Rigidity of Bott-Samelson-Demazure-Hansen variety for F 4 and G 2
aa r X i v : . [ m a t h . AG ] A ug RIGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETYFOR F AND G S.SENTHAMARAI KANNAN AND PINAKINATH SAHA
Abstract.
Let G be a simple algebraic group of adjoint type over C , whose root systemis of type F . Let T be a maximal torus of G and B be a Borel subgroup of G containing T. Let w be an element of Weyl group W and X ( w ) be the Schubert variety in the flagvariety G/B corresponding to w. Let Z ( w, i ) be the Bott-Samelson-Demazure-Hansenvariety (the desingularization of X ( w )) corresponding to a reduced expression i of w. In this article, we study the cohomology modules of the tangent bundle on Z ( w , i ) , where w is the longest element of the Weyl group W. We describe all the reduced expres-sions of w in terms of a Coxeter element such that Z ( w , i ) is rigid (see Theorem 7.1).Further, if G is of type G , there is no reduced expression i of w for which Z ( w , i ) isrigid (see Theorem 8.2). introduction Let G be a simple algebraic group of adjoint type over the field C of complex numbers.We fix a maximal torus T of G and let W = N G ( T ) /T denote the Weyl group of G withrespect to T . We denote by R the set of roots of G with respect to T and by R + ⊂ R aset of positive roots. Let B + be the Borel subgroup of G containing T with respect to R + .Let w denote the longest element of the Weyl group W . Let B be the Borel subgroup of G opposite to B + determined by T , i.e. B = n w B + n − w , where n w is a representative of w in N G ( T ). Note that the roots of B is the set R − := − R + of negative roots. We use thenotation β < β ∈ R − . Let S = { α , . . . , α n } denote the set of all simple roots in R + ,where n is the rank of G . For simplicity of notation, the simple reflection s α i correspondingto a simple root α i is denoted by s i . For w ∈ W , let X ( w ) := BwB/B denote the Schubertvariety in
G/B corresponding to w . Given a reduced expression w = s i s i · · · s i r of w , withthe corresponding tuple i := ( i , . . . , i r ), we denote by Z ( w, i ) the desingularization of theSchubert variety X ( w ), which is now known as Bott-Samelson-Demazure-Hansen variety.This was first introduced by Bott and Samelson in a differential geometric and topologicalcontext (see [2]). Demazure in [6] and Hansen in [8] independently adapted the constructionin algebro-geometric situation, which explains the reason for the name. For the sake ofsimplicity, we will denote any Bott-Samelson-Demazure-Hansen variety by a BSDH-variety.The construction of the BSDH-variety Z ( w, i ) depends on the choice of the reducedexpression i of w . In [5], the automorphism groups of these varieties were studied. There,the following vanishing results of the tangent bundle T Z ( w,i ) on Z ( w, i ) were proved (see [5,Section 3]): (1) H j ( Z ( w, i ) , T Z ( w,i ) ) = 0 for all j ≥ G is simply laced, then H j ( Z ( w, i ) , T Z ( w,i ) ) = 0 for all j ≥ i of w . While computing thefirst cohomology module H ( Z ( w, i ) , T Z ( w,i ) ) for non simply laced group, we observed thatthis cohomology module very much depend on the choice of a reduced expression i of w .It is a natural question to ask that for which reduced expressions i of w , the cohomologymodule H ( Z ( w, i ) , T Z ( w,i ) ) does vanish ? In [4] a partial answer is given to this questionfor w = w when G = P Sp (2 n, C ) . In [16] a partial answer is given to this question for w = w when G = P SO (2 n + 1 , C ) . In this article, we give partial answers to this questionfor w = w when G is of type F , G . Recall that a Coxeter element is an element of the Weyl group having a reduced ex-pression of the form s i s i · · · s i n such that i j = i l whenever j = l (see [12, p.56, Sec-tion 4.4]). Note that for any Coxeter element c ∈ W , the Weyl group correspondingto the root system of type F (respectively, G ) there is a decreasing sequence of in-tegers 4 ≥ a > a > . . . > a k = 1 (respectively, 2 ≥ a > . . . > a k = 1) such that c = k Q j =1 [ a j , a j − − a := 5 (respectively, a := 3), [ i, j ] := s i s i +1 · · · s j for i ≤ j .In this paper we prove the following theorems. Theorem 1.1.
Assume that G is of type F . Then, H j ( Z ( w , i ) , T ( w ,i ) ) = 0 for all j ≥ if and only if a = 3 or a = 2 . Theorem 1.2.
Assume that G is of type G . Then, H ( Z ( w , i r ) , T ( w ,i r ) ) = 0 for r = 1 , . By the above results, we conclude that if G is of type F (respectively, G ) and i =( i , i , i , i , i , i ) (respectively, i = ( i , i )) is a reduced expression of w as above, thenthe BSDH-variety Z ( w , i ) is rigid (respectively, non rigid).The organization of the paper is as follows: In Section 2, we recall some preliminarieson BSDH-varieties. We deal with G which is of type F , in the later sections 3, 4, 5, 6 and7. In Section 3, we prove H ( w, α j ) = 0 for j = 1 , w ∈ W. In Section 4 (respectively,Section 5) we compute the weight spaces of H (respectively, H ) of the relative tangentbundle of BSDH-varieties associated to some elements of the Weyl group. In Section 6,we prove surjectivity results of some maps from cohomology module of tangent bundleon BSDH variety to cohomology module of relative tangent bundle on BSDH variety. InSection 7, we prove Theorem 1.1 using the results from the previous sections. In Section8, we prove Theorem 1.2. 2. preliminaries In this section, we set up some notation and preliminaries. We refer to [3], [9], [10], [14]for preliminaries in algebraic groups and Lie algebras.Let G be a simple algebraic group of adjoint type over C and T be a maximal torus of G .Let W = N G ( T ) /T denote the Weyl group of G with respect to T and we denote the set IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G of roots of G with respect to T by R . Let B + be a Borel subgroup of G containing T . Let B be the Borel subgroup of G opposite to B + determined by T . That is, B = n B + n − ,where n is a representative in N G ( T ) of the longest element w of W . Let R + ⊂ R bethe set of positive roots of G with respect to the Borel subgroup B + . Note that the set ofroots of B is equal to the set R − := − R + of negative roots.Let S = { α , . . . , α n } denote the set of simple roots in R + . For β ∈ R + , we also use thenotation β >
0. The simple reflection in W corresponding to α i is denoted by s i .Let g be the Lie algebra of G . Let h ⊂ g be the Lie algebra of T and b ⊂ g be the Liealgebra of B . Let X ( T ) denote the group of all characters of T . We have X ( T ) ⊗ R = Hom R ( h R , R ), the dual of the real form of h . The positive definite W -invariant form on Hom R ( h R , R ) induced by the Killing form of g is denoted by ( , ). We use the notation h , i to denote h µ, α i = µ,α )( α,α ) , for every µ ∈ X ( T ) ⊗ R and α ∈ R . We denote by X ( T ) + the setof dominant characters of T with respect to B + . Let ρ denote the half sum of all positiveroots of G with respect to T and B + . For any simple root α , we denote the fundamentalweight corresponding to α by ω α . For 1 ≤ i ≤ n, let h ( α i ) ∈ h be the fundamental coweightcorresponding to α i . That is ; α i ( h ( α j )) = δ ij , where δ ij is Kronecker delta.For a simple root α ∈ S, we denote by n α , a representative of s α in N G ( T ) , and P α theminimal parabolic subgroup of G containing B and n α . We recall that the BSDH-varietycorresponds to a reduced expression i of w = s i s i · · · s i r defined by Z ( w, i ) = P α i × P α i × · · · × P α ir B × B × · · · × B where the action of B × B × · · · × B on P α i × P α i × · · · × P α ir is given by( p , p , . . . , p r )( b , b , . . . , b r ) = ( p · b , b − · p · b , . . . , b − r − · p r · b r ) , p j ∈ P α ij , b j ∈ B for1 ≤ j ≤ r, and i = ( i , i , . . . , i r ) (see [6, Definition 1, p.73], [3, Definition 2.2.1, p.64]).We note that for each reduced expression i of w, Z ( w, i ) is a smooth projective variety.We denote by φ w , the natural birational surjective morphism from Z ( w, i ) to X ( w ) . Let f r : Z ( w, i ) −→ Z ( ws i r , i ′ ) denote the map induced by the projection P α i × P α i × · · · × P α ir −→ P α i × P α i × · · · × P α ir − , where i ′ = ( i , i , . . . , i r − ) . Thenwe observe that f r is a P α ir /B ≃ P -fibration.For a B -module V, let L ( w, V ) denote the restriction of the associted homogeneous vectorbundle on G/B to X ( w ) . By abuse of notation, we denote the pull back of L ( w, V ) via φ w to Z ( w, i ) also by L ( w, V ) , when there is no confusion. Since for any B -module V thevector bundle L ( w, V ) on Z ( w, i ) is the pull back of the homogeneous vector bundle from X ( w ) , we conclude that the cohomology modules H j ( Z ( w, i ) , L ( w, V )) ≃ H j ( X ( w ) , L ( w, V ))for all j ≥ i. Hence we denote H j ( Z ( w, i ) , L ( w, V )) by H j ( w, V ) . In particular, if λ is character of B, then we denote the cohomology modules H j ( Z ( w, i ) , L λ ) by H j ( w, λ ) . We recall the following short exact sequence of B -modules from [5], we call it SES. If l ( w ) = l ( s γ w ) + 1 , γ ∈ S, then we have S.SENTHAMARAI KANNAN AND PINAKINATH SAHA (1) H ( w, V ) ≃ H ( s γ , H ( s γ w, V )) . (2) 0 → H ( s γ , H ( s γ w, V )) → H ( w, V ) → H ( s γ , H ( s γ w, V )) → . Let α be a simple root and λ ∈ X ( T ) be such that h λ, α i ≥ . Let C λ denote onedimensional B -module associated to λ. Here, we recall the following result due to Demazure[7, p.271] on short exact sequence of B -modules: Lemma 2.1.
Let α be a simple root and λ ∈ X ( T ) be such that h λ, α i ≥ . Let ev : H ( s α , λ ) −→ C λ be the evaluation map. Then we have (1) If h λ, α i = 0 , then H ( s α , λ ) ≃ C λ . (2) If h λ, α i ≥ , then C s α ( λ ) ֒ → H ( s α , λ ) , and there is a short exact sequene of B -modules: → H ( s α , λ − α ) −→ H ( s α , λ ) / C s α ( λ ) −→ C λ → . Further more, H ( s α , λ − α ) = 0 when h λ, α i = 1 . (3) Let n = h λ, α i . As a B -module, H ( s α , λ ) has a composition series ⊆ V n ⊆ V n − ⊆ · · · ⊆ V = H ( s α , λ ) such that V i /V i +1 ≃ C λ − iα for i = 0 , , . . . , n − and V n = C s α ( λ ) . We define the dot action by w · λ = w ( λ + ρ ) − ρ, where ρ is the half sum of positiveroots. As a consequence of exact sequences of Lemma 2.1, we can prove the following.Let w ∈ W , α be a simple root, and set v = ws α . Lemma 2.2. If l ( w ) = l ( v ) + 1 , then we have (1) If h λ, α i ≥ , then H j ( w, λ ) = H j ( v, H ( s α , λ )) for all j ≥ . (2) If h λ, α i ≥ , then H j ( w, λ ) = H j +1 ( w, s α · λ ) for all j ≥ . (3) If h λ, α i ≤ − , then H j +1 ( w, λ ) = H j ( w, s α · λ ) for all j ≥ . (4) If h λ, α i = − , then H j ( w, λ ) vanishes for every j ≥ . The following consequence of Lemma 2.2 will be used to compute the cohomology mod-ules in this paper. Now onwards we will denote the Levi subgroup of P α ( α ∈ S ) containing T by L α and the subgroup L α ∩ B by B α . Let π : ˜ G −→ G be the universal cover. Let ˜ L α (respectively, ˜ B α ) be the inverse image of L α (respectively, B α ). Lemma 2.3.
Let V be an irreducible L α -module. Let λ be a character of B α . Then wehave (1) As L α -modules, H j ( L α /B α , V ⊗ C λ ) ≃ V ⊗ H j ( L α /B α , C λ ) . (2) If h λ, α i ≥ , then H ( L α /B α , V ⊗ C λ ) is isomorphic as an L α -module to the tensorproduct of V and H ( L α /B α , C λ ) . Further, we have H j ( L α /B α , V ⊗ C λ ) = 0 forevery j ≥ . (3) If h λ, α i ≤ − , then H ( L α /B α , V ⊗ C λ ) = 0 , and H ( L α /B α , V ⊗ C λ ) is isomorphicto the tensor product of V and H ( L α /B α , C s α · λ ) . (4) If h λ, α i = − , then H j ( L α /B α , V ⊗ C λ ) = 0 for every j ≥ . IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G Proof.
