Multiplicative formulas in Cohomology of G/P and in quiver representations
aa r X i v : . [ m a t h . AG ] D ec Multipli ative formulas in Cohomology of
G/P and in quiver representationsN. Ressayre ∗ November 8, 2018Abstra t1 Introdu tionConsider a partial (cid:29)ag variety X whi h is not a grassmaninan. Consider alsoits ohomology ring H ∗ ( X, Z ) endowed with the base formed by the Poin arédual lasses of the S hubert varieties. In [Ri ar℄, E. Ri hmond showed thatsome oe(cid:30) ient stru ture of the produ t in H ∗ ( X, Z ) are produ ts of twosu h oe(cid:30) ients for smaller (cid:29)ag varieties.Consider now a quiver without oriented y le. If α and β denote twodimension-ve tors, α ◦ β denotes the number of α -dimensional subrepresen-tations of a general α + β -dimensional representation. In [DW06℄, H. Derksenand J. Weyman expressed some numbers α ◦ β as produ ts of two smallersu h numbers.The aim of this work is to prove two generalisations of the two aboveresults by the same way.We now explain our result about ohomology of the generalized (cid:29)ag va-rieties. Let G be a semi-simple group, T ⊂ B ⊂ Q ⊂ G be a maximal torus,a Borel subgroup and a paraboli subgroup respe tively. In [BK06℄, P. Bel-kale and S. Kumar de(cid:28)ned a new produ t (asso iative and ommutative) onthe ohomology group H ∗ ( G/Q, Z ) denoted by ⊙ . Any oe(cid:30) ient stru -ture of ⊙ in the base of S hubert lasses is either zero or the orresponding oe(cid:30) ient stru ture for the up produ t. ∗ ressayremath.univ-montp2.fr 1et now P ⊂ Q be se ond paraboli subgroups of G and L denote theLevi subgroup of P ontaining T . We obtain the following:Theorem A Any oe(cid:30) ient stru ture of (H ∗ ( G/Q, Z ) , ⊙ ) in the base ofS hubert lasses is the produ t of su h two oe(cid:30) ients for (H ∗ ( G/P, Z ) , ⊙ ) and (H ∗ ( L/ ( L ∩ Q ) , Z ) , ⊙ ) respe tively.A tualy, Theorem 4 below is more pre ise and expli it than Theorem A.This result was already obtained in [Ri ar℄ when G = SL n , Q is any paraboli subgroup and P is the maximal paraboli subgroup orresponding to the lin-ear subspa e in G/Q of minimal dimension.Let Q be a quiver. The Ringle form (see Se tion 4.1) is denoted by h· , ·i .We prove the following:Theorem B Let α, β and γ be three dimension-ve tors. We assume that h α, β i = h α, γ i = h β, γ i = 0 . Then, ( α + β ◦ γ ) . ( α ◦ β ) = ( α ◦ β + γ ) . ( β ◦ γ ) . Note that, Theorem 5 is more general than Theorem B, sin e s dimension-ve tors o ur. Moreover, we give in Theorem B a geometri interpretationof the produ t.If Q has no oriented y le, we obtain the following orollary:Theorem C We assume that Q has no oriented y le. Let α, β and γ bethree dimension-ve tors. We assume that h α, β i = h α, γ i = 0 and β ◦ γ = 1 .Then, α ◦ β + γ = ( α ◦ β ) . ( α ◦ γ ) .This result is not really stated in [DW06℄. However, the proof of [DW06,Theorem 7.14℄ implies it. Note that the proof of Theorem B is really di(cid:27)en-rent from those of [DW06, Theorem 7.14℄. Indeed, the numbers α ◦ β havetwo non trivially equivalent interpretations (see [DSW07℄): as a number ofpoints in a generi (cid:28)ber of a morphism or as a dimension of the subspa e ofinvariant ve tors in a representation. Here we use the (cid:28)rst hara terisation.Derksen and Weyman used the se ond one.In Se tion 2, we onsider more generally a semi-simple group G a tingon a variety X . 2 Degree of dominant pairs2.1 De(cid:28)nitionsLet G be a redu tive group a ting on a smooth variety X . Let λ be a oneparameter subgroup of G . Let G λ or L denote the entralizer of λ in G . We onsider the usual paraboli subgroup asso iated to λ with Levi subgroup L : P ( λ ) = n g ∈ G : lim t → λ ( t ) .g.