Let (fw )w∈Sn be an element of im. From the exact sequence (9), we deduce that for each w ∈ Sn and each pair p, q with ? 1 ≤ p < q ≤ n, hA (w ) > hA (wspq ), there exists αw,pq ∈ HK (Sw,pq ) with αw,pq |w = fw and αw,pq |wspq = fwspq . A simple argument (using again a Mayer-Vietoris sequence) shows that ? there exists α ∈ HK (N ) such that α|Sw,pq = αw,pq . This implies (fw )w∈Sn = ?? (α). Remarks. 1. It is likely that the main results of this section, namely Proposition 3.2 and Proposition 3.7, can be proved with the methods of [Ha-He-Ho]. 2. A GKM type description similar to the one given in Proposition 3.7, for (Z/2Z)n equivariant cohomology with coe?cients in Z/2Z can be deduced from [Bi-Ho-Gu, Theorem C] (we recall that F ln (H) can be regarded as the real locus of a coadjoint orbit of the group SU (2n)). The following result will complete the proof of Theorem 1.1. We consider the tautological vector bundles Li over F ln (H), 1 ≤ i ≤ n. That is, the ?bre of Li over (L1 , . . . , Ln ) ∈ F ln (H) is Li .

? Lemma 3.8. Take w ∈ Sn = (F ln (H)K ), and identify HK ({w }) = H ? (BK ) = Z[u1 , . . . , un ]. Then the equivariant Euler class eK (Lν ) restricted to w is equal to uw(ν ) , for all ν ∈ {1, . . . , n}.

Proof. The cohomology class eK (Lν )|w is the equivariant Euler class of the space Lν |w = H with the K action (γ1 , . . . , γn ).h = γw(ν ) h, for all (γ1, . . . , γn ) ∈ K = Sp(1)n and h ∈ H. The result follows from Lemma 3.4. Now we are ready to prove our main result. Proof of Theorem 1.1. Let F ln (C) be the space of ?ags in Cn , which can be equipped in a natural way with the action of the torus T := (S 1 )n . The idea of the proof is to compare the ? ? equivariant cohomology rings HT (F ln (C)) and HK (F ln (H)). The ?rst one can be computed using the GKM theory (cf. [Ho-Gu-Za]), as follows. Like in the quaternionic case explained here, the ?xed point set F ln (C)T can be identi?ed with the symmetric group Sn , and the ? ? restriction map HT (F ln (C)) → HT (Sn ) is injective. Moreover, the image of the latter map

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?p ? u ?q , consists of all sequences of polynomials (fw )w∈Sn such that fw ? fspq w is divisible by u for all w ∈ Sn and 1 ≤ p < q ≤ n, where we have identi?ed H ? (BT ) = Z[? u1 , . . . , u ?n ] ?ν ) of the tautological com(compare to Proposition 3.7). The equivariant Euler classes eT (L ?ν )|w = u plex line bundles over F ln (C) have the property that eT (L ?w(ν ) , for all 1 ≤ ν ≤ n (compare to Lemma 3.8). On the other hand, we have the Borel-type description of ? HT (F ln (C)), namely

? HT (F ln (C)) ? Z[x1 , . . . , xn , u ?1 , . . . , u ?n ]/ (1 + x1 ) . . . (1 + xn ) = (1 + u ?1 ) . . . (1 + u ?n ) , ?ν ) → xν , 1 ≤ ν ≤ n. Theorem 1.1 follows. via eT (L

4. Appendix A. The Atiyah-Bott lemma We will prove the following version of the Atiyah and Bott’s [At-Bo] lemma. We recall that all cohomology rings are with integer coe?cients (unless otherwise speci?ed). Lemma 4.1. Let V be an even-dimensional real vector space, with the linear action of the group K := Sp(1)n . Assume that the only ?xed point of the action of T := (S 1 )n ? (Sp(1))n on V is 0. If one regards V as a vector bundle over the point 0, then the equivariant Euler ? ({0}) = H ? (BK ) is di?erent from zero (hence it is not a zero divisor). class eK (V ) ∈ HK