Proof (1): By [14, Proposition 4.8, p.53, I] and [14, Proposition 5.12, p.77, I], forall j ≥
0, we have the following isomorphism of L α -modules: H j ( L α /B α , V ⊗ C λ ) ≃ V ⊗ H j ( L α /B α , C λ ) . Proof of (2), (3) and (4) follows from Lemma 2.2 by taking w = s α and the fact that L α /B α ≃ P α /B . (cid:3) Recall the structure of indecomposable B α -modules and e B α -modules (see [1, Corollary9.1, p.130]). Lemma 2.4. (1)
Any finite dimensional indecomposable e B α -module V is isomorphicto V ′ ⊗ C λ for some irreducible representation V ′ of e L α and for some character λ of e B α . (2) Any finite dimensional indecomposable B α -module V is isomorphic to V ′ ⊗ C λ forsome irreducible representation V ′ of e L α and for some character λ of e B α . reduced expressions Now onwards we will assume that G is of type F . Note that longest element w of theWeyl group W of G is equal to − identity. We recall the following Proposition from [17,Proposition 1.3, p.858]. We use the notation as in [17].
Proposition 3.1.
Let c ∈ W be a Coxeter element, let ω i be the fundamental weightcorresponding to the simple root α i . Then there exists a least positive integer h ( i, c ) suchthat c h ( i,c ) ( ω i ) = w ( ω i ) . Now we can deduce the following:
Lemma 3.2.
Let c ∈ W be a Coxeter element. Then, we have (1) w = c . (2) For any sequence i = ( i , i , · · · , i ) of reduced expressions of c ; the sequence i =( i , i , · · · , i ) is a reduced expression of w . Proof.
Proof of (1): Let η : S −→ S be the involution of S defined by i → i ∗ , where i ∗ is given by ω i ∗ = − w ( ω i ) . Since G is of type F , w = − identity , and hence ω i ∗ = ω i for every i. Therefore, we have i = i ∗ for every i. Let h be the Coxeter number. By [17,Proposition 1.7], we have h ( i, c ) + h ( i ∗ , c ) = h. Since h = 2 | R + | / i = i ∗ , we have h ( i, c ) = h/ , as | R + | = 24 . By Proposition 3.1, we have c ( ω i ) = − ω i for all 1 ≤ i ≤ . Since { ω i : 1 ≤ i ≤ } forms an R -basis of X ( T ) ⊗ R , itfollows that c = − identity. Hence, we have w = c . The assertion (2) follows from thefact that l ( c ) = 4 and l ( w ) = | R + | = 24 . (see [9, p.66, Table 1]). (cid:3) Lemma 3.3.
Let v ∈ W and α ∈ S . Then H ( s j , H ( v, α )) = 0 for j = 1 , . Proof. If H ( s j , H ( v, α )) µ = 0 , then there exists an indecomposable ˜ L α j -summand V of H ( v, α ) such that H ( s j , V ) µ = 0 . By Lemma 2.4, we have V ≃ V ′ ⊗ C λ for some character S.SENTHAMARAI KANNAN AND PINAKINATH SAHA λ of ˜ B α j and for some irreducible ˜ L α j -module V ′ . Since H ( s j , V ) µ = 0 , from Lemma 2.3(3)we have h λ, α j i ≤ −
2. If α is a short root, then H ( w, α ) = 0 for all w ∈ W (see [15,Corollary 5.6, p.778]). Hence we may assume that α is a long root. Then there exists w ∈ W such that w ( α ) = α . Thus H ( v, α ) ⊆ H ( vw, α ) . Again, since α is highest longroot, H ( w , α ) = g −→ H ( vw, α ) is surjective. Let µ ′ be the lowest weight of V. Thenby the above argument µ ′ is a root. Therefore we have µ ′ = µ + λ, where µ is the lowestweight of V ′ . Hence, we have h µ ′ , α j i ≤ −
2. Since α j is a long root and µ ′ is a root, wehave h µ ′ , α j i = − , , . This is a contradiction. Thus we have H ( s j , H ( v, α )) µ = 0 . (cid:3) cohomology modules H ( w, α i )Let w r = ( s s s s ) r s s for 1 ≤ r ≤ . In this section we compute various cohomologymodules H ( w, α i ) for some elements w ∈ W and i = 2 , . Lemma 4.1. (1) H ( w , α ) = 0 . (2) H ( w r , α ) = 0 for r = 4 , . Proof.
We have w = [1 , . By using SES we have H ( s s , α ) = C h ( α ) ⊕ C − α ⊕ C − ( α + α ) . Since h α , α i = 0 , by using SES we have H ( s s s , α ) = H ( s s , α ) . Since h− α , α i = 2 and h− ( α + α ) , α i = 2 , then by using SES we have H ( s s s s , α )= C h ( α ) ⊕ ( C − α ⊕ C − ( α + α ) ⊕ C − ( α +2 α ) ) ⊕ ( C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α +2 α ) ) . (4 . . C h ( α ) ⊕ C − α is indecomposable two dimensional ˜ B α -module, by Lemma 2.4 wehave C h ( α ) ⊕ C − α = V ⊗ C − ω , where V is the standard two dimensional irreducible˜ L α -module.Thus by Lemma 2.3(4), we have H ( ˜ L α / ˜ B α , C h ( α ) ⊕ C − α ) = 0 . Since h− ( α + α ) , α i = − , h− ( α + α ) , α i = − , by Lemma 2.2(4) we have H ( ˜ L α / ˜ B α , C − ( α + α ) ) = 0 and H ( ˜ L α / ˜ B α , C − ( α + α ) ) = 0 . Since h− ( α + 2 α ) , α i = 0 , h− ( α + α + α ) , α i = 0 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α +2 α ) ) = C − ( α +2 α ) and H ( ˜ L α / ˜ B α , C − ( α + α + α ) ) = C − ( α + α + α ) . Since h− ( α + α + 2 α ) , α i = 1 , we have H ( ˜ L α / ˜ B α , C − ( α + α +2 α ) ) = C − ( α + α +2 α ) ⊕ C − ( α +2 α +2 α ) . Thus we have H ( s s s s s , α ) = C − ( α +2 α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α +2 α ) ⊕ C − ( α +2 α +2 α ) . (4 . . IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G Since h− ( α + α + α ) , α i = − , h− ( α + α + 2 α ) , α i = − , by using Lemma 2.3(4)we have H ( ˜ L α / ˜ B α , C − ( α + α + α ) ) = 0 , and H ( ˜ L α / ˜ B α , C − ( α + α +2 α ) ) = 0 . Since h− ( α + 2 α + 2 α ) , α i = 0 , by using Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α +2 α +2 α ) ) = C − ( α +2 α +2 α ) . Since h− ( α + 2 α ) , α i = 1 , by using Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α +2 α ) ) = C − ( α +2 α ) ⊕ C − ( α + α +2 α ) . Therefore we have H ( w , α )= C − ( α +2 α +2 α ) ⊕ C − ( α +2 α ) ⊕ C − ( α + α +2 α ) . (4 . . H ( w , α ) = H ([1 , , H ( w , α )) . Note that the computations of the module H ([1 , , H ( w , α )) is independent of thechoice of a reduced expression of [1 , . We consider the reduced expression s s s s s s s s , of [1 , to compute H ([1 , , H ( w , α )) . Since h− ( α + 2 α ) , α i = − , h− ( α + α + 2 α ) , α i = − , by using Lemma 2.3(3) wehave H ( ˜ L α / ˜ B α , C − ( α +2 α ) ) = 0and H ( ˜ L α / ˜ B α , C − ( α + α +2 α ) ) = 0 . Since h− ( α + 2 α + 2 α ) , α i = 0 , by using Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α +2 α +2 α ) ) = C − ( α +2 α +2 α ) . Thus from the above discussion we have H ( s w , α ) = C − ( α +2 α +2 α ) . Since h− ( α + 2 α + 2 α ) , α i = 2 , by using SES and Lemma 2.3(2) we have H ( s s w , α ) = C − ( α +2 α +2 α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +2 α +2 α ) . Since h− ( α + 2 α + 2 α ) , α i = 0 , h− ( α + 2 α + 2 α + α ) , α i = 1 , h− ( α + 2 α + 2 α +2 α ) , α i = 2 , by using Lemma 2.3(2) we have H ( s s s w , α ) = C − ( α +2 α +2 α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +2 α +2 α ) ⊕ C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) . S.SENTHAMARAI KANNAN AND PINAKINATH SAHA
Since h− ( α + 2 α + 2 α ) , α i = − , h− ( α + 2 α + 2 α + α ) , α i = − , h− ( α + 2 α +2 α + 2 α ) , α i = − , h− ( α + 2 α + 3 α + α ) , α i = 0 , h− ( α + 2 α + 3 α + 2 α ) , α i = 0 , h− ( α + 2 α + 4 α + 2 α ) , α i = 1 , by using Lemma 2.3(2), Lemma 2.3(4) we have H ( s s s s w , α )= C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) ⊕ C − ( α +3 α +4 α +2 α ) . Since C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) is two dimensional indecomposable ˜ B α -module, thus by Lemma 2.4(1) we have C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) = V ⊗ C − ω , where V is the standard two di-mensional irreducible ˜ L α -module.Thus by Lemma 2.3(4) we have H ( ˜ L α / ˜ B α , C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) ) = 0 . Since h− ( α + 2 α + 3 α + α ) , α i = − , and h− ( α + 3 α + 4 α + 2 α ) , α i = 0 , byLemma 2.3(2), Lemma 2.3(4) we have H ( s s s s s w , α ) = C − ( α +3 α +4 α +2 α ) . Since h− ( α + 3 α + 4 α + 2 α ) , α i = − , by Lemma 2.3(4) we have H ( s s s s s s w , α ) = 0 . Thus by using SES and Lemma 2.3(2) we have H ( s s s s s s s w , α ) = 0 . Again by using SES and Lemma 2.3(2) we have H ( w , α ) = H ([1 , , H ( w , α )) = 0 . Proof of (2) follows from (1). (cid:3)
Recall that ω = α + 2 α + 3 α + 2 α . Now onwards we replace α + 2 α + 3 α + 2 α by ω . Lemma 4.2. (1) H ( w s , α ) = C − ω + α . (2) H ( w s , α ) = C − ω . Proof.
Proof of (1): Using SES we have H ( s , α ) = C − α ⊕ C h ( α ) ⊕ C α . Since h α , α i = − , by using SES and Lemma 2.3 we have H ( s s , α ) = C h ( α ) ⊕ C − α ⊕ C − ( α + α ) . Further, since h α , α i = 0 and h− ( α + α ) , α i = 1 , by using SES and Lemma 2.3 wehave H ( s s s , α ) = C h ( α ) ⊕ C − α ⊕ C − ( α + α ) ⊕ C − ( α + α + α ) . Note that the computations of the module H ([1 , , H ( s s s , α )) is independent of thechoice of a reduced expression of [1 , . We consider the reduced expression s s s s s s s s of [1 , to compute H ([1 , , H ( s s s , α )) . Since C h ( α ) ⊕ C − α is two dimensional ˜ B α -module, by Lemma 2.4(1) we have C h ( α ) ⊕ C − α = V ⊗ C − ω where V is the standard two dimensional irreducible ˜ L α -module.Thus by using Lemma 2.3(4) we have H ( ˜ L α / ˜ B α , C h ( α ) ⊕ C − α ) = 0 . IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G Since h− ( α + α ) , α i = 0 , h− ( α + α + α ) , α i = 0 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α + α ) ) = C − ( α + α ) and H ( ˜ L α / ˜ B α , C − ( α + α + α ) ) = C − ( α + α + α ) . Thus from the above discussion we have H ( s s s s , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) . Since h− ( α + α ) , α i = 1 , h− ( α + α + α ) , α i = 1 , by using Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α + α ) ) = C − ( α + α ) ⊕ C − ( α + α + α ) and H ( ˜ L α / ˜ B α , C − ( α + α + α ) ) = C − ( α + α + α ) ⊕ C − ( α + α + α + α ) . Thus from the above discussion we have H ( s s s s s , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α + α ) . Since h− ( α + α ) , α i = 0 , and h− ( α + α + α ) , α i = 0 , by using Lemma 2.3(2) wehave H ( ˜ L α / ˜ B α , C − ( α + α ) ) = C − ( α + α ) and H ( ˜ L α / ˜ B α , C − ( α + α + α ) ) = C − ( α + α + α ) . Since h− ( α + α + α ) , α i = 1 , and h− ( α + α + α + α ) , α i = 1 , by using Lemma2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α + α + α ) ) = C − ( α + α + α ) ⊕ C − ( α +2 α + α ) and H ( ˜ L α / ˜ B α , C − ( α + α + α + α ) ) = C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) . Thus from the above discussion we have H ( s s s s s s , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) . Since h− ( α + α ) , α i = − , h− ( α + α + α ) , α i = − , by using Lemma 2.3(4) we have H ( ˜ L α / ˜ B α , C − ( α + α ) ) = 0and H ( ˜ L α / ˜ B α , C − ( α + α + α ) ) = 0 . Since h− ( α + 2 α + α ) , α i = 0 , h− ( α + α + α ) , α i = 0 , h− ( α + α + α + α ) , α i = 0 , and h− ( α + α + 2 α + α ) , α i = 1 , by using Lemma 2.3(2) we have H ( s s s s s s s , α )= C − ( α +2 α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . Since C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) is the standard two dimensional irreducible˜ L α -module, h− ( α + α + α ) , α i = 0 , h− ( α + 2 α + 2 α + α ) , α i = 1 , and h− ( α +2 α + α ) , α i = − , by using similar arguments as above and using Lemma 2.3(2), Lemma2.3(4) we have H ( s s s s s s s s , α )= C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ ⊕ C − ( α +2 α +3 α + α ) . Since h− ( α + 2 α + 2 α + α ) , α i = 0 , h− ( α + 2 α + 3 α + α ) , α i = 0 , h− ( α + α + α ) , α i = − , h− ( α + α + α + α ) , α i = − , h− ( α + α + 2 α + α ) , α i = − , byusing similar arguments as above and using Lemma 2.3(2), Lemma 2.3(4) we have H ( s s s s s s s s s , α ) = C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) . Since h− ( α + 2 α + 3 α + α ) , α i = 0 , h− ( α + 2 α + 2 α + α ) , α i = − , by using Lemma2.3(2), Lemma 2.3(4) we have H ( s s s s s s s s s s , α ) = C − ( α +2 α +3 α + α ) . Since h− ( α + 2 α + 3 α + α ) , α i = 0 , by using Lemma 2.3(2) we have H ( s s s s s s s s s s s , α ) = C − ( α +2 α +3 α + α ) . Thus we have H ( w s , α ) = C − ( α +2 α +3 α + α ) = C − ω + α . Proof of (2): By the proof of (1) we have H ( w s , α ) = C − ω + α . Since h− ω + α , α i = 1 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ω + α ) = C − ω + α ⊕ C − ω . Therefore we have H ( s w s , α ) = C − ω + α ⊕ C − ω . Since h− ω + α , α i = − , by Lemma 2.3(4) we have H ( ˜ L α / ˜ B α , C − ω + α ) = 0 . Since h− ω , α i = 0 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ω ) = C − ω . Thus from above disscussion we have H ( s s w s , α ) = C − ω . Since α , α are othogonal to ω , by Lemma 2.3(2) we have IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G H ( w s , α ) = C − ω . (cid:3) Corollary 4.3. (1) H ( s w s , α ) = C − ( α +2 α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . (2) H ( s w s , α ) = C − ω ⊕ C − ω + α . (3) H ( s w r s , α ) = 0 for r = 3 , , . Proof.