λ ( t ) − exists in G o . Let C be an irredu ible omponent of the (cid:28)x point set X λ of λ in G . Wealso onsider the Bialini ky-Birula ell C + asso iated to C : C + = { x ∈ X | lim t → λ ( t ) x ∈ C } . Then, C is stable by the a tion of L and C + by the a tion of P ( λ ) .Consider over G × C + the a tion of G × P ( λ ) given by the formula(with obvious notation): ( g, p ) . ( g ′ , y ) = ( gg ′ p − , py ) . Consider the quotient G × P ( λ ) C + of G × C + by the a tion of { e } × P ( λ ) . The lass of a pair ( g, y ) ∈ G × C + in G × P ( λ ) C + is denoted by [ g : y ] .The a tion of G × { e } indu es an a tion of G on G × P ( λ ) C + . More-over, the (cid:28)rst proje tion G × C + −→ G indu es a G -equivariant map π : G × P ( λ ) C + −→ G/P ( λ ) whi h is a lo ally trivial (cid:28)bration with (cid:28)ber C + .In parti ular, we have dim( G × P ( λ ) C + ) = dim( G/P ( λ )) + dim( C + ) . Consider also the G -equivariant map η : G × P ( λ ) C + −→ X, [ g : y ] gy .We (cid:28)nally obtain: G × P ( λ ) C + η ✲ X.G/P ( λ ) π ❄ It is well known that the map G × P ( λ ) C + −→ G/P ( λ ) × X [ g : y ] ( gP ( λ ) , gy ) (1)3s an immersion; its image is the set of the ( gP ( λ ) , x ) ∈ G/P ( λ ) × X su hthat g − x ∈ Y . Note that this fa t an be used to prove that G × P ( λ ) C + a tually exists.De(cid:28)nition. We set δ ( G, X, C, λ ) = dim( X ) − dim( G/P ( λ )) − dim( C + )= codim( C + , X ) − codim( P ( λ ) , G ) . If δ ( G, X, C, λ ) = 0 and η is dominant, it indu es a (cid:28)nite (cid:28)eld extension: k ( X ) ⊂ k ( G × P ( λ ) C + ) . We denote by d ( G, X, C, λ ) the degree of thisextension. If δ ( G, X, C, λ ) = 0 or η is not dominant, we set d ( G, X, C, λ ) = 0 .More generally, we de(cid:28)ne the degree of any morphism to be the degreeof the indu ed extension if it is (cid:28)nite and zero otherwise.2.2 A produ t formula for d ( G, X, C, λ ) Let T be a maximal torus of G and x be a (cid:28)xed point of T in X . We keepnotation of Se tion 2.1 and assume that the image of λ is ontained in T and x ∈ C . We set P = P ( λ ) .Let λ ε be another one parameter subgroup of T . Set P ε = P ( λ ε ) . Con-sider the irredu ible omponent C ε of X λ ε whi h ontains x and C + ε = { x ∈ X : lim t → λ ε ( t ) x ∈ C } . We assume that:(i) P ε ⊂ P ,(ii) C + ε ⊂ C + , and(iii) C ε ⊂ C .Remark. Noti e that the set of the λ ε whi h satisfy these three assump-tions generated an open onvex one in the ve tor spa e ontaining the oneparameters subgroups of T as a latti e.Now, we want to ompare η and η ε . We introdu e the natural morphism: η L : L × P ε ∩ L ( C + ε ∩ C ) −→ C. This map is a map η as in Se tion 2.1 with G = L , X = C , C = C ε and λ = λ ε . In parti ular, we have de(cid:28)ned δ ( L, C, C + ε ∩ C, λ ε ) and d ( L, C, C + ε ∩ C, λ ε ) .We an now state our main resultTheorem 1 With above notation, we have:4i) δ ( G, X, C ε , λ ε ) = δ ( L, C, C ε , λ ε ) + δ ( G, X, C, λ ) , and(ii) If δ ( L, C, C ε , λ ε ) = δ ( G, X, C, λ ) = 0 , then d ( G, X, C ε , λ ε ) = d ( L, C, C ε , λ ε ) · d ( G, X, C, λ ) . Y L = L × P ε ∩ L ( C + ε ∩ C ) and Y P = P × P ε C + ε . We onsider the natural morphism η P : Y P −→ C + , and [Id : η P ] : G × P Y P −→ G × P C + , [ g : [ p : x ]] [ g : px ] . Lemma 1 With above notation, we have:(i) the map G × P Y P −→ G × P ε C + ε , [ g : [ p : x ]] [ gp : x ] is anisomorphism denoted by ι ; moreover,(ii) η ε ◦ ι = η ◦ ([Id : η P ]) .Proof. The morphism ι ommutes with the two proje tions on G/P . More-over, the restri tion of ι over P/P is the losed immersion P × P ε C + ε −→ G × P ε C + ε . It follows (see for example [Res04, Appendi e℄) that ι is anisomorphism.The morphisms η ε ◦ ι and η ◦ ([Id : η P ]) are G -equivariant and extend theimmersion of C + ε in X . They have to be equal. (cid:3) η P . Consider the two following limitmorphisms: Λ P : P −→ Lp lim t → λ ( t ) pλ ( t − ) and Λ + : C + −→ Cy lim t → λ ( t ) y. The omputation λ ( t ) px = λ ( t ) pλ ( t − ) λ ( t ) x implies the easyLemma 2 We have: Λ + ( px ) = Λ P ( p )Λ + ( x ) .5.3.3 (cid:22) Re all that Λ + : C + −→ C is a ve tor bundle. The pullba kof this ve tor bundle by η L is η ∗ L ( C + ) = { ([ l : x ] , y ) ∈ Y L × C + | lx = Λ + ( y ) } , endowed with the (cid:28)rst proje tion p on Y L . Consider the following diagram: Y P Θ : [ p : x ] ([Λ P ( p ) : Λ + ( x )] , px ) ✲ η ∗ L ( C + ) Y L ✛ p [ p : x ] → [ Λ P ( p ) : Λ + ( x ) ] ✲ L/ ( P ε ∩ L ) . ❄ (2)Lemma 3 The above diagram is ommutative, and the top horizontal map Θ is an isomorphism.Proof. First, note that the map Y P −→ Y L in