? ? Proof. The inclusion HK ({ 0 } ) → H T ({0}) maps eK (V ) to eT (V ) (because if E := EK = ET , then the line bundle E ×T V over E/T is the pullback of E ×K V over E/K via the natural map ? ? E/T → E/K ). On the other hand, the natural map HK ({ 0 } ) → H T ({0}) is injective. The ? ? ? reasons are as follows: ?rst, the map H (BK, Q) = HK ({0}, Q) → HT ({0}, Q) = H ?(BT, Q) is injective (its image is actually the set of all elements invariant under the Weyl group action); second, both H ?(BK ) = H ? ((HP ∞ )n ) and H ? (BT ) = H ? ((CP ∞ )n ) are torsion ? free. So it is su?cient to show that eT (V ) ∈ HT ({0}) = R[u1 , . . . , un ] is di?erent from zero. Since the representation of T on V has no nonzero ?xed points, we have V = ⊕m i=1 Li , where Li are 1-dimensional complex representations of T . Thus we have T m T T eT (V ) = cT m (V ) = cm (⊕i=1 Li ) = c1 (L1 ) . . . c1 (Lm ), T T where cT m and c1 denote the T -equivariant Chern classes. Each Chern class c1 (Li ) is di?erent from zero, since the 1-dimensional complex representations of T are labeled by the character group Hom(T, S 1 ), and the map Hom(T, S 1 ) → H 2 (BT ) given by L → cT 1 (L) is a linear isomorphism (see for instance [Hu, chapter 20, section 11]). This ?nishes the proof.

5. Appendix B. Height functions on isoparametric submanifolds The goal of this section is to provide a proof of Proposition 2.1, and also to achieve a better understanding of the spheres Sw,pq , which are important objects of our paper. We will place ourselves in the more general context of isoparametric submanifolds. We recall (see for instance [Pa-Te, chapter 6]) that an n-dimensional submanifold M ? Rn+k which is closed, complete with respect to the induced metric, and full (i.e. not contained in any a?ne subspace) is called isoparametric if any normal vector at a point of M can be extended to a parallel normal vector ?eld ξ on M with the property that the eigenvalues of the shape

QUATERNIONIC FLAG MANIFOLDS

11

operators Aξ(x) (i.e. the principal curvatures) are independent of x ∈ M , as values and multiplicities. It follows that for x ∈ M , the set {Aξ(x) : ξ (x) ∈ ν (M )x } is a commutative family of selfadjoint endomorphisms of Tx (M ), and so it determines a decomposition of Tx (M ) as a direct sum of common eigenspaces E1 (x), E2 (x), ..., Er (x). There exist normal vectors η1 (x), η2 (x), ..., ηr (x) such that Aξ(x) |Ei(x) = ξ (x), ηi (x) idEi (x) , for all ξ (x) ∈ νx (M ), 1 ≤ i ≤ r . By parallel extension in the normal bundle we obtain the vector ?elds η1 , . . . , ηr . The eigenspaces from above give rise to the distributions E1 , . . . , Er on M , which are called the curvature distributions. The numbers mi = rankEi , 1 ≤ i ≤ r , are the multiplicities of M . We ?x a point x0 ∈ M and we consider the normal space ν0 := νx0 (M ). In the a?ne space x0 + ν0 we consider the hyperplanes ?i (x0 ) := {x0 + ξ (x0 ) : ξ (x0 ) ∈ ν0 , ηi (x0 ), ξ (x0 ) = 1}, 1 ≤ i ≤ r ; one can show that they have a unique intersection point, call it c0 , which is independent of the choice of x0 . Moreover, M is contained in a sphere with center at c0 . We do not lose any generality if we assume that M is contained in the unit sphere S n+k?1, hence c0 is just the origin 0 and x0 + ν0 = ν0 (because x0 ∈ ν0 ). One shows that the group of linear transformations of ν0 generated by the re?ections about ?1 (x0 ), . . . , ?r (x0 ) is a Coxeter group. We denote it by W and call it the Weyl group of M . We have ν0 ∩ M = W.x0 . For each i ∈ {1, . . . , r }, the distribution Ei is integrable and the leaf through x ∈ M of Ei is a round distance sphere Si (x), of dimension mi , whose center is the orthogonal projection of x on ?i (x). These are called the curvature spheres. We note that Si (x) contains x and si x as antipodal points. We have the following result. Proposition 5.1. Let a ∈ ν0 contained in (the interior of ) the same Weyl chamber as x0 and let ha : M → R, ha (x) = a, x be the corresponding height function. The following is true. (a) The gradient of ha at any x ∈ M is aTx (M ) , that is, the orthogonal projection of a on the tangent space Tx M . (b) Crit(ha ) = W.x0 and the negative space of the hessian of ha at x ∈ Crit(ha ) is i Ei (x), where i runs over those indices in {1, . . . , r } with the property ha (x) > ha (si x). (c) For any x ∈ Crit(ha ), and i ∈ {1, . . . , n} such that ha (x) > ha (si x), the meridians of Si (x) going from x to si x are gradient lines of the function ha : M → R with respect to the submanifold metric on M . Proof. The points (a) and (b) are proved in [Pa-Te, chapter 6]. We only need to prove (c). This follows immediately from the fact that for any z ∈ Si (x), the vector ?(ha )(z ) =