Proof of (1): we have H ( s s s , α ) = C h ( α ) ⊕ C − α ⊕ C − ( α + α ) ⊕ C − ( α + α + α ) . Since h− α , α i = 1 , h− ( α + α ) , α i = 1 , and h− ( α + α + α ) , α i = 1 , by using SESand Lemma 2.3(2) we have H ( s s s s , α )= C h ( α ) ⊕ C − α ⊕ C − ( α + α ) ⊕ C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α + α ) . Since C h ( α ) ⊕ C − α is two dimensional ˜ B α -module, by Lemma 2.4(1) we have C h ( α ) ⊕ C − α = V ⊗ C − ω where V is the standard two dimensional ˜ L α -module.Thus by using Lemma 2.3(4) we have H ( ˜ L α / ˜ B α , C h ( α ) ⊕ C − α ) = 0 . Since h− ( α + α ) , α i = − , by Lemma 2.3(4) we have H ( ˜ L α / ˜ B α , C − ( α + α ) ) = 0 . Since h− ( α + α ) , α i = 0 , h− ( α + α + α ) , α i = 0 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α + α ) ) = C − ( α + α ) and H ( ˜ L α / ˜ B α , C − ( α + α + α ) ) = C − ( α + α + α ) . Since h− ( α + α + α ) , α i = 1 , h− ( α + α + α + α ) , α i = 1 , by Lemma 2.3(2) wehave H ( ˜ L α / ˜ B α , C − ( α + α + α ) ) = C − ( α + α + α ) ⊕ C − ( α +2 α + α ) and H ( ˜ L α / ˜ B α , C − ( α + α + α + α ) ) = C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) . Thus combining the above discussion we have H ( s s s s s , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) . (4 . . h− ( α + α ) , α i = − , h− ( α + α + α ) , α i = − , by Lemma 2.3(4) we have H ( ˜ L α / ˜ B α , C − ( α + α ) ) = 0and H ( ˜ L α / ˜ B α , C − ( α + α + α ) ) = 0 . Since h− ( α + α + α ) , α i = 0 , h− ( α +2 α + α ) , α i = 0 , and h− ( α + α + α + α ) , α i =0 , by Lemma 2.3(4) we have H ( ˜ L α / ˜ B α , C − ( α + α + α ) ) = C − ( α + α + α ) H ( ˜ L α / ˜ B α , C − ( α +2 α + α ) ) = C − ( α +2 α + α ) and H ( ˜ L α / ˜ B α , C − ( α + α + α + α ) ) = C − ( α + α + α + α ) . Since h− ( α + α + 2 α + α ) , α i = 1 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α + α +2 α + α ) ) = C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . Thus combining the above discussion we have H ( s s s s s s , α )= C − ( α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . Since h− ( α + α + α ) , α i = − h− ( α + α + α + α ) , α i = − , by Lemma 2.3(4)we have H ( ˜ L α / ˜ B α , C − ( α + α + α ) ) = 0and H ( ˜ L α / ˜ B α , C − ( α + α + α + α ) ) = 0 . Since C − ( α + α +2 α + α ) ⊕ C − ( α +2 α + α ) is the standard two dimensional irreducible ˜ L α -module, by using Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α + α +2 α + α ) ⊕ C − ( α +2 α + α ) ) = C − ( α + α +2 α + α ) ⊕ C − ( α +2 α + α ) . Since h− ( α + 2 α + 2 α + α ) , α i = 0 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α +2 α +2 α + α ) ) = C − ( α +2 α +2 α + α ) . Thus combining the above discussion we have H ( w s , α )= C − ( α +2 α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . Since h− ( α + 2 α + α ) , α i = 0 , h− ( α + α + 2 α + α ) , α i = 0 , and h− ( α + 2 α +2 α + α ) , α i = 0 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α +2 α + α ) ) = C − ( α +2 α + α ) H ( ˜ L α / ˜ B α , C − ( α + α +2 α + α ) ) = C − ( α + α +2 α + α ) and IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G H ( ˜ L α / ˜ B α , C − ( α +2 α +2 α + α ) ) = C − ( α +2 α +2 α + α ) . Therefore we have H ( s w s , α )= C − ( α +2 α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . Proof of (2): By Lemma 4.2(1) we have H ( w s , α ) = C − ω + α . Since h− ω + α , α i = 1 , by using SES and Lemma 2.3(2) we have H ( s w s , α ) = C − ω + α ⊕ C − ω . Proof of (3): By the Lemma 4.2(3) we have H ( w s , α ) = C − ω . (cid:3) Since h− ω , α i = − , by Lemma 2.3(4) we have H ( s w s , α ) = 0 . By using SES repeatedly we have H ( s w r s , α ) = 0 for r = 4 , . Corollary 4.4. (1) H ( s s s s s , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) . (2) H ( s s w s , α ) = C − ( α +2 α +2 α + α ) ⊕ C − ω + α . (3) H ( s s w s , α ) = C − ω . Proof.
Proof of (1): Proof follows from (4 . . . Proof of (2): Proof follows from the Corollary 4.3(1).Proof of (3): Proof follows from the Corollary 4.3(2). (cid:3)
Corollary 4.5. (1) H ( s s s s s s , α ) = C − ( α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . (2) H ( s s s w s , α ) = C − ω + α . (3) H ( s s s w s , α ) = C − ω . Proof.
Proof of (1): Proof follows from Corollary 4.4(1).Proof of (2): Proof follows from Corollary 4.4(2).Proof of (3): Proof follows from Corollary 4.4(3). (cid:3)
Corollary 4.6. (1) H ( s s s s s s , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) . (2) H ( s s s w s , α ) = C − ( α +2 α +2 α + α ) ⊕ C − ω + α ⊕ C − ω . ‘Proof. Proof of (1): By the Corollary 4.4(1) we have H ( s s s s s , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) . Since C − ( α + α ) ⊕ C − ( α + α + α ) is the standard two dimensional irreducible ˜ L α -module, byLemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α + α ) ⊕ C − ( α + α + α ) ) = C − ( α + α ) ⊕ C − ( α + α + α ) . Also, since C − ( α + α + α ) ⊕ C − ( α + α + α + α ) is the standard two dimensional irreducible˜ L α -module, by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ) = C − ( α + α + α ) ⊕ C − ( α + α + α + α ) . Since h− ( α + 2 α + α ) , α i = 0 and h− ( α + α + 2 α + α ) , α i = 0 , by Lemma 2.3(2)we have H ( ˜ L α / ˜ B α , C − ( α +2 α + α ) ) = C − ( α +2 α + α ) and H ( ˜ L α / ˜ B α , C − ( α + α +2 α + α ) ) = C − ( α + α +2 α + α ) . Thus combining the above discussion we have H ( s s s s s s , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) . Proof of (2): By Corollary 4.4(2) we have H ( s s w s , α ) = C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) . Since h− ( α + 2 α + 2 α + α ) , α i = 0 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α +2 α +2 α + α ) ) = C − ( α +2 α +2 α + α ) . Further, since h− ( α + 2 α + 3 α + α ) , α i = 1 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α +2 α +3 α + α ) ) = C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +3 α +2 α ) . Thus combining the above discussion we have H ( s s s w s , α ) = C − ( α +2 α +2 α + α ) ⊕ C − ω + α ⊕ C − ω , since ω = α + 2 α + 3 α + 2 α . (cid:3) Corollary 4.7. (1) H ( s s s s s s s , α ) = C − ( α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . (2) H ( s s s s w s , α ) = C − ω + α ⊕ C − ω . IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G Proof.
Proof of (1): By Corollary 4.5(1) we have H ( s s s s s s , α )= C − ( α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . Since C − ( α + α + α ) ⊕ C − ( α + α + α + α ) is the standard two dimenional irreducible ˜ L α -module, by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ) = C − ( α + α + α ) ⊕ C − ( α + α + α + α ) . Moreover, Since h− ( α + 2 α + α ) , α i = 0 , h− ( α + α + 2 α + α ) , α i = 0 and h− ( α + 2 α + 2 α + α ) , α i = 0 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α +2 α + α ) ) = C − ( α +2 α + α ) H ( ˜ L α / ˜ B α , C − ( α + α +2 α + α ) ) = C − ( α + α +2 α + α ) and H ( ˜ L α / ˜ B α , C − ( α +2 α +2 α + α ) ) = C − ( α +2 α +2 α + α ) . Thus combining the above discussion we have H ( s s s s s s s , α )= C − ( α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . Proof of (2): By Corollary 4.5(2) we have H ( s s s w s , α ) = C − ω + α . Since h− ω + α , α i = 1 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ω + α ) = C − ω + α ⊕ C − ω . Thus we have H ( s s s s w s , α ) = C − ω + α ⊕ C − ω . (cid:3) Corollary 4.8. (1) H ( s s s s s s s s , α ) = C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ω + α . (2) H ( s s s s s w s , α ) = C − ω . Proof.
Proof of (1): By Corollary 4.7(1) we have H ( s s s s s s s , α )= C − ( α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . Since h− ( α + 2 α + α ) , α i = − , by Lemma 2.3(4) we have H ( ˜ L α / ˜ B α , C − ( α +2 α + α ) ) = 0 . Since h− ( α + α + α ) , α i = 0 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α + α + α ) ) = C − ( α + α + α ) . Since C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) is the standard two dimensional irreducible ˜ L α -module, by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ) = C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) . Since h− ( α + 2 α + 2 α + α ) , α i = 1 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α +2 α +2 α + α ) ) = C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) . Thus combining the aove discussion we have H ( s s s s s s s , α )= C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) . Proof of (2): By Corollary 4.7(2) we have H ( s s s s w s , α ) = C − ω + α ⊕ C − ω . Since h− ω + α , α i = − , by Lemma 2.3(4) we have H ( ˜ L α / ˜ B α , C − ω + α ) = 0 . Further, since h− ω , α i = 0 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ω ) = C − ω . Thus from the above discussion we have H ( s s s s s w s , α ) = C − ω . (cid:3) Corollary 4.9. (1) H ( s s s s s , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α +2 α + α ) . (2) H ( s s s s s s s s s , α ) = C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ω + α ⊕ C − ω . Proof.