Diagram 2 is well de(cid:28)ned byLemma 2. Diagram 2 is obviously ommutative.Sin e all the morphisms in Diagram 2 are L -equivariant, [Res04, Appen-di e℄ implies that it is su(cid:30) ient to prove that Θ is an isomorphism whenrestri ted over the lass of e in L/ ( P ε ∩ L ) . The (cid:28)ber in Y L over thispoint if C ∩ C + ε . Sin e P u ⊂ P uε , the (cid:28)ber in Y P identify with C + ε , by x ∈ C + ε [ e : x ] . The (cid:28)ber in η ∗ L ( C + ) also identify with C + ε in su h a waythe restri tion of Θ be omes the identity. It follows that Θ is an isomor-phism. (cid:3) P η P ✲ C + η ∗ L ( C + ) ✲ Θ ˜ ✲ Y L ❄ η L ✲ ✲ C. Λ + ❄ It follows that dim( C + ) − dim( Y P ) = δ ( L, C, C ε , λ ε ) and d ( L, C, C ε , λ ε ) equals the degree of η P .Moreover, by Lemma 1, we have the following ommutative diagram: G × P ε C + ε ˜ ✲ G × P Y P G × P C + [ id : η P ] ❄ X.η ❄ η ε ✲ The (cid:28)rst assertion follows immediately. Let d denote the degree of [ id : η P ] that is the degree of η P . Sin e d = d ( L, C, C ε , λ ε ) , we have to prove that d ( G, X, C ε , λ ε ) = d.d ( G, X, C, λ ) . We (cid:28)rstly assume that d ( G, X, C ε , λ ε ) =0 . Sin e δ (( G, X, C ε , λ ε ) = 0 , η ε is not dominant. So, η or [ id : η P ] is notdominant. It follows that either d ( G, X, C, λ ) or d is zero. The assertionfollows.We now assume that d ( G, X, C ε , λ ε ) = 0 , that is that η ε is dominant.Sin e the image of η ε is ontained in the image of η , η is dominant. Sin e η ε is dominant, the dimension of the losure of the image of [ id : η P ] at leastthose of X . Sin e δ ( L, C, C ε , λ ε ) = δ ( G, X, C, λ ) = 0 , this implies that η P is dominant. Now, the se ond assertion is simply the multipli ative formulafor the degree of a double extension (cid:28)eld. (cid:3) Y is a smooth variety of dimension n , T Y denotes its tangentbundle. The line bundle V n T Y over Y will be alled the determinant bundleand denoted by D etY . If ϕ : Y −→ Y ′ is a morphism between smoothvariety, we denote by T ϕ : T Y −→ T Y ′ its tangent map, and by D etϕ : D etY −→ D etY ′ its determinant.2.4.2 (cid:22) Consider η : G × P ( λ ) C + −→ X as in Se tion 2.1.De(cid:28)nition. We say that ( G, X, C, λ ) is generi aly (cid:28)nite if d ( G, X, C, λ ) = 0 .We say that ( G, X, C, λ ) is well generi aly (cid:28)nite if it is generi aly (cid:28)nite andthere exists x ∈ C su h that T η [ e : x ] is invertible.2.4.3 (cid:22) Consider the restri tion of T η to C : T η | C + : T ( G × P C + ) | C + −→ T ( X ) | C + , and the restri tion of D etη to C + : D etη | C + : D et ( G × P C + ) | C + −→ D et ( X ) | C + . Sin e η is G -equivariant, the morphism D etη | C + is P -equivariant; it an bethought as a P -invariant se tion of the line bundle D := D et ( G × P C + ) ∗| C + ⊗D et ( X ) | C + over C + . For any x ∈ C , K ∗ a ts linearly via λ on the (cid:28)ber D x over x in D : this a tion is given by a hara ter of K ∗ , that is an interger m .Moreover, this integer does not depends on x in C : we denote by µ D ( C, λ ) this interger.Lemma 4 We assuma that X is smooth. The, the following are equivalent:(i) ( G, X, C, λ ) is well generi aly (cid:28)nite;(ii) ( G, X, C, λ ) is generi aly (cid:28)nite and µ D ( C, λ ) = 0 .Proof. Let us assume that ( G, X, C, λ ) is well generi aly (cid:28)nite and let x ∈ C be su h that T η x is invertible. Then, D etη x is a non zero K ∗ -(cid:28)xed point in D x : the a tion of K ∗ on the line D x must be trivial.Let us now assume that ( G, X, C, λ ) is generi aly (cid:28)nite and µ D ( C, λ ) = 0 .Sin e the base (cid:28)eld is assumed to be of hara teristi zero, the exists a pointin G × P ( λ ) C + where the T η is invertible. Sin e η is G -equivariant, one an(cid:28)nd su h a point y in C + . In parti ular, D etη | C + is a non zero P ( λ ) -invariantse tion of D . Sin e µ D ( C, λ ) = 0 , [Res07, Proposition 5℄ implies that D etη | C is non identa aly zero. (cid:3) X is smooth.Let us also assume that ( G, X, C ε , λ ε ) is well generi aly (cid:28)nite.Then, ( G, X, C, λ ) and ( L, C, C ε , λ ε ) are well generi aly (cid:28)nite.Proof. If V is a ve tor spa e endowed