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j =i

aTz M is tangent to Si (z ) (because a ∈ νx0 (M ) = νx (M ) is perpendicular to j =i Ej (z ), cf. [Pa-Te, Theorem 6.2.9 (iv) and Proposition 6.2.6]).

Ej (x) =

Finally, Proposition 2.1 can be deduced from Proposition 5.1 by taking into account that F ln (H) is an isoparametric submanifold of Hn with the following properties (cf. [Pa-Te, Example 6.5.6] for the symmetric space SU (2n)/Sp(n)): ? the normal space to x0 := Diag(r1 , . . . , rn ) is ν0 = d ? the Weyl group W is Sn acting in the obvious way on d ? the curvature spheres through wx0 are the orbits Kpq .w , where 1 ≤ p < q ≤ n; thus all multiplicities are equal to 4. References

[Bi-Ho-Gu] D. Biss, T. Holm, and V. Guillemin, The mod 2 equivariant cohomology of real loci, Adv. Math. 185 (2004), 370–399 [Bo-Sa] R. Bott and H. Samelson Applications of the theory of Morse to symmetric spaces, Amer. J. Math. 80 (1958), 964–1029 [Gu-Gi-Ka] V. Guillemin, V. Ginzburg, and Y. Karshon, Moment Maps, Cobordism, and Hamiltonian Group Actions, Amer. Math. Soc., Mathematical Surveys and Monographs 98, 2002 [Ha-Ho-He] M. Harada, A. Henriques, and T. Holm, Computation of generalized equivariant cohomologies of Kac-Moody ?ag varieties, Adv. Math. 197 (2005), 198–221 [Ho-Gu-Za] T. Holm, V. Guillemin, and C. Zara, A GKM description of the equivariant cohomology ring of a homogeneous space, Jour. Alg. Comb. 23 (2006), 21–41 [Hs-Pa-Te] W.Y. Hsiang, R. Palais, and C.-L. Terng Topology of isoparametric submanifolds, J. Di?. Geom. 27 (1988), 423–460 [Hu] D. Husemoller, Fibre Bundles (Third Edition), Springer-Verlag, 1994 [Ma1] A.-L. Mare, Equivariant cohomology of real ?ag manifolds, Di?. Geom. Appl. 23 (2006), 223–229 [Ma2] A.-L. Mare, Connectivity and Kirwan surjectivity for isoparametric submanifolds, Inter. Math. Res. Not., 2005, Issue 55, 3427-3443 [Pa-Te] R. Palais and C.-L. Terng, Critical Point Theory and Submanifold Geometry, Lecture Notes in Mathematics 1353, Springer Verlag 1988 Department of Mathematics and Statistics, University of Regina, College West 307.14, Regina SK, Canada S4S 0A2 E-mail address : mareal@math.uregina.ca

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