Proof of (1):It is easy to see that H ( s , α ) = C − α ⊕ C h ( α ) ⊕ C α . Since h− α , α i = 1 , by using Lemma 2.3(2), Lemma 2.3(4) we have H ( s s , α ) = C h ( α ) ⊕ C − α ⊕ C − ( α + α ) . Since h− α , α i = 1 , by using Lemma 2.3(2) we have H ( s s s , α ) = C h ( α ) ⊕ C − α ⊕ C − ( α + α ) ⊕ C − ( α + α ) ⊕ C − ( α + α + α ) . Since C h ( α ) ⊕ C − α is the two dimensionl indecomposable ˜ B α -module, by Lemma 2.4(1)we have C h ( α ) ⊕ C − α = V ⊗ C − ω IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G where V is the standard two dimensional irreducible ˜ L α - module. Thus by Lemma 2.3(4)we have H ( ˜ L α / ˜ B , C h ( α ) ⊕ C − α ) = 0 . Also, since h− ( α + α ) , α i = − , by Lemma 2.3(4) we have H ( ˜ L α / ˜ B , C − ( α + α ) ) = 0 . Since h− ( α + α ) , α i = 0 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B , C − ( α + α ) ) = C − ( α + α ) . Since h− ( α + α + α ) , α i = 1 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B , C − ( α + α + α ) ) = C − ( α + α + α ) ⊕ C − ( α +2 α + α ) . Thus combining the above discussion we have H ( s s s s , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α +2 α + α ) . Since C − ( α + α ) ⊕ C − ( α + α + α ) is the standard two dimensional irreducible ˜ L α -module, byLemma 2.3(2) we have H ( ˜ L α / ˜ B , C − ( α + α ) ⊕ C − ( α + α + α ) ) = C − ( α + α ) ⊕ C − ( α + α + α ) . Further, since h− ( α + 2 α + α ) , α i = 0 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B , C − ( α +2 α + α ) ) = C − ( α +2 α + α ) . Therefore we have H ( s s s s s , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α +2 α + α ) . Proof of (2): By Corollary 4.8(1) we have H ( s s s s s s s , α )= C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) . Since C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) , C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) are thetwo dimensional irreducible ˜ L α -modules and h− ( α + α + α ) , α i = 0 , by Lemma 2.3(2)we have H ( s s s s s s s s , α )= C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) . Since C − ( α + α + α ) ⊕ C − ( α + α + α + α ) is the standard two dimensional irreducible ˜ L α -module and h− ( α + 2 α + 2 α + α ) , α i = 0 , by Lemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ) = C − ( α + α + α ) ⊕ C − ( α + α + α + α ) . Moreover, since h− ( α + α + 2 α + α ) , α i = 0 and h− ( α + 2 α + 2 α + α ) , α i = 0 , byLemma 2.3(2) we have H ( ˜ L α / ˜ B α , C − ( α + α +2 α + α ) ) = C − ( α + α +2 α + α ) and H ( ˜ L α / ˜ B α , C − ( α +2 α +2 α + α ) ) = C − ( α +2 α +2 α + α ) . Since h− ( α + 2 α + 3 α + α ) , α i = 1 , by Lemma 2.3 we have H ( ˜ L α / ˜ B α , C − ( α +2 α +3 α + α ) ) = C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +3 α +2 α ) . Therefore combinng the above discussion we have H ( s s s s s s s s s , α )= C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ω + α ⊕ C − ω since ω = α + 2 α + 3 α + 2 α . (cid:3) computions of relative tangent bundles H ( w, α )In this section we compute cohomology modules H ( w, α ) corresponding to some specialWeyl group elements. Lemma 5.1. (1) H ( w r , α ) = 0 for r = 1 , , . (2) H ( w , α ) = C − ω + α . (3) H ( w , α ) = C − ω . Proof.
It is easy to see H ( s , α ) = 0 . Note that we have H ( s , α ) = C h ( α ) ⊕ C − α ⊕ C α . Since h− α , α i = 1 , by using Lemma 2.3(2), Lemma 2.3(4) we have H ( s , H ( s , α )) = 0 . Since H ( s , α ) = 0 , by using Lemma 2.3(1) we have H ( s , H ( s , α )) = 0 . Thus by using SES and the above discussion we have H ( s s , α ) = 0 . By using SES and Lemma 2.3(2) we have H ( s s , α ) = C h ( α ) ⊕ C − α ⊕ C − ( α + α ) .Since h α , α i = 0 , by using Lemma 2.3(2) we have H ( s , H ( s s , α )) = 0and H ( s s s , α ) = C h ( α ) ⊕ C − α ⊕ C − ( α + α ) . (5 . . H ( s s , α ) = 0 , by using Lemma 2.3(1) we have IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G H ( s , H ( s s , α )) = 0 . Thus by using SES and the above discussion we have H ( s s s , α ) = 0 . (5 . . h− α , α i = 2 , h− ( α + α ) , α i = 2 , by using (5 . . , and Lemma 2.3(2) we have H ( s , H ( s s s , α )) = 0 . Further, by (5 . .
2) we have H ( s , H ( s s s , α )) = 0 . Thus by using SES we have H ( s s s s , α ) = 0 . (5 . . H ( s , H ( s s s s , α )) = 0 . By using Lemma 3.3 we have H ( s , H ( s s s s , α )) = 0 . Thus by SES we have H ( s s s s s , α ) = 0 . (5 . . H ( s , H ( s s s s s , α )) = 0 . By Lemma 3.3 we have H ( s , H ( s s s s s , α )) = 0 . Therefore by using SES we have H ( w , α ) = 0 . Since H ( w , α ) = 0 , we have H ( s , H ( w , α )) = 0 . Recall that by (4 . .
3) we have H ( w , α ) = C − ( α +2 α +2 α ) ⊕ C − ( α +2 α ) ⊕ C − ( α + α +2 α ) . Since h− ( α + 2 α ) , α i = 2 , h− ( α + α + 2 α ) , α i = 2 , h− ( α + 2 α + 2 α ) , α i = 2 , by using Lemma 2.3(2) we have H ( s , H ( w , α )) = 0 . Thus by using SES and the above discussion we have H ( s w , α ) = 0 (5 . . and H ( s w , α ) = ( C − ( α +2 α +2 α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +2 α +2 α ) ) ⊕ ( C − ( α +2 α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α +2 α +2 α ) ) ⊕ ( C − ( α + α +2 α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α + α +2 α +2 α ) ) . Since H ( s w , α ) = 0 , we have H ( s , H ( s w , α )) = 0 . Since h− ( α +2 α +2 α ) , α i = 0 , h− ( α +2 α +2 α ) , α i = 0 , h− ( α + α +2 α +2 α ) , α i =0 , h− ( α +2 α +2 α + α ) , α i = 1 , h− ( α +2 α +2 α +2 α ) , α i = 2 , h− ( α +2 α ) , α i = − , h− ( α + α +2 α ) , α i = − , h− ( α +2 α + α ) , α i = − , h− ( α + α +2 α + α ) , α i = − , by using Lemma 2.3 we have H ( s , H ( s w , α )) = C − ( α + α ) ⊕ C − ( α + α + α ) . Thus by using SES and the above discussion we have H ( s s w , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) , (5 . . H ( s s w , α ) = C − ( α +2 α +2 α ) ⊕ C − ( α +2 α +2 α ) ⊕ C − ( α + α +2 α +2 α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +2 α +2 α ) ⊕ C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) . (5 . . H ( s s w , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) , by using Lemma 2.3 we have H ( s , H ( s s w , α )) = C − ( α + α + α ) . By Lemma 3.3 we have H ( s , H ( s s w , α )) = 0 . Thus using SES and above discussion we have H ( s s s w , α )= C − ( α + α + α ) . (5 . . C − ( α + α +2 α +2 α ) ⊕ C − ( α +2 α +2 α +2 α ) is the standard two dimensional irreducible˜ L α -module and h− ( α + 2 α + 2 α ) , α i = 0 , h− ( α + 2 α + 3 α + α ) , α i = 0 , h− ( α +2 α + 3 α + 2 α ) , α i = 0 , h− ( α + 2 α + 4 α + 2 α ) , α i = 1 , h− ( α + 2 α + 2 α ) , α i = − , h− ( α + 2 α + 2 α + α ) , α i = − , by using SES and Lemma 2.3 we have H ( s s s w , α ) = C − ( α +2 α +2 α ) ⊕ C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +3 α +2 α ) ⊕ C − ( α + α +2 α +2 α ) ⊕ C − ( α +2 α +2 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) ⊕ C − ( α +3 α +4 α +2 α ) . (5 . . h− ( α + α + α ) , α i = − , by using Lemma 2.3(4) we have H ( s , H ( s s s w , α )) = 0 . Further, by Lemma 3.3 we have H ( s , H ( s s s w , α )) = 0 . IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G Thus using SES we have H ( w , α ) = 0 . Since C − ( α +2 α +2 α ) ⊕ C − ( α + α +2 α +2 α ) is the standard two dimensional irreducible ˜ L α -module and h− ( α +2 α +2 α +2 α ) , α i = 0 , h− ( α +2 α +3 α + α ) , α i = 0 , h− ( α +2 α +3 α +2 α ) , α i = 0 , and h− ( α +2 α +4 α +2 α ) , α i = 0 , h− ( α +3 α +4 α +2 α ) , α i = 1 , by using SES and Lemma 2.3 we have H ( w , α ) = C − ( α +2 α +2 α ) ⊕ C − ( α + α +2 α +2 α ) ⊕ C − ( α +2 α +2 α +2 α ) ⊕ C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) ⊕ C − ( α +3 α +4 α +2 α ) ⊕ C − (2 α +3 α +4 α +2 α ) . Since H ( w , α ) = 0 , we have H ( s , H ( w , α )) = 0 . Since C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +3 α +2 α ) is the standard two dimensional irreducible˜ L α -module and h− ( α + 2 α + 4 α + 2 α ) , α i = 0 , h− ( α + 3 α + 4 α + 2 α ) , α i = 0 , h− (2 α + 3 α + 4 α + 2 α ) , α i = 0 , h− ( α + 2 α + 2 α ) , α i = − , h− ( α + α + 2 α +2 α ) , α i = − , h− ( α + 2 α + 2 α + 2 α ) , α i = − , by using SES and Lemma 2.3 we have H ( s , H ( w , α )) = C − ( α +2 α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . Thus from the above discussion we have H ( s w , α )= C − ( α +2 α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) (5 . . H ( s w , α ) = C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) ⊕ C − ( α +3 α +4 α +2 α ) ⊕ C − (2 α +3 α +4 α +2 α ) . Since h− ( α + 2 α + 2 α + α ) , α i = 1 , h− ( α + 2 α + α ) , α i = − , and h− ( α + α +2 α + α ) , α i = − , by using Lemma 2.3 we have H ( s , H ( s w , α )) = C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) . Since h− ( α + 3 α + 4 α + 2 α ) , α i = 0 , h− (2 α + 3 α + 4 α + 2 α ) , α i = 0 , h− ( α +2 α + 3 α + α ) , α i = − , and C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) = V ⊗ C − ω (where V is the standard two dimensional irreducible ˜ L α -module), by Lemma 2.3 we have H ( s , H ( s w , α )) = 0 . and H ( s s w , α )= C − ( α +3 α +4 α +2 α ) ⊕ C − (2 α +3 α +4 α +2 α ) . (5 . . H ( s s w , α )= C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) . (5 . . h− ( α + 2 α + 3 α + α ) , α i = 0 , and h− ( α + 2 α + 2 α + α ) , α i = − , by usingLemma 2.3 we have H ( s , H ( s s w , α )) = C − ( α +2 α +3 α + α ) . By Lemma 3.3 we have H ( s , H ( s s w , α )) = 0 . Thus from the above discussion we have H ( s s s w , α )= C − ( α +2 α +3 α + α ) = C − ω + α . (5 . . h− (2 α + 3 α + 4 α + 2 α ) , α i = 0 and h− ( α + 3 α + 4 α + 2 α ) , α i = − , byusing SES and Lemma 2.3 we have H ( s s s w , α )= C − (2 α +3 α +4 α +2 α ) . (5 . . h− ( α + 2 α + 3 α + α ) , α i = 0 , by using Lemma 2.3(2) we have H ( s , H ( s s s w , α )) = C − ( α +2 α +3 α + α ) . By Lemma 3.3 we have H ( s , H ( s s s w , α )) = 0 . Thus from the above discussion we have H ( w , α ) = C − ω + α since ω = α + 2 α + 3 α + 2 α . This proves (2).Since we have H ( w , α ) = 0 (see Lemma 4.1), by using SES we have H ( s w , α ) = H ( s , H ( w , α )) . Since h− ω + α , α i = 1 , by using Lemma 2.3(2) we have H ( s w , α ) = C − ω + α ⊕ C − ω . (5 . . H ( w , α ) = 0 (see Lemma 4.1), by using SES we have H ( s s w , α ) = H ( s , H ( s w , α )) . Since h− ω , α i = 0 and h− ω + α , α i = − , by using Lemma 2.3(2), Lemma 2.3(4) wehave H ( s s w , α ) = C − ω . (5 . . H ( w , α ) = 0 (see Lemma 4.1) and α , α are orthogonal to ω , by Lemma2.3(2) we have H ( w , α ) = C − ω . This gives the proof of (3).Since we have H ( w , α ) = 0 (see Lemma 4.1) and h− ω , α i = − , by using Lemma2.3(4) we have H ( s w , α ) = 0 . IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G Since by Lemma 4.1 we have H ( w , α ) = 0 , and H ( s w , α ) = 0 , by using SES re-peatedly we have H ( w , α ) = 0 . This completes the proof of (1). (cid:3)
Corollary 5.2. (1) H ( s s s , α ) = 0 . (2) H ( s w r , α ) = 0 for r = 1 , , . (3) H ( s w , α ) = C − ( α +2 α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . (4) H ( s w , α ) = C − ω ⊕ C − ω + α . Proof.