with a linear a tion of a one parametersubgroup λ we denote by V λ< the set of v ∈ V su h that lim t → λ ( t − ) v = 0 .Let x ∈ C ε be su h that T η ε is invertible at [ e : x ] . Consider the subtorus S of dimension two ontaining the images of λ and λ ε . It (cid:28)xes x . The tangentmap of η ε at the point [ e : x ] indu es a S -equivariant linear isomoprhim: θ : g / p ε ≃ g λ ε < −→ ( T x X ) λ ε < . By assumption, g λ< ⊂ g λ ε < and ( T x X ) λ< ⊂ ( T x X ) λ ε < . Sin e θ is S -equivariant, it follows that it indu es an isomorphismbetween g λ< and ( T x X ) λ< . In parti ular, δ ( G, X, C, λ ) = 0 .Now, the se ond assertion of Lemma 1 implies that T [ e : x ] η is invertible.It follows that ( G, X, C, λ ) is well generi aly (cid:28)nite.Sin e δ ( G, X, C ε , λ ε ) = 0 , Theorem 1 implies that δ ( L, C, C ε , λ ε ) = 0 .Now, Lemma 1 implies that T [ e : x ] η P is invertible. By Lemma 3, it followsthat T [ e : x ] η L is invertible. Then, ( L, C, C ε , λ ε ) is well generi aly (cid:28)nite. (cid:3) P be a paraboli subgroup of the semisimple group G . Let T ⊂ B ⊂ P be a maximal torus and a Borel subgroup of G . Let W denotethe Weyl group of T and G . For w ∈ W , we set X ( w ) = BwP/P , X ◦ ( w ) = BwP/P and denote by [ X ( w )] ∈ H ∗ ( G/P, Z ) the Poin aré dual lass of X ( w ) in ohomology. Let w , · · · , w s ∈ W be su h that P i codim X ( w i ) =dim G/P . Let c be the non negative integer su h that [ X ( w )] . · · · . [ X ( w s )] = c [pt] . Let λ be a one parameter subgroup of T su h that P = P ( λ ) . Consider X = ( G/B ) s and the following T -(cid:28)xed point x = ( w − B/B, · · · , w − s B/B ) .Let C denote the irredu ible omponent of X λ ontaining x . An easy on-sequen e of Kleiman's transversality Theorem (see [Kle76℄) is the followinglemma whi h express c has a degree.Lemma 5 We have: δ ( G, X, C, λ ) = 0 and c = d ( G, X, C, λ ) .9roof. See [Res07, proof of Lemma 14℄. (cid:3) H ∗ ( G/P, Z ) in the basis of S hubert lasses in terms of maps η 's as in Se -tion 2. We are now going to dis uss Levi-movability, a notion introdu ed in[BK06℄:De(cid:28)nition. Let w i ∈ W su h that P i codim( X ( w i ) , X ) = dim( X ) . Then, ( X ( w ) , · · · , X ( w s )) is said to be Levi-movable if for generi l , · · · , l s ∈ L the interse tion l X ◦ ( w ) ∩ · · · ∩ l s X ◦ ( w s ) is transverse at P/P .Lemma 6 The following are equivalent:(i) ( X ( w ) , · · · , X ( w s )) is Levi-movable;(ii) there exists y ∈ C su h that the tangent map T [ e : y ] η of η at [ e : y ] isinvertible.Proof. Let y ∈ C and l , · · · , l s ∈ L su h that y = ( l w − B/B, · · · , l s w − s B/B ) .Sin e η extends the immersion of C + in C + ; the tangent map T η [ e : y ] re-stri ts to the identity on T [ e : y ] C + . In parti ular, it indu es a linear map: T η [ e : y ] : N [ e : y ] ( C + , G × P C + ) −→ N y ( C + , X ) su h that T [ e : y ] η is an isomorphism if and only if T η [ e : y ] is. By π , N [ e : y ] ( C + , G × P C + ) identi(cid:28)es with T e G/P that is with g / p . Moreover, N y ( C + , X ) equals L i N l i w − i B/B ( P l i w − i B/B, G/B ) whi h identi(cid:28)es with ⊕ i g / ( p + l i w − i b w i l i ) .Moreover, after omposing by these isomorphisms T η [ e : y ] is the anoni almap g / p −→ ⊕ i g / ( p + l i w − i b w i l i ) . The lemma follows immediately. (cid:3) G bea semisimple group and P be a paraboli subgroup of G . We hoose a Levisubgroup L of P and denote by U its unipotent radi al. We are interestedin the a tion of L on the Lie algebra u of U .Let T be a maximal torus of L and B be a Borel subgroup of G ontaining T . Let g denote the Lie algebra of G . Let ∆ ⊂ Φ + ⊂ Φ (resp. ∆ L ⊂ Φ + L ⊂ Φ L ) be the set of simple roots, positive roots and roots of G (resp. L ) for T orresponding of B (resp. B ∩ L ). For any α ∈ Φ , we denote by u α the linegenerated by the eigenve tors in g of weight α .10in e u has no multipli ity for the a tion of T , it has no multipli ityfor the a tion of L : we have a anoni al de omposition of u as a sum ⊕ i V i of irredu ible L -modules. Sin e T ⊂ L , ea h V i is a sum of u α for some α ∈ Φ + − Φ + L : the de omposition u = ⊕ i V i orresponds to a partition Φ + − Φ + L = F i Φ i .Let β and β ′ be two positive roots. We write β = X α ∈ ∆ L c α α + X α ∈ ∆ − ∆ L d α α, (3)with c α and d α in N . We also write β ′ in the same way with some c ′ α and d ′ α .We write β ≡ β ′ if and only if P α ∈ ∆ − ∆ L d α α = P α ∈ ∆ − ∆ L d ′ α α . The relation ≡ is obviously an equivalen e relation. Let S denote the set of equivalen e lasses in Φ + − Φ + L for ≡ . We an now rephrase the main result of [ABS90℄:Theorem 3 (Azad-Barry-Seitz) For any s ∈ S , V s := ⊕ α ∈ s u α is anirredu ible L -module. In parti ular, F i Φ i is the partition in equivalen e lasses for ≡ .For any α ∈ Φ , we denote by Ker α the Kernel of the hara ter α of T . Let Z be the enter of L . Let Z ◦ denote the neutral omponent of Z and X ( Z ◦ ) denote the group of hara ters of Z ◦ . Under the a tion of Z ◦ , u de ompose as V = ⊕ χ ∈ X ( Z ◦ ) V χ , where V χ is the ve tor subspa e of weight χ . Sin e Z ◦ is entral in L , ea h V χ is L -stable.Note that Z ◦ ⊂ Z ⊂ T ; and more pre isely Z = [ α ∈ ∆ L Ker α. It follows that for β as in Equation 3, the restri tion β | Z ◦ of β to Z ◦ equals P α ∈ ∆ − ∆ L d α α | Z ◦ . Moreover, the family ( α | Z ◦ ) α ∈ ∆ − ∆ L is free in the rationalve tor ve tor spa e ontaining the hara ters of the torus Z ◦ . We obtain that β ≡ β ′ ⇐⇒ β | Z ◦ = β ′| Z ◦ . In parti ular, ea h V s is one V χ with above notation. In parti ular, we have:Corollary 1 Ea h V χ (with χ ∈ X ( Z ◦ ) ) is an irredu ible L -module.11.3 A multipli ative formula for stru ture oe(cid:30) ients of ⊙ Q ⊂ P be two paraboli subgroups of the semisimplegroup G . Let T ⊂ B ⊂ Q be a maximal torus and a Borel subgroup of G .Let L denote the Levi subgroup of P ontaining T . Let W (resp. W P )denote the Weyl group of T and G (resp. L ).For any w ∈ W , w − Bw ∩ L is a Borel subgroup of L ontaining T . So,there exists a unique w ∈ W P su h that w − ( B ∩ L ) w = w − Bw ∩ L. (4)To any w ∈ W , we now asso iated three S hubert varieties in G/P , G/Q and
L/L ∩ Q respe tively: X G/P ( w ) = BwP/P , X
G/Q ( w ) = BwQ/Q and X L/L ∩ Q ( w ) = ( L ∩ B ) w ( L ∩ Q/L ∩ Q ) . Theorem 4 Let w , · · · , w s ∈ W . We assume that P i codim X G/Q ( w i ) =dim G/Q and ( X G/Q ( w ) , · · · , X G/Q ( w s )) is Levi-movable. Then, we have:(i) P i codim X G/P ( w i ) = dim G/P and P i codim X L/L ∩ Q ( w i ) = dim L/ ( L ∩ Q ) ;(ii) ( X G/P ( w ) , · · · , X G/P ( w s )) and ( X L/L ∩ Q ( w ) , · · · , X L/L ∩ Q ( w s )) are Levi-movable.Moreover, by Assertion (i) we an de(cid:28)ne three integers by the formulas: [ X G/Q ( w )] . · · · . [ X G/Q ( w s )] = c G/Qw , ··· ,w s [pt] , [ X G/P ( w )] . · · · . [ X G/P ( w s )] = c G/Pw , ··· ,w s [pt] and[ X L/L ∩ Q ( w )] . · · · . [ X L/L ∩ Q ( w s )] = c L/L ∩ Qw , ··· ,w s [pt] . Then, we have: c G/Qw , ··· ,w s = c G/Pw , ··· ,w s .c L/L ∩ Qw , ··· ,w s . Proof. We begin the proof by making some remarks about the tangent spa e T Q/Q
G/Q of G/Q at Q/Q . Let L Q denote the Levi subgroup of Q ontaining T and Z ◦ denote its onne ted enter. We de ompose T Q/Q
G/Q as a sum ⊕ χ ∈ X ( Z ◦ ) V χ of eigenve tor spa es for the a tion of the torus Z ◦ . Note that12 Q/Q
P/Q ⊂ T Q/Q
G/Q is stable by the a tion of L Q . Now, Corollary 1implies that there exists S ⊂ X ( Z ◦ ) su h that T Q/Q
P/Q = ⊕ χ ∈ S V χ . (5)Let l ∈ L Q and w ∈ W . We set Y ◦ ( w ) = w − BwQ/Q . One easily he ksthat lY ◦ ( w ) is stable by the a tion of Z ◦ . Sin e Q/Q ∈ lY ◦ ( w ) is a point(cid:28)xed by Z ◦ , Z ◦ a ts on T Q/Q lY ◦ ( w ) . In parti ular, T Q/Q lY ◦ ( w ) = ⊕ χ ∈ X ( Z ◦ ) V χ ∩ T Q/Q lY ◦ ( w ) . (6)Sin e ( X G/Q ( w ) , · · · , X G/Q ( w s )) is Levi-movable, there exist l , · · · , l s ∈ L Q su h that T Q/Q l Y ◦ ( w ◦ ) ⊕ · · · ⊕ T Q/Q l s Y ◦ ( w ◦ s ) = T Q/Q
G/Q. (7)Consider now the G -equivariant proje tion π : G/Q −→ G/P . Note thatthe Kernel of the tangent map T Q/Q π of π at Q/Q is T Q/Q
P/Q . So, Equa-tions 5 and 6 imply that for any i = 1 , · · · , s , T Q/Q indu es an isomorphismfrom T Q/Q l i Y ◦ ( w i ) ∩ ⊕ χ S V χ onto T Q/Q l i π ( Y ◦ ( w i )) . Now, Equation 7 im-ply that ⊕ i T P/P l i π ( Y ◦ ( w i )) = T P/P