Proof of (1) follows from (5 . . r = 1 , proof follows from (5 . . . For r = 4 , . . . . . (cid:3) Corollary 5.3. (1) H ( s s s s , α ) = 0 . (2) H ( s s w , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) . (3) H ( s s w , α ) = C − ( α +2 α +2 α + α ) ⊕ C − ω + α . (4) H ( s s w , α ) = C − ω . (5) H ( s s w r , α ) = 0 for r = 4 , . Proof.
Proof of (1) follows from (5 . . . Proof of (2) follows from (5 . . . Proof of (3) follows from (5 . . . Proof of (4) follows from (5 . . . Proof of (5): By Lemma 4.1 we have H ( w r , α ) = 0 for r = 4 , . Therefore H ( s w r , α ) =0 for r = 4 , . Hence we have H ( s , H ( s w r , α )) = 0 for r = 4 , . On the other hand, byCorollary 5.2(2) we have H ( s w r , α ) = 0 for r = 4 , . Therefore H ( s , H ( s w r , α )) = 0for r = 4 , . Thus by SES we have H ( s s w r , α ) = 0 for r = 4 , . (cid:3) Corollary 5.4. (1) H ( s s s s s , α ) = 0 . (2) H ( s s s w , α ) = C − ( α + α + α ) . (3) H ( s s s w , α ) = C − ω + α . (4) H ( s s s w , α ) = C − ω . (5) H ( s s s w , α ) = 0 . Proof.
Proof of (1) follows from (5 . . . Proof of (2) follows from (5 . . . Proof of (3) follows from (5 . . . Proof of (4): By Lemma 4.1 we have H ( w , α ) = 0 . Therefore H ( s s w , α ) =0 . Hence we have H ( s , H ( s s w , α )) = 0 . On the other hand, by Corollary 5.3(4)we have H ( s s w , α ) = C − ω . Since ω is orthogonal to α , by Lemma2.3(2) we have H ( s , H ( s s w , α )) = C − ω . Thus by SES we have H ( s s s w , α ) = C − ω . Proof of (5): By Lemma 4.1(2) we have H ( w , α ) = 0 . Therefore H ( s s w , α ) = 0 . Hence we have H ( s , H ( s s w , α )) = 0 . On the otherhand, by Corollary 5.3(5) we have H ( s s w , α ) = 0 . Therefore H ( s , H ( s s w , α )) =0 . Thus by SES we have H ( s s s w , α ) = 0 . (cid:3) Corollary 5.5. (1) H ( s s s s s , α ) = 0 . (2) H ( s s s w , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α +2 α + α ) . (3) H ( s s s w , α ) = C − ( α +2 α +2 α + α ) ⊕ C − ω + α ⊕ C − ω . (4) H ( s s s w r , α ) = 0 for r = 3 , . Proof.
Proof of (1): By (4 . .
1) if H ( s s s s , α ) µ = 0 , then we have h µ, α i ≥ . Thususing Lemma 2.3(3) we have H ( s , H ( s s s s , α )) = 0 . On the other hand, by using Corollary 5.3(1) we have H ( s , H ( s s s s , α )) = 0 . Hence we have H ( s s s s s , α ) = 0 . Proof of (2): By (5 . .
7) the ˜ B α -indecomposable summands V of H ( s s w , α ) forwhich H ( s , V ) = 0 are C − ( α +2 α +2 α ) and C − ( α + α +2 α +2 α ) . Thus using Lemma 2.3(3)we have H ( s , H ( s s w , α )) = C − ( α +2 α + α ) ⊕ C − ( α + α +2 α + α ) . On the other hand, by using Corollary 5.3(2) and Lemma 2.3(2) we have H ( s , H ( s s w , α )) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α + α ) . Hence we have H ( s s s w , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α +2 α + α ) . Proof of (3): By (5 . .
11) we have if H ( s s w , α ) µ = 0 , then h µ, α i = 0 . Thus usingLemma 2.3(3) we have H ( s , H ( s s w , α )) = 0 . On the other hand, by using Corollary 5.3(3) and Lemma 2.3(2) we have H ( s , H ( s s w , α )) = C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +3 α +2 α ) . Hence we have
IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G H ( s s s w , α ) = C − ( α +2 α +2 α + α ) ⊕ C − ω + α ⊕ C − ω . Proof of (4): By Lemma 4.1 we have H ( w r , α ) = 0 for r = 3 , . Therefore H ( s s w r , α ) = 0 for r = 3 , . Hence we have H ( s , H ( s s w r , α )) = 0 for r = 3 , . On the other hand, by Corollary 5.3(4), Corollary 5.3(5), we have H ( s , H ( s s w r , α )) =0 for r = 3 , . Thus by using SES we have H ( s s s w r , α ) = 0 for r = 3 , . (cid:3) Corollary 5.6. (1) H ( s s s s s s , α ) = 0 . (2) H ( s s s s w , α ) = C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . (3) H ( s s s s w , α ) = C − ω + α ⊕ C − ω . (4) H ( s s s s w r , α ) = 0 for r = 3 , . Proof.
Proof of (1): By Lemma 3.3 we have H ( s , H ( s s s s s , α )) = 0 . On the other hand, by using Corollary 5.5(1) we have H ( s , H ( s s s s s , α )) = 0 . Hence we have H ( s s s s s s , α ) = H ( s s s s s s , α ) = 0 . Proof of (2): By Corollary 5.5(2) we have H ( s , H ( s s s w , α ))= C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . Now the proof of (2) follows from Lemma 3.3 and SES.Proof of (3): By Corollary 5.5(3), using SES, and Lemma 2.3 we have H ( s , H ( s s s w , α )) = C − ω + α ⊕ C − ω . Now the proof of (3) follows from Lemma 3.3 and SES.Proof of (4): By Lemma 3.3 we have H ( s , H ( s s s w r , α )) = 0 for r = 3 , . On theother hand, by Corollary 5.5(5) we have H ( s , H ( s s s w r , α )) = 0 for r = 3 , . Thusby using SES we have H ( s s s s w r , α ) = 0 for r = 3 , . (cid:3) Lemma 5.7. (1) H ( s s s s s s s , α ) = C − ( α + α ) . (2) H ( s s s s s w , α ) = C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ω + α . (3) H ( s s s s s w , α ) = C − ω . (4) H ( s s s s s w r , α ) = 0 for r = 3 , . Proof.
Proof of (1): Recall from (4 . .
2) that H ( s s s s s , α ) = C − ( α +2 α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α +2 α ) ⊕ C − ( α +2 α +2 α ) . Since h− ( α + 2 α ) , α i = 2 , h− ( α + α + α ) , α i = 1 , h− ( α + α + 2 α ) , α i = 2 , and h− ( α + 2 α + 2 α ) , α i = 2 , by using SES and Lemma 2.3(2) we have H ( s s s s s s , α )= C − ( α +2 α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α +2 α +2 α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α + α +2 α +2 α ) ⊕ C − ( α +2 α +2 α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +2 α +2 α ) . (5 . . C − ( α + α + α ) ⊕ C − ( α + α +2 α ) is the indecomposable ˜ B α -module, by using Lemma2.4(1) we have C − ( α + α + α ) ⊕ C − ( α + α +2 α ) = V ⊗ C − ω (where V is the standard twodimensional irreducible ˜ L α -module), and h− ( α + 2 α ) , α i = − , by using SES andLemma 2.3(3) we have H ( s , H ( s s s s s s , α )) = C − ( α + α ) . By using SES and Corollary 5.6(1) we have H ( s , H ( s s s s s s , α )) = 0 . Thus we have H ( s s s s s s s , α ) = C − ( α + α ) . Proof of (2): Recall from (5 . .
9) that H ( s s s w , α ) = C − ( α +2 α +2 α ) ⊕ C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +3 α +2 α ) ⊕ C − ( α + α +2 α +2 α ) ⊕ C − ( α +2 α +2 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) ⊕ C − ( α +3 α +4 α +2 α ) . Since C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +3 α +2 α ) is the standard two dimensional irreducible˜ L α -module, h− ( α + 2 α + 4 α + 2 α ) , α i = 0 , h− ( α + 3 α + 4 α + 2 α ) , α i = 0 , h− ( α + 2 α + 2 α ) , α i = − , h− ( α + α + 2 α + 2 α ) , α i = − , and h− ( α + 2 α +2 α + 2 α ) , α i = − , by using Lemma 2.3 we have H ( s s s s w , α )= C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) ⊕ C − ( α +3 α +4 α +2 α ) . (5 . . H ( s s s s w , α )= C − ( α + α + α ) ⊕ ⊕ C − ( α + α + α + α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) . Since C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) is indeomposable ˜ B α -module, by Lemma2.4(1) we have C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) = V ⊗ C − ω where V is the standard two dimensional irreducible ˜ L α -module.Further, since h− ( α + 3 α + 4 α + 2 α ) , α i = 0 , h− ( α + 2 α + 3 α + α ) , α i = − , by using Lemma 2.3 we have H ( s , H ( s s s s w , α )) = 0 . IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G Since C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) is the standard two dimensional irreduible ˜ L α -module, h− ( α + α + α ) , α i = 0 , h− ( α +2 α +2 α + α ) , α i = 1 , h− ( α +2 α + α ) , α i = − , by using Lemma 2.3(2) we have H ( s , H ( s s s s w , α ))= C − ( α + α + α ) ⊕ ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) . Thus we have H ( s s s s s w , α )= C − ( α + α + α ) ⊕ ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) . Proof of (3): Recall from (5 . .
14) that H ( s s s w , α ) = C − (2 α +3 α +4 α +2 α ) = C − ω . Since α is orthogonal to ω , by using Lemma 2.3(2) we have H ( s s s s w , α ) = C − (2 α +3 α +4 α +2 α ) = C − ω . Since α is orthogonal to ω , by using Lemma 2.3(2) we have H ( s , H ( s s s s w , α )) = 0 . On the other hand, by Corollary 5.6(3) we have H ( s s s s w , α ) = C − ω + α ⊕ C − ω . Since h− ω , α i = 0 and h− ω + α , α i = − , by using Lemma 2.3 we have H ( s , H ( s s s s w , α )) = C − ω . Thus we have H ( s s s s s w , α ) = C − ω . Proof of (4): By Lemma 4.1 we have H ( w r , α ) = 0 for r = 3 , . Therefore we have H ( s s s s w r , α ) = 0 for r = 3 , . Hence we have H ( s , H ( s s s s w r , α )) = 0 for r = 3 , . On the other hand, Corollary 5.6(4) we have H ( s , H ( s s s s w r , α )) = 0 for r = 3 , . Thus by using SES we have H ( s s s s s w r , α ) = 0 for r = 3 , . (cid:3) Lemma 5.8. (1) H ( s s s s , α ) = 0 . (2) H ( s s s s s s s s , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α +2 α + α ) . (3) H ( s s s s s s w , α ) = C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ω + α ⊕ C − ω . (4) H ( s s s s s s w r , α ) = 0 for r = 2 , . Proof.
Proof of (1): By using SES it is easy to see that H ( s s s , α ) = C h ( α ) ⊕ C − α ⊕ C − ( α + α ) ⊕ C − ( α + α ) and H ( s s s , α ) = C α + α . Since H ( s s s , α ) µ = 0 implies h µ, α i ≥ , by using Lemma 2.3(2) we have H ( s , H ( s s s , α )) = 0 . Since h α + α , α i = − , by using Lemma 2.3(4) we have H ( s , H ( s s s , α )) = 0 . Therefore by using SES we have H ( s s s s , α ) = 0 . Proof of (2): By the Corollary 5.7(1) we have H ( s s s s s s s , α ) = C − ( α + α ) . Since h− ( α + α ) , α i = 1 , by using Lemma 2.3(2) we have H ( s , H ( s s s s s s s , α )) = C − ( α + α ) ⊕ C − ( α + α + α ) . Recall from (5 . .
1) that H ( s s s s s s , α ) = C − ( α +2 α ) ⊕ C − ( α +2 α + α ) ⊕ C − ( α +2 α +2 α ) ⊕ C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α + α +2 α +2 α ) ⊕ C − ( α +2 α +2 α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +2 α +2 α ) . Since C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) is the standard two dimensional irreducible ˜ L α -module, h− ( α + 2 α + 2 α ) , α i = 0 , h− ( α + α + 2 α + 2 α ) , α i = 0 , h− ( α + 2 α +2 α ) , α i = 0 , h− ( α + 2 α + 2 α + α ) , α i = 1 , h− ( α + 2 α + 2 α + 2 α ) , α i = 2 , h− ( α +2 α ) , α i = 0 , h− ( α +2 α + α ) , α i = − , h− ( α +2 α +2 α ) , α i = 0 , and C − ( α + α + α ) ⊕ C − ( α + α +2 α ) = V ⊗ C − ω (where V is the standard twi dimensional irreducible ˜ L α ), byusing SES and Lemma 2.3 we have H ( s s s s s s s , α )= C − ( α +2 α +2 α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α + α +2 α +2 α ) ⊕ C − ( α +2 α +2 α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +2 α +2 α ) ⊕ C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) . (5 . . B α -indecomposable summands V of H ( s s s s s s s , α ) for which H ( s , V ) = 0is C − ( α +2 α +2 α ) . Thus by using SES and Lemma 2.3(3) we have H ( s , H ( s s s s s s s , α )) = C − ( α +2 α + α ) . Therefore by using SES we have H ( s s s s s s s s , α ) = C − ( α + α ) ⊕ C − ( α + α + α ) ⊕ C − ( α +2 α + α ) . Proof of (3): Recall that from Corollary 5.7(2) we have H ( s s s s s w , α )= C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) . Since C − ( α + α + α ) ⊕ C − ( α + α + α + α ) is the standard two dimensional irreducible ˜ L α -module, h− ( α + α + 2 α + α ) , α i = 0 , h− ( α + 2 α + 2 α + α ) , α i = 0 , and h− ( α +2 α + 3 α + α ) , α i = 1 , by using SES and Lemma 2.3 we have IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G H ( s , H ( s s s s s w , α ))= C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +3 α +2 α ) . On the other hand, from (5 . .
2) we have H ( s s s s w , α )= C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) ⊕ C − ( α +3 α +4 α +2 α ) . Since C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) = V ⊗ C − ω , where V is the standard twodimensional irreducible ˜ L α -module, h− ( α + 3 α + 4 α + 2 α ) , α i = 0 , and h− ( α + 2 α +3 α + α ) , α i = − , by using SES and Lemma 2.3 we have H ( s s s s s w , α )= C − ( α +3 α +4 α +2 α ) . Since h− ( α + 3 α + 4 α + 2 α ) , α i = 0 , by using SES and Lemma 2.3 we have H ( s , H ( s s s s s w , α )) = 0 . Therefore by SES we have H ( s s s s s s w , α )= C − ( α + α + α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +3 α +2 α ) . Proof of (4): For r = 2 , we recall that from (5 . .
14) that H ( s s s w , α ) = C − ω . Since α , α are orthogonal to ω , by using SES we have H ( s s s s s w , α ) = C − ω . Further,using the orthogonality of α and ω we have H ( s , H ( s s s s s w , α )) = 0 . On theother hand, by Corollary 5.7(3) we have H ( s , H ( s s s s s w , α )) = 0 . Thus we have H ( s s s s s s w , α ) = 0 . For r = 3 , By Lemma 4.1 we have H ( w , α ) = 0 . Therefore H ( s s s s s w , α ) = 0 . Hence we have H ( s , H ( s s s s s w , α )) = 0 . On the otherhand,by Corollary 5.7(4) we have H ( s , H ( s s s s s w , α )) = 0 . Thus by using SES wehave H ( s s s s s s w r , α ) = 0 . (cid:3) We denote v r = [1 , r for 1 ≤ r ≤ τ r = [1 , r ≤ r ≤ . Lemma 5.9.
We have (1) H i ( τ r , α ) = 0 for all i ≥ , ≤ r ≤ . (2) H i ( v r , α ) = 0 for all i ≥ , ≤ r ≤ . Proof.
Proof of (1): By [15, Corollary 6.4, p.780] we have H i ( τ r , α ) = 0 for all i ≥ , r ≥ . Note that H i ( s s s s s , α ) = H i ( s s s , α ) = H i ( s s s , α ) = 0 for i = 0 , H i ( v r , α ) = 0 for all i ≥ , r ≥ . We note that H i ( s s s s s , α ) = H i ( s s s s s , α )= H i ( s s s s s , α ) = 0 (5 . . for i = 0 , ≤ r ≤ , we have v r = us s s s s for some u ∈ W such that l ( v r ) = l ( u ) + 5 . Thus by using SES repeatedly we have the required result. (cid:3)
Corollary 5.10.
We have the following: (1) H i ( s τ r , α ) = 0 for i ≥ , ≤ r ≤ .H i ( s v r , α ) = 0 for i ≥ , ≤ r ≤ . (2) H i ( s s τ r , α ) = 0 for i ≥ , ≤ r ≤ .H i ( s s v r , α ) = 0 for i ≥ , ≤ r ≤ . (3) H i ( s s s τ r , α ) = 0 for i ≥ , ≤ r ≤ .H i ( s s s v r , α ) = 0 for i ≥ , ≤ r ≤ . (4) H i ( s s s τ r , α ) = 0 for i ≥ , ≤ r ≤ .H i ( s s s v r , α ) = 0 for i ≥ , ≤ r ≤ . (5) H i ( s s s s τ r , α ) = 0 for i ≥ , ≤ r ≤ .H i ( s s s s v r , α ) = 0 for i ≥ , ≤ r ≤ . (6) H i ( s s s s s s τ r , α ) = 0 for i ≥ , ≤ r ≤ .H i ( s s s s s s v r , α ) = 0 for i ≥ , ≤ r ≤ . Proof.
Proofs of (1): By using SES and Lemma 5.9(1) we have H i ( s τ r , α ) = 0 for all1 ≤ r ≤ , i ≥ . By (5 . .
1) we have H i ( s v , α ) = 0 for i ≥ . On the other hand by using SES andLemma 5.9(2) we have H i ( s v r , α ) = 0 for i ≥ , ≤ r ≤ . Thus combining we have H i ( s v r , α ) = 0 for i ≥ , ≤ r ≤ . Proofs of (2) , (3) , (4) , (5) , and (6) follow by using SES and (1) . (cid:3) Surjectivity of some maps
Let w ∈ W and let w = s i s i · · · s i r be a reduced expression for w and let i =( i , i , . . . , i r ). Let τ = s i s i · · · s i r − and i ′ = ( i , i , . . . , i r − ) . Recall the following long exact sequence of B -modules from [5] (see [5, Proposition 3.1,p.673]): 0 → H ( w, α i r ) −→ H ( Z ( w, i ) , T ( w,i ) ) −→ H ( Z ( τ, i ′ ) , T ( τ,i ′ ) ) −→ H ( w, α i r ) −→ H ( Z ( w, i ) , T ( w,i ) ) −→ H ( Z ( τ, i ′ ) , T ( τ,i ′ ) ) −→ H ( w, α i r ) −→ H ( Z ( w, i ) , T ( w,i ) ) −→ H ( Z ( τ, i ′ ) , T ( τ,i ′ ) ) −→ H ( w, α i r ) −→ · · · By [15, Corollary 6.4, p.780], we have H j ( w, α i r ) = 0 for every j ≥ . Thus we have thefollowing exact sequence of B -modules:0 → H ( w, α i r ) −→ H ( Z ( w, i ) , T ( w,i ) ) −→ H ( Z ( τ, i ′ ) , T ( τ,i ′ ) ) −→ H ( w, α i r ) −→ H ( Z ( w, i ) , T ( w,i ) ) −→ H ( Z ( τ, i ′ ) , T ( τ,i ′ ) ) −→ IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G Let w = s j s j · · · s j N be a reduced expression of w . Let w = s j s j · · · s j r , i =( j , j , . . . , j r ) , and j = ( j , j , . . . , j N ) . Lemma 6.1.
The natural homomorphism f : H ( Z ( w , j ) , T ( w ,j ) ) −→ H ( Z ( w, i ) , T ( w,i ) ) of B -modules is injective if and only if w − ( α ) < . Proof.
Suppose w − ( α ) < . By [5, Lemma 6.2, p.667], we have H ( Z ( w, i ) , T ( w,i ) ) − α = 0 . By [5, Theorem 7.1], H ( Z ( w , i ) , T ( w ,i ) ) is a parabolic subalgebra of g and hence thereis a unique B -stable line in H ( Z ( w , i ) , T ( w ,i ) ) , namely g − α . Therefore we conclude thatthe natural homomorphism H ( Z ( w , i ) , T ( w ,i ) ) → H ( Z ( w, i ) , T ( w,i ) )is injective.Conversely, suppose the natural homomorphism H ( Z ( w , i ) , T ( w ,i ) ) → H ( Z ( w, i ) , T ( w,i ) )is injective. Then by [5, Lemma 6.2, p.667], we have w − ( α ) < . (cid:3) Lemma 6.2.
The natural homomorphism f : H ( Z ( w , j ) , T ( w ,j ) ) −→ H ( Z ( w, i ) , T ( w,i ) ) of B -modules is surjective.Proof. (see [4, Lemma 7.1, p.459]). (cid:3) For 1 ≤ r ≤ j r be the reduced expression of τ r = [1 , r s , i r = ( j r ,
2) be thereduced expression of w r = [1 , r s s , and l r = ( i r ,
3) be the reduced expression of w r s =[1 , r s s s . Lemma 6.3. (1)
We have dim H ( Z ( τ , j ) , T ( τ ,j ) ) − ω = 2 . Further, the natural map H ( Z ( τ , j ) , T ( τ ,j ) ) −→ H ( w , α ) is surjective. (2) We have dim H ( Z ( τ , j ) , T ( τ ,j ) ) − ω + α = 2 . Further, the natural map H ( Z ( τ , j ) , T ( τ ,j ) ) −→ H ( w , α ) is surjective.Proof. Proof of (1): Since w − ( α ) < , by Lemma 6.1 we conclude that the naturalhomomorphism H ( Z ( w , i ) , T ( w ,i ) ) → H ( Z ( w , i ) , T ( w ,i ) )is injective.Since α is a short simple root, by [15, Corollary 5.6, p.778] we have H ( w r s , α ) = 0for r = 4 , . On the other hand, by Lemma 5.1 we have H ( w , α ) = 0 , and by Lemma5.9 H ( v r , α ) = 0 , and H ( τ r , α ) = 0 for r = 4 , . Thus from above observations and using LES the natural map H ( Z ( w , i ) , T ( w ,i ) ) → H ( Z ( w , i ) , T ( w ,i ) ) (6 . . H ( Z ( w , i ) , T ( w ,i ) ) is parabolic subalgebra of g . Hence for any µ ∈ X ( T ) \ { } , we have dim H ( Z ( w , i ) , T ( w ,i ) ) µ ≤ . By using LES repeatedly and using Lemma 5.9 we have H ( Z ( τ , j ) , T ( τ ,j ) ) = H ( Z ( w s , l ) , T ( l ,l ) ) . (6 . . → H ( w s , α ) → H ( Z ( w s , l ) , T ( w s ,l ) ) → H ( Z ( w , i ) , T ( w ,i ) ) → . . B -modules.Since w − ( α ) < , by using Lemma 6.1 we conclude that the natural homomorphism H ( Z ( w , i ) , T ( w ,i ) ) → H ( Z ( w , i ) , T ( w ,i ) ) (6 . . . .
4) we have dim H ( Z ( w , i ) , T ( w ,i ) ) − ω ≥ . Hence by Lemma 4.2(2),(6 . .
3) we havedim H ( Z ( w s , l ) , T ( w s ,l ) ) − ω ≥ . (6 . . . .
1) we have dim H ( Z ( w , i ) , T ( w ,i ) ) − ω ≤ . Therefore by using LES we see thatdim H ( Z ( τ , j ) , T ( τ ,j ) ) − ω ≤ . Thus by (6 . . , (6 . .
5) we have dim H ( Z ( τ , j ) , T ( τ ,j ) ) − ω = 2 . Therefore by LES thenatural map H ( Z ( τ , j ) , T ( τ ,j ) ) − ω −→ H ( w , α ) − ω is surjective. Hence by Lemma5.1(3) the natural map H ( Z ( τ , j ) , T ( τ ,j ) ) −→ H ( w , α ) is surjective.Proof of (2): By using LES repeatedly and using Lemma 5.9 we have H ( Z ( τ , j ) , T ( τ ,j ) ) = H ( Z ( w s , l ) , T ( w s ,l ) ) . (6 . . → H ( w s , α ) → H ( Z ( w s , l ) , T ( w s ,l ) ) → H ( Z ( w , i ) , T ( w ,i ) ) → . . B -modules.Since w − ( α ) < , by using Lemma 6.1 we conclude that the natural homomorphism H ( Z ( w , i ) , T ( w ,i ) ) → H ( Z ( w , i ) , T ( w ,i ) ) (6 . . . .
8) we have dim H ( Z ( w , i ) , T ( w ,i ) ) − ω + α ≥ . Hence by Lemma 4.2(1),(6 . .
7) we havedim H ( Z ( w s , l ) , T ( w s ,l ) ) − ω + α ≥ . (6.3.9) IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G By Lemma 5.1, we have H ( w , α ) − ω + α = 0 . Since α is a short simple root, by [15,Corollary 5.6, p.778] we have H ( w s , α ) = 0 . On the other hand, by Lemma 5.9 we have H ( v , α ) = 0 and H ( τ , α ) = 0 . Thus by using LES and from above discussion we havethe natural map H ( Z ( w , i ) , T ( w ,i ) ) − ω + α −→ H ( Z ( w , i ) , T ( w ,i ) ) − ω + α is surjective.Thus by using (6 . .