G/P . Moreover, L Q is ontained in L ;Assertions (i) and (ii) of the theorem follows for G/P .Re all that X is the variety ( G/B ) s and x = ( w − B/B, · · · , w − s B/B ) .Let λ (resp. λ ε ) be a one parameter subgroup of T su h that P ( λ ) (resp. P ( λ ε ) ) equals P and Q . Let C (resp. C ε ) denote the irredu ible omponentof X λ (resp. X λ ε ) ontaining x . With notation of Se tion 2, Lemma 5 im-plies that δ ( G, X, C ε , λ ε ) and δ ( G, X, C, λ ) equal zero. Theorem 1 impliesthat δ ( L, C, C ε , λ ε ) = 0 . Assertion (i) for L/L ∩ Q follows. Now, the se ondassertion of Theorem 1 with Lemma 5 imply the last formula of the theorem.It remains to prove that ( X L/L ∩ Q ( w ) , · · · , X L/L ∩ Q ( w s )) is Levi-movable.Sin e ( X G/Q ( w ) , · · · , X G/Q ( w s )) is Levi-movable, Lemma 6 shows that thereexists y ∈ C ε su h that T [ e : y ] η ε is invertible. Now, Lemmas 1 and 3 implythat T [ e : y ] η L is invertible. So, Lemma 6 allows to on lude. (cid:3) Remark. In the ase when G = SL n , Theorem 4 was already obtained in[Ri ar℄ for a lot of pairs Q ⊂ P .3.3.2 (cid:22) If one know how to ompute in (H ∗ ( G/P, Z ) , ⊙ ) for any max-imal P and any G , then Theorem 4 an be used to ompute the stru ture oe(cid:30) ients of (H ∗ ( G/Q, Z ) , ⊙ ) for any paraboli subgroup Q . To illustratethis prin iple, we state an analogous to [Ri ar, Corollary 23℄:13orollary 2 Let G = Sp n . The non-zero oe(cid:30) ients stru tures of the ring (H ∗ ( G/B, Z ) , ⊙ ) are all equal to .Proof. The proof pro eeds by indu tion on n . Let c be a non-zero oe(cid:30) ientstru ture of (H ∗ ( G/B, Z ) , ⊙ ) . Let ( w , w , w ) be su h that [ X ( w )] . [ X ( w )] . [ X ( w )] = c [pt] . Sin e c is non-zero, ( X ( w ) , X ( w ) , X ( w )) is Levi-movable.Consider the stabilizer P in G of a line in K n . Theorem 4 applied with B ⊂ P shows that c is the produ t of oe(cid:30) ient stru ture of (H ∗ ( G/P, Z ) , ⊙ ) and one of (H ∗ (Sp n − /B, Z ) , ⊙ ) . The fa t that G/P is a proje tive spa eand the indu tion allow to on lude. (cid:3) Q be a quiver (that is, a (cid:28)nite oriented graph) with vertexes Q andarrows Q . An arrow a ∈ Q has initial vertex ia and terminal one ta . Arepresentation R of Q is a family ( V ( s )) s ∈ Q of (cid:28)nite dimensional ve torspa es and a family of linear maps u ( a ) ∈ Hom( V ( ia ) , V ( ta )) indexed by a ∈ Q . The dimension ve tor of R is the family (dim( V ( s ))) s ∈ Q ∈ N Q .Let us (cid:28)x α ∈ N Q and a ve tor spa e V ( s ) of dimension α ( s ) for ea h α ∈ Q . Set Rep(
Q, α ) = M a ∈ Q Hom( V ( ia ) , V ( ta )) . Consider also the groups:
GL( α ) Y s ∈ Q GL( V ( s )) and SL( α ) Y s ∈ Q SL( V ( s ) . Let α, β ∈ Z Q . The Ringle form is de(cid:28)ned by h α, β i = X s ∈ Q α ( s ) β ( s ) − X a ∈ Q α ( ia ) β ( ta ) . Assume now that α, β ∈ N Q . Following Derksen-S ho(cid:28)eld-Weyman (see [DSW07℄),we de(cid:28)ne α ◦ β to be the number of α -dimensional subrepresentation of ageneral representation of dimension α + β if it is (cid:28)nite, and otherwise.14.2 Dominant pairs4.2.1 (cid:22) Let λ be a one parameter subgroups of GL( α ) . For any i ∈ Z and s ∈ Q , we set V i ( s ) = { v ∈ V ( s ) | λ ( t ) v = t i v } and α i ( s ) = dim V i ( s ) . Obvi-ously, almost all α i are zero; and, α = P i ∈ Z α i . Moreover, λ is determinedup to onjuga y by the α i 's.The paraboli subgroup P ( λ ) of GL( α ) asso iated to λ is the set of ( g ( s )) s ∈ Q su h that for all i ∈ Z g ( s )( V i ( s )) ⊂ ⊕ j ≤ i V j ( s ) .Now, Rep(
Q, α ) λ is the set of the ( u ( a )) a ∈ Q 's su h that for any a ∈ Q and for any i ∈ Z , u ( a )( V i ( ia )) ⊂ V i ( ta ) . It is isomorphi to Q i Rep(
Q, α i ) .In parti ular, it is irredu ible and denoted by C from now on.Moreover, C + is the set of the ( u ( a )) a ∈ Q 's su h that for any a ∈ Q andfor any i ∈ Z , u ( a )( V i ( ia )) ⊂ ⊕ j ≤ i V j ( ta ) .Consider the morphism η λ : G × P ( λ ) C + −→ Rep(