1) and above surjectivity we have dim H ( Z ( w , i ) , T ( w ,i ) ) − ω + α ≤ . Therefore by using LES we see that dim H ( Z ( τ , j ) , T ( τ ,j ) ) µ ≤ . Thus by (6 . . , (6 . . H ( Z ( τ , j ) , T ( τ ,j ) ) µ = 2 . Therefore by LES the natural map H ( Z ( τ , j ) , T ( τ ,j ) ) − ω + α −→ H ( w , α ) − ω + α is surjective. Hence by Lemma 5.1(2)the natural map H ( Z ( τ , j ) , T ( τ ,j ) ) −→ H ( w , α ) is surjective. (cid:3) Lemma 6.4. (1)
Let µ = − ω , − ω + α . Then we have dim H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) µ = 2 . Further,the natural map H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) −→ H ( s w , α ) is surjective. (2) Let µ = − ( α + 2 α + α ) , − ( α + α + 2 α + α ) , − ( α + 2 α + 2 α + α ) . Then we havedim H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) µ = 2 . Further, the natural map H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) −→ H ( s w , α ) is surjective.Proof. Since ( s w ) − ( α ) < , by Lemma 6.1 we conclude that the natural homomorphism H ( Z ( w , (4 , l )) , T ( w , (4 ,l ) ) → H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) )is injective.Since α is a short simple root, by [15, Corollary 5.6, p.778] we have H ( s w r s , α ) = 0for r = 3 , , . On the other hand, by Corollary 5.2 we have H ( s w r , α ) = 0 for r = 4 , , and by Corollary 5.10(1) we have H ( s v r , α ) = 0 and H ( s τ r , α ) = 0 for r = 4 , . Thus from above observations and using LES the natural map H ( Z ( w , (4 , l )) , T ( w , (4 ,l )) ) → H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) (6 . . H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) )= H ( Z ( s w s , (4 , l )) , T ( s w s , (4 ,l )) ) . (6 . . → H ( s w s , α ) → H ( Z ( s w s , (4 , l )) , T ( s w s , (4 ,l )) ) → H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) → . . B -modules. On the other hand, since ( s w ) − ( α ) < , by using Lemma 6.1, weconclude that the natural homomorphism H ( Z ( w , (4 , l )) , T ( w , (4 ,l ) ) → H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) (6 . . is injective.Let µ = − ω , − ω + α . Thus by (6 . . , we have dim H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) µ ≥ . Hence by (6 . . , by Corollary 4.3(2) we havedim H ( Z ( s w s , (4 , l )) , T ( s w s , (4 ,l )) ) µ ≥ . (6 . . . . , dim H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) µ ≤ . By using LES we have dim H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) µ ≤ . Thus by (6 . . , (6 . . H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) µ = 2 . Therefore by LES the natural map H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) µ −→ H ( s w , α ) µ is surjective. Hence by Corollary 5.2(4)the natural map H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) −→ H ( s w , α ) is surjective.Proof of (2): By using LES repeatedly and using Corollary 5.10(1) we have H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) )= H ( Z ( s w s , (4 , l )) , T ( s w s , (4 ,l )) ) . (6 . . → H ( s w s , α ) → H ( Z ( s w s , (4 , l )) , T ( s w s , (4 ,l )) ) → H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) → , (6 . . B -modules.Let µ = − ( α + 2 α + α ) , − ( α + α + 2 α + α ) , − ( α + 2 α + 2 α + α ) . Since H ( s w , α ) µ = 0 , by Corollary 5.2, (5 . .
5) the same weight appears in H ( s w , α ) , i.e. H ( s w , α ) µ = 0 . This implies H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) µ = 0 . Thus by (6 . . , Corollary 4.3(1) we havedim H ( Z ( s w s , (4 , l )) , T ( s w s , (4 ,l )) ) µ ≥ . (6 . . H ( s w , α ) µ = 0 , by Corollary 5.2 we have H ( s w , α ) µ = 0 . Since α is a shortsimple root, by [15, Corollary 5.6, p.778] we have H ( s w s , α ) = 0 . On the other hand,by using Corollary 5.10(1) we have H ( s v , α ) = 0 and H ( s τ , α ) = 0 . Thus by usingLES and from above discussion we have the natural map H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) µ −→ H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) µ is surjective.By (6 . .
1) and above surjectivity we have dim H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) µ ≤ . By using LES we see that dim H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) µ ≤ . Thus by (6 . . , (6 . . H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) µ = 2 . Therefore by LES the natural map H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) µ −→ H ( s w , α ) µ is surjective. Hence by Corollary 5.2(3)the natural map H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) −→ H ( s w , α ) is surjective. (cid:3) Lemma 6.5. (1)
We have dim H ( Z ( s s τ , (3 , , j )) , T ( s s τ , (3 , ,j )) ) − ω = 2 . Further, the natural map H ( Z ( s s τ , (3 , , j )) , T ( s s τ , (3 , ,j )) ) −→ H ( s s w , α ) is surjective. (2) Let µ = − ( α + 2 α + 2 α + α ) , − ( α + 2 α + 3 α + α ) . Then we havedim H ( Z ( s s τ , (3 , , j )) , T ( s s τ , (3 , ,j )) ) µ = 2 . Further, the natural map
IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G H ( Z ( s s τ , (3 , , j )) , T ( s s τ , (3 , ,j )) ) −→ H ( s s w , α ) is surjective. (3) Let µ = − ( α + α ) , − ( α + α + α ) . Then we havedim H ( Z ( s s τ , (3 , , j )) , T ( s s τ , (3 , ,j )) ) µ = 2 . Further, the natural map H ( Z ( s s τ , (3 , , j )) , T ( s s τ , (3 , ,j )) ) −→ H ( s s w , α ) is surjective.Proof. Proofs of Lemma 6.5(1), Lemma 6.5(2), Lemma 6.5(3) are similar to that of Lemma6.4 with using [15, Corollary 5.6, p,778], Corollary 5.3 and Corollary 5.10(2) appropriately. (cid:3)
Lemma 6.6. (1)
We have dim H ( Z ( s s s τ , (2 , , , j )) , T ( s s s τ , (2 , , ,j )) ) − ω = 2 . Further, the naturalmap H ( Z ( s s s τ , (2 , , , j )) , T ( s s s τ , (2 , , ,j )) ) −→ H ( s s s w , α ) is surjective. (2) We have dim H ( Z ( s s s τ , (2 , , , j )) , T ( s s s τ , (2 , , ,j )) ) − ω + α = 2 . Further, the nat-ural map H ( Z ( s s s τ , (2 , , , j )) , T ( s s s τ , (2 , , ,j )) ) −→ H ( s s s w , α ) is surjective. (3) We have dim H ( Z ( s s s τ , (2 , , , j )) , T ( s s s τ , (2 , , ,j )) ) − ( α + α + α ) = 2 . Further, thenatural map H ( Z ( s s s τ , (2 , , , j )) , T ( s s s τ , (2 , , ,j )) ) −→ H ( s s s w , α ) is surjective.Proof. Proofs of Lemma 6.6(1), Lemma 6.6(2), Lemma 6.6(3) are similar to that of Lemma6.4 with using [15, Corollary 5.6, p,778], Corollary 5.4 and Corollary 5.10(3) appropriately. (cid:3)
Lemma 6.7. (1)
Let µ = − ( α + 2 α + 2 α + α ) , − ω + α , − ω . Then we havedim H ( Z ( s s s τ , (4 , , , j )) , T ( s s s τ , (4 , , ,j )) ) µ = 2 . Further, the natural map H ( Z ( s s s τ , (4 , , , j )) , T ( s s s τ , (4 , , ,j )) ) −→ H ( s s s w , α ) is surjective. (2) Let µ = − ( α + α ) , − ( α + α + α ) , − ( α + α + α ) , − ( α + α + α + α ) , − ( α +2 α + α ) , − ( α + α + 2 α + α ) . Then we havedim H ( Z ( s s s τ , (4 , , , j )) , T ( s s s τ , (4 , , ,j )) ) µ = 2 . Further, the natural map H ( Z ( s s s τ , (4 , , , j )) , T ( s s s τ , (4 , , ,j )) ) −→ H ( s s s w , α ) is surjective.Proof. Proofs of Lemma 6.7(1), Lemma 6.7(2), are similar to that of Lemma 6.4 with using[15, Corollary 5.6, p,778], Corollary 5.5 and Corollary 5.10(4) appropriately. (cid:3)
Lemma 6.8. (1)
Let µ = − ω + α , − ω . Then we havedim H ( Z ( s s s s τ , (4 , , , , j )) , T ( s s s s τ , (4 , , , ,j )) ) µ = 2 . Further, the natural map H ( Z ( s s s s τ , (4 , , , , j )) , T ( s s s s τ , (4 , , , ,j )) ) −→ H ( s s s s w , α ) is surject-ive. (2) Let µ = − ( α + α + α ) , − ( α + α + α + α ) , − ( α + 2 α + α ) , − ( α + α + 2 α + α ) , − ( α + 2 α + 2 α + α ) . Then we have dim H ( Z ( s s s s τ , j ′ ) , T ( s s s s τ ,j ′ ) ) µ = 2 . Further, the natural map H ( Z ( s s s s τ , (4 , , , , j )) , T ( s s s s τ , (4 , , , ,j )) ) −→ H ( s s s s w , α ) is surject-ive.Proof. Proofs of Lemma 6.8(1), Lemma 6.8(2), are similar to that of Lemma 6.4 with using[15, Corollary 5.6, p,778], Corollary 5.6 and Corollary 5.10(5) appropriately. (cid:3)
Lemma 6.9.
Let j ′ = (4 , , , , , , j ) and j ′ = (4 , , , , , , . (1) Let
Λ = {− ( α + α + α ) , − ( α + α + α + α ) , − ( α + α + 2 α + α ) , − ( α + 2 α +2 α + α ) , − ( α + 2 α + 3 α + α ) , − ( α + 2 α + 3 α + 2 α ) } . Then we havedim H ( Z ( s s s s s s τ , j ′ ) , T ( s s s s s s τ ,j ′ ) ) µ = 2 for all µ ∈ Λ . Further, the naturalmap H ( Z ( s s s s s s τ , j ′ ) , T ( s s s s s s τ ,j ′ ) ) −→ H ( s s s s s s w , α ) is surjective. (2) Let
Π = {− ( α + α ) , − ( α + α + α ) , − ( α + 2 α + α ) } . Then we havedim H ( Z ( s s s s s s s , j ′ ) , T ( s s s s s s s ,j ′ ) ) µ = 2 for all µ ∈ Π . Further, the naturalmap H ( Z ( s s s s s s s , j ′ ) , T ( s s s s s s s ,j ′ ) ) −→ H ( s s s s s s s s , α ) is surjective.Proof. Let u = s s s s s s τ and u = s s s s s s s . Note that w = s s s s s s w s s s . Let i ′ be this reduced expression of w . By Lemma 4.1(2) and Corollary 5.2(2) we have H i ( s w , α ) = 0 for i ≥ . Since s commutes with s , s , we have H i ( s w , α )= H i ( s w s s s s , α )= H i ( s w s s s , α ) for i ≥ . Thus we have H i ( s w s s s , α ) for i ≥ . Therefore by using SES we have H i ( s s s s s s w s s s , α ) = 0 for i ≥ . Since s commutes with s we have H i ( s s s s s s w s s , α ) = H i ( s s s s s s w s , α ) for i ≥ .H i ( s s s s s s w s , α ) = H i ( s s s s s s [1 , s s s , α ) = 0 for i ≥ H i ( s s s s s s w s s , α ) = 0 for i ≥ . By Lemma 5.8(4) we have H ( s s s s s s w r , α ) = 0 for r = 2 , . Since α is a short simple root, by [15, Corollary5.6, p.778] we have H ( s s s s s s w r s , α ) = 0 for r = 1 , , . On the other hand, byusing Corollary 5.10(6) we have H ( s s s s s s v r , α ) = 0 , and H ( s s s s s s τ r , α ) = 0for r = 2 , . Thus by using LES and the above discussion we have the natural map H ( Z ( w , i ′ ) , T ( w ,i ′ ) ) → H ( Z ( s s s s s s w , ( j ′ , , T ( s s s s s s w , ( j ′ , ) . (6 . . H ( Z ( u , j ′ ) , T ( u ,j ′ ) ) = H ( Z ( us s , ( j ′ , , , T ( us s , ( j ′ , , ) . (6 . . → H ( us s , α ) → H ( Z ( us s ) , T ( us s , ( j ′ , , ) → H ( Z ( us ) , T ( us , ( j ′ , ) → . (6.9.3)of B -modules.Let Λ = {− ( α + 2 α + 2 α + α ) , − ( α + 2 α + 3 α + α ) , − ( α + 2 α + 3 α + 2 α ) } . Let Λ = {− ( α + α + α ) , − ( α + α + α + α ) , − ( α + α + 2 α + α ) } . By (5 . .
1) we have
IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G H ( s s s s s s s , α )= C − ( α +2 α +2 α ) ⊕ C − ( α + α + α + α ) ⊕ C − ( α + α +2 α + α ) ⊕ C − ( α + α +2 α +2 α ) ⊕ C − ( α +2 α +2 α ) ⊕ C − ( α +2 α +2 α + α ) ⊕ C − ( α +2 α +3 α + α ) ⊕ C − ( α +2 α +2 α +2 α ) ⊕ C − ( α +2 α +3 α +2 α ) ⊕ C − ( α +2 α +4 α +2 α ) . Thus by using SES we see that H ( us , α ) µ = 0 for all µ ∈ Λ . By using LES and Lemma 5.8(2) we have an exact sequence0 → H ( us , α ) µ → H ( Z ( us , ( j ′ , , T ( us , ( j ′ , ) µ → H ( Z ( u, j ′ ) , T ( u, ( j ′ )) ) µ → µ ∈ Λ . Note that H ( u, α ) = H ( s s s s , α ) . Now it is easy to see that H ( s s s s , α ) µ = 0for µ ∈ Λ . Therefore we have H ( Z ( u, j ′ ) , T ( u,j ′ ) ) µ = 0 , for all µ ∈ Λ . Thus combiningabove discussion we have H ( Z ( us , ( j ′ , , T ( us , ( j ′ , ) µ = 0 for all µ ∈ Λ . (6 . . . . , (6 . .