Q, α ) . Note that, P ( λ ) , C and C + only depend on the list (ordered by the index i ) of non-zero α i 's.4.2.2 (cid:22) The last observation allows the followingDe(cid:28)nition. A de omposition of the ve tor dimension α , is a family ( β , · · · , β s ) of non-zero ve tor dimensions su h that α = β + · · · + β s . We denote thede omposition by α = β ˜+ · · · ˜+ β s .Any one parameter subgroup λ indu es a de omposition of α = β ˜+ · · · ˜+ β s where the β j 's are the non-zero α i 's ordered by the index i . Note that, upto onjuga y, P ( λ ) , C and C + only depend on this de omposition. In par-ti ular, one an de(cid:28)ne (up to onjuga y) the map η β ˜+ ··· ˜+ β s asso iated to ade omposition of α .4.2.3 (cid:22) Consider a de omposition α = β ˜+ β with two dimension-ve tors and the asso iated morphism η . In this se tion, we olle t some easyproperties of η . Let ( u, v ) ∈ Rep(
Q, β ) × Rep(
Q, β ) = C ⊂ Rep(
Q, α ) = X .Sin e η extends the immersion of C + in X , the tangent map T ( u,v ) η indu esthe identity on T ( u,v ) C + . In parti ular, it indu es a linear map T η [ e :( u,v )] : N [ e :( u,v )] ( C + , G × P C + ) −→ N ( u,v ) ( C + , Rep(
Q, α )) . Moreover, N [ e :( u,v )] ( C + , G × P C + ) identi(cid:28)es with ⊕ s ∈ Q Hom( V ( s ) , W ( s )) and N ( u,v ) ( C + , Rep(
Q, α )) with ⊕ a ∈ Q Hom( V ( ia ) , W ( ta )) . A dire t om-putation gives the following 15emma 7 With the above identi(cid:28) ation, we have: T η [ e :( u,v )] ( X s ∈ Q ϕ ( s )) = X a ∈ Q v ( a ) ϕ ( ta ) − ϕ ( ha ) u ( a ) . In parti ular, the Kernel of
T η ( u,v ) is Hom( u, v ) and its Image is Ext( u, v ) .The quantities δ ( η ) and d ( η ) are also parti ularly interesting:Lemma 8 Consider a de omposition α = β ˜+ β and the asso iated map η .Then,(i) δ ( η ) = −h β , β i , and(ii) d ( η ) = β ◦ β .Proof. By the dis ussion pre eding Lemma 7, δ ( η ) equals the di(cid:27)eren e be-tween the dimension of ⊕ a ∈ Q Hom( V ( ia ) , W ( ta )) and of ⊕ s ∈ Q Hom( V ( s ) , W ( s )) .The (cid:28)rst assertion follows.Let u ∈ Rep(
Q, α ) . Using Immersion 1, one identi(cid:28)es the (cid:28)ber η − ( u ) with the set u -stable subspa es of V of dimension β . In parti ular, η − ( u ) identi(cid:28)es with the set of β -dimensional subrepresentations of u . Sin e the hara teristi of k is assumed to be zero, when u is generi this numbersequals d ( η ) on one hand and β ◦ β on the other one. (cid:3) If Y is a smooth variety of dimension n , T Y denotes its tangent bundle.The line bundle V n T Y over Y will be alled the determinant bundle anddenoted by D etY . If ϕ : Y −→ Y ′ is a morphism between smooth variety, wedenote by D etϕ : D etY −→ D etY ′ the determinant of its tangent map T ϕ .We onsider now the restri tion of D etη to C + : it is a P -invariant se tionof the P -linearized line bundle D et over C + de(cid:28)ned by D et = D et ( G × P C + ) ∗| C + ⊗ D et ( X ) | C + .Re all that for any s ∈ Q , we have (cid:28)xed a ve tor spa e V ( s ) of dimension α ( s ) . Let us (cid:28)x, for any s ∈ Q a de omposition V ( s ) = V ( s ) ⊕ V ( s ) su hthat dim V i ( s ) = β i ( s ) for i = 1 , . Consider the one parameter subgroup λ of GL( α ) de(cid:28)ned by λ ( s )( t ) stabilizes the de ompostion V ( s ) ⊕ V ( s ) , equalsto Id when restri ted to V ( s ) and t Id when restre ted to V ( s ) . It satis(cid:28)es P ( λ ) = P , Rep(
Q, α ) λ = C and C + ( λ ) = C + .Lemma 9 We assume that h β , β i = 0 . The one parameter subgroup λ a
tstrivially on D et | C . 16roof. Sin
e C is an a(cid:30)ne spa
e, λ a
ts by the same
hara
ter on ea
h (cid:28)berof D et | C . Sin
e η extend the identity on C + , its
hara
ter is the di(cid:27)eren
ebetween the weights of λ in N ( C + , X ) ≃ ⊕ a ∈ Q Hom( V ( ia ) , V ( ta )) and in N ( C + , G × P C + ) ≃ T e G/P ≃ ⊕ s ∈ Q Hom( V ( s ) , V ( s )) . So, this
hara
ter equals: X a ∈ Q β ( ia ) β ( ta ) − X s ∈ Q β ( s ) β ( s ); that is, −h β , β i . The lemma follows. (cid:3) d ( η β ˜+ ··· ˜+ β s ) Here
omes the main result of this se
tion:Theorem 5 Let α = β ˜+ · · · ˜+ β s be a de
omposition of α su
h that for all i < j , h β i , β j i = 0 .Then, δ ( η β ˜+ ··· ˜+ β s ) = 0 and d ( η β ˜+ ··· ˜+ β s ) = ( β ◦ α − β ) . ( β ◦ α − β − β ) . · · · . ( β s − ◦ β s )= ( α − β s ◦ β s ) . ( β − β s − β s − ◦ β s − ) . · · · . ( β ◦ β ) . Proof. By Se
tion 4.2.1, the
odimension of C + in G × P C + is X i Q, α ) is X i Q, α ) of weight t j t − i . In parti
ular,these two eigensubspa
es have the same dimension. But, a dire
t
omputionshows that the di(cid:27)eren
e between these two dimension is pre
isely h β i , β j i .Assertion (i) follows.Conversely, let us assume that Assertion (i) follows. Sin
e d ( η β ˜+ ··· ˜+ β s ) =0 , there exists a point G × P C + where the tangent map of η β ˜+ ··· ˜+ β s isinvertible. Sin
e η is G -equivariant, its determinant is not identi
aly zeroon C + . Using the fa
t for all i < j h β i , β j i = 0 , a dire
t
omputation (likein the proof of Lemma 9) shows that Z a
ts trivialy on D et | C . By [Res07,Proposition 5℄, the determinant of η is not identi
aly zero on C . Assertion (ii)follows. (cid:3) Ext(u , v) for generi α and β dimensionalrepresentations u and v is denoted by ext( α, β ) .Corollary 3 We assume that Q has no oriented
y
le. Let α, β and γ bethree dimension-ve
tors. We assume that h α, β i = h α, γ i = 0 and β ◦ γ = 1 .Then, α ◦ β + γ = ( α ◦ β ) . ( α ◦ γ ) .Proof. Theorem 5 applied to α ˜+ β ˜+ γ gives: ( α + β ◦ γ ) . ( α ◦ β ) = ( α ◦ β + γ ) . ( β ◦ γ ) = ( α ◦ β + γ ) , sin
e ( β ◦ γ ) = 1 . If α ◦ β = 0 , the
orollary follows. Now assume that α ◦ β = 0 . Lemma 10 implies that the determinant of η α ˜+ β is not identi
allyzero on C . But, Lemma 7 implies that ext( α, β ) = 0 . Now, the
orollary isa dire
t
onsequen
e of Lemma 11 below. (cid:3) Lemma 11 We assume that Q has no oriented
y
le. Let α, β and γ bethree ve
tor dimensions. We assume that β ◦ γ = 1 and ext( α, β ) = 0 .Then, α + β ◦ γ = α ◦ γ .Proof. In [DSW07℄, Derksen-S
ho(cid:28)eld-Weyman prove that α ◦ γ equals thedimension of K [Rep( Q, γ )] h α, ·i . Consider the multipli
ation morphism: m : K [Rep( Q, γ )] h α, ·i ⊗ K [Rep( Q, γ )] h β, ·i −→ K [Rep( Q, γ )] h α + β, ·i . We
laim that m is an isomorphism. The lemma will follow dire
tly. Sin
e dim( K [Rep( Q, γ )] h β, ·i ) = 1 and K [Rep( Q, γ )] has no zero-divisors, m is in-je
tive.Sin
e ext( α, β ) = 0 , η α ˜+ β is dominant. But, it is proper; so, it is surje
-tive.In [DW00℄, Derksen-Weyman prove that K [Rep( Q, γ )] h α + β, ·i is generatedby fun
tions c V asso
iated to various α + β -dimensional representation V (see also [DZ01℄). Sin
e η α ˜+ β is surje
tive, there exists an α -dimensionalsubrepresentation V ′ of V . By [DW00, Lemma 1℄, c V = c V ′ .c V/V ′ . It followsthat m is surje
tive. (cid:3) Referen
es[ABS90℄ H. Azad, M. Barry, and G. Seitz, On the stru
ture of paraboli
subgroups, Com. in Algebra 18 (1990), no. 2, 551(cid:21)562.19BK06℄ Prakash Belkale and Shrawan Kumar, Eigenvalue problem and anew produ
t in
ohomology of (cid:29)ag varieties, Invent. Math. 166(2006), no. 1, 185(cid:21)228.[DSW07℄ Harm Derksen, Aidan S
ho(cid:28)eld, and Jerzy Weyman, On the num-ber of subrepresentations of a general quiver representation, J.Lond. Math. So
. (2) 76 (2007), no. 1, 135(cid:21)147.[DW00℄ Harm Derksen and Jerzy Weyman, Semi-invariants of quivers andsaturation for Littlewood-Ri
hardson
oe(cid:30)
ients, J. Amer. Math.So
. 13 (2000), no. 3, 467(cid:21)479 (ele
troni
).[DW06℄ Harm Derksen and Jerzy Weyman, The
ombinatori
s of quiverrepresentations, 2006.[DZ01℄ M. Domokos and A. N. Zubkov, Semi-invariants of quivers as de-terminants, Transform. Groups 6 (2001), no. 1, 9(cid:21)24.[Kle76℄ Steven L. Kleiman, Problem 15: rigorous foundation of S
hu-bert's enumerative
al
ulus, Mathemati
al developments arisingfrom Hilbert problems (Pro
. Sympos. Pure Math., Northern Illi-nois Univ., De Kalb, Ill., 1974), Amer. Math. So
., Providen
e, R.I., 1976, pp. 445(cid:21)482. Pro
. Sympos. Pure Math., Vol. XXVIII.[Res04℄ Ni
olas Ressayre, Sur les orbites d'un sous-groupe sphérique dansla variété des drapeaux, Bull. de la SMF 132 (2004), 543(cid:21)567.[Res07℄ , Geometri
invariant theory and generalized eigenvalueproblem, Preprint (2007), no. arXiv:0704.2127, 1(cid:21)45.[Ri
ar℄ Edward Ri
hmond, A partial horn re
ursion in the
ohomologyof (cid:29)ag varieties, Journal of Algebrai
Combinatori
s (To appear),1(cid:21)15. - ♦♦