4) and Corollary 4.9(2) we havedim H ( Z ( us s , ( j ′ , , , T ( us s , ( j ′ , , ) µ ≥ µ ∈ Λ . (6 . . . .
1) we have H ( Z ( s s s s s s w , ( j ′ , , T ( s s s s s s w , ( j ′ , ) µ ≤ µ ∈ Λ . Therefore by using LES, Lemma 5.8(3) we have dim H ( Z ( u , j ′ ) , T ( u ,j ′ ) ) µ ≤ µ ∈ Λ . Thus by (6 . . , (6 . .
5) we have dim H ( Z ( u , j ′ ) , T ( u ,j ′ ) ) µ = 2 for all µ ∈ Λ . By using LES we have H ( Z ( u , j ′ ) , T ( u ,j ′ ) ) µ −→ H ( s s s s s s w , α ) µ is surjective for all µ ∈ Λ. Henceby Lemma 5.8(3) the natural map H ( Z ( u , j ′ ) , T ( u ,j ′ ) ) −→ H ( s s s s s s w , α ) issurjective.Proof of (2): It is easy to see that H ( u, α ) = H ( s s s s , α ) = 0 and H ( u, α ) = H ( s s s s , α ) µ = 0 for all µ ∈ Π . Further, we have H i ( s s s s s s , α ) = H i ( s s s s s s , α ) = 0 for all i ≥ H ( Z ( u, j ′ ) , T ( u,j ′ ) ) µ = H ( Z ( s s s s s , (4 , , , , , T ( s s s s s , (4 , , , , ) µ (6 . . µ ∈ Π . By using LES and [15, Corollary 5.6, p.778] we have an exact sequence0 → H ( s s s s s , α ) → H ( Z ( s s s s s ) , T ( s s s s s , (4 , , , , ) → H ( Z ( s s s s ) , T ( s s s s , (4 , , , ) → . (6 . . H ( s s s s , α ) µ = 0 for all µ ∈ Π . Therefore we have H ( Z ( s s s s ) , T ( s s s s , (4 , , , ) µ = 0 for all µ ∈ Π . Thus from (6 . .
7) and Corollary4.9(1) we havedim H ( Z ( s s s s s ) , T ( s s s s s , (4 , , , , ) µ ≥ µ ∈ Π . (6 . . α is a short simple root, by [15, Corollary 5.6, p.778] we have H ( us s , α ) = 0 . By using Corollary 5.10(6) we have H ( s s s s s s τ , α ) = 0 and H ( s s s s s s v , α ) = 0 . By Lemma 5.8 we have H ( s s s s s s w , α ) µ = 0 for all µ ∈ Π . Thus combining above discussion we have the natural map H ( Z ( s s s s s s w , ( j ′ , , T ( s s s s s s w , ( j ′ , ) µ → H ( Z ( us , ( j ′ , , T ( us , ( j ′ , ) µ , is surjective for all µ ∈ Π . Now, using (6 . .
1) and above surjectivity we have H ( Z ( us , ( j ′ , , T ( us , ( j ′ , ) µ ≤ µ ∈ Π . Further, by Lemma 5.8(2) dim H ( us , α ) µ = 1 for all µ ∈ Π . Therefore by using LES0 −→ H ( us , α ) −→ H ( Z ( us , ( j ′ , , T ( us , ( j ′ , ) −→ H ( Z ( u, j ′ ) , T ( u,j ′ ) ) −→ H ( us , α ) −→ H ( Z ( us , ( j ′ , , T ( us , ( j ′ , ) −→ H ( Z ( u, j ′ ) , T ( u,j ′ ) ) −→ H ( Z ( u, j ′ ) , T ( u,j ′ ) ) µ ≤ µ ∈ Π . Therefore by (6 . . , (6 . .
8) we have dim H ( Z ( s s s s s ) , T ( s s s s s , (4 , , , , ) µ = 2 forall µ ∈ Π . Therefore H ( Z ( u, j ′ ) , T ( u,j ′ ) ) µ → H ( us , α ) µ is surjective for all µ ∈ Π . Hence by Lemma 5.8(2) the natural map H ( Z ( u, j ′ ) , T ( u,j ′ ) ) → H ( us , α ) is surjective. (cid:3) main theorem In this section we prove the main theorem. Let c be a Coxeter element of W. Then thereexists a decreasing sequence 4 ≥ a > a > · · · > a k = 1 of positive integers such that c = [ a , a , a − · · · [ a k , a k − − , where for i ≤ j denotes [ i, j ] = s i s i +1 · · · s j . Theorem 7.1. H j ( Z ( w , i ) , T ( w ,i ) ) = 0 for all j ≥ if and only if a = 3 or a = 2 . Proof.
From [5, Proposition 3.1, p. 673], we have H j ( Z ( w , i ) , T ( w ,i ) ) = 0 for all j ≥ . Itis enough to prove the following: H ( Z ( w , i ) , T ( w ,i ) ) = 0 if and only if c is of the form[ a , a , a − · · · [ a k , a k − −
1] with a = 3 or a = 2 . Proof of ( = ⇒ ): If a = 3 , and a = 2 , then c = s s s s , Let u = s s s . Then c = us . Let j = (3 , ,
2) be the sequence corresponding to u. Then using LES, we have:0 → H ( u, α ) → H ( Z ( u, j ) , T ( u,j ) ) → H ( Z ( s s , (3 , , T ( s s , (3 , ) → H ( u, α ) f −→ H ( Z ( u, j ) , T ( u,j ) ) → H ( Z ( s s , (3 , , T ( s s , (3 , ) → . We see that H ( u, α ) = C α + α , H ( s , α ) α + α = 0 , and H ( s s , α ) α + α = 0 . Therefore by LES we have H ( Z ( s s , (3 , , T ( s s , (3 , ) α + α = 0 . Hence f is non zerohomomorphism. Hence H ( Z ( u, j ) , T ( u,j ) )) = 0 . By Lemma 6.2, the natural homomorphism H ( Z ( w , i ) , T ( w ,i ) ) −→ H ( Z ( u, j ) , T ( u,j ) )is surjective.Hence we have H ( Z ( w , i ) , T ( w ,i ) ) = 0 . IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G Proof of ( ⇐ = ) : Assume that a = 3 or a = 2 . We prove the result by studying caseby case. Note that by using Lemma 2.3(4) we have H ( w , α i ) = 0 for i = 1 , , , . In eachof the following cases we use these appropriately.
Case 1: c = s s s s . Then in this case we have w = v = [1 , . By using LES and[15, Corollary 5.6, p.778] we have H ( Z ( w , i ) , T ( w ,i ) ) = H ( Z ( w , i ) , T ( w ,i ) ) . By using LES, Lemma 5.1, Lemma 5.9, and [15, Corollary 5.6, p.778] we have H ( Z ( w , i ) , T ( w ,i ) ) = H ( Z ( w , i ) , T ( w ,i ) ) . By using LES and Lemma 6.3(1) we have H ( Z ( w , i ) , T ( w ,i ) ) = H ( Z ( τ , j ) , T ( τ ,j ) ) . By using LES, Lemma 5.9, and [15, Corollary 5.6, p.778] we have H ( Z ( τ , j ) , T ( τ ,j ) ) = H ( Z ( w , i ) , T ( w ,i ) ) . By using LES and Lemma 6.3(2) we have H ( Z ( w , i ) , T ( w ,i ) ) = H ( Z ( τ , j ) , T ( τ ,j ) ) . By using LES, Lemma 5.9, and [15, Corollary 5.6, p.778] we have H ( Z ( τ , j ) , T ( τ ,j ) ) = H ( Z ( w , i ) , T ( w ,i ) ) . By using LES, Lemma 5.1, Lemma 5.9, and [15, Corollary 5.6, p.778] we have H ( Z ( w , i ) , T ( w ,i ) ) = H ( Z ( w , i ) , T ( w ,i ) ) . By using LES, Lemma 5.1, Lemma 5.9, and [15, Corollary 5.6, p.778] we have H ( Z ( w , i ) , T ( w ,i ) ) = H ( Z ( s s , (1 , , T ( s s , (1 , ) . We see that H ( s , α ) = 0 , H ( s s , α ) = 0 . Thus by using LES we have H ( Z ( s s , (1 , , T ( s s , (1 , ) = 0 . Thus combining all we have H ( Z ( w , i ) , T ( w ,i ) ) = 0 . Case 2: c = s s s s . Then in this case we have w = s w s . By using LES and [15, Corollary 5.6, p.778] we have H ( Z ( w , (4 , l )) , T ( w , (4 ,l )) ) = H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i ) )) . By using LES, Corollary 5.2, Corollary 5.10(1), and [15, Corollary 5.6, p.778] we have H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) = H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) . By using LES, Corollary 5.2, Corollary 5.10(1), and [15, Corollary 5.6, p.778] we have H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) = H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) . By using LES and Lemma 6.4(1) we have H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) = H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) . By using LES, Corollary 5.10(1), and [15, Corollary 5.6, p.778] we have H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) = H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) . By using LES, Lemma 6.4(2) we have H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) = H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) . By using LES, Corollary 5.10(1), and [15, Corollary 5.6, p.778] we have H ( Z ( s τ , (4 , j )) , T ( s τ , (4 ,j )) ) = H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) . By using LES, Corollary 5.2, Corollary 5.10(1), and [15, Corollary 5.6, p.778] we have H ( Z ( s w , (4 , i )) , T ( s w , (4 ,i )) ) = H ( Z ( s s s , (4 , , , T ( s s s , (4 , , ) . We see that H ( s s , α ) = 0 , H ( s s s , α ) = 0 . Thus by using LES we have H ( Z ( s s s , (4 , , , T ( s s s , (4 , , ) = 0 . Thus combining all we have H ( Z ( w , (4 , l )) , T ( w , (4 ,l )) ) = 0 . Case 3: c = s s s s . Then we have w = s s w . By using LES, Corollary 5.3, Corollary 5.10(2), and [15, Corollary 5.6, p.778] we have H ( Z ( s s w , (3 , , i )) , T ( s s w , (3 , ,i )) ) = H ( Z ( s s w , (3 , , i )) , T ( s s w , (3 , ,i )) ) . By using LES, Corollary 5.3, Corollary 5.10(2), and [15, Corollary 5.6, p.778] we have H ( Z ( s s w , (3 , , i )) , T ( s s w , (3 , ,i )) ) = H ( Z ( s s w , (3 , , i )) , T ( s s w , (3 , ,i )) ) . By using LES and Lemma 6.5(1) we have H ( Z ( s s w , (3 , , i )) , T ( s s w , (3 , ,i )) ) = H ( Z ( s s τ , (3 , , j )) , T ( s s τ , (3 , ,j )) ) . By using LES, Corollary 5.10(2), and [15, Corollary 5.6, p.778] we have H ( Z ( s s τ , (3 , , j )) , T ( s s τ , (3 , ,j )) ) = H ( Z ( s s w , (3 , , i )) , T ( s s w , (3 , ,i )) ) . By using LES, Lemma 6.5(2) we have H ( Z ( s s w , (3 , , i )) , T ( s s w , (3 , ,i )) ) = H ( Z ( s s τ , (3 , , j )) , T ( s s τ , (3 , ,j )) ) . By using LES, Corollary 5.10(2) and [15, Corollary 5.6, p.778] we have H ( Z ( s s τ , (3 , , j )) , T ( s s τ , (3 , ,j )) ) = H ( Z ( s s w , (3 , , i )) , T ( s s w , (3 , ,i )) ) . By using LES, Lemma 6.5(3) we have H ( Z ( s s w , (3 , , i )) , T ( s s w , (3 , ,i )) ) = H ( Z ( s s τ , (3 , , j )) , T ( s s τ , (3 , ,j )) ) . By using LES, Corollary 5.10(2), and [15, Corollary 5.6, p.778] we have H ( Z ( s s τ , (3 , , j )) , T ( s s τ , (3 , ,j )) ) = H ( Z ( s s s s , (3 , , , , T ( s s s s , (3 , , , ) . IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G We see that H ( s s , α ) = 0 (see [15, Corollary 5.6, p.778]), H ( s s s , α ) = 0 , H ( s s s s , α ) =0 . Thus by using LES we have H ( Z ( s s s s , (3 , , , , T ( s s s s , (3 , , , ) = 0 . Thus combining all we have H ( Z ( w , (3 , , i )) , T ( w , (3 , ,i )) ) = 0 . Case 4: c = s s s s . Then w = s s s τ . Let t = s s s . By using LES, Corollary 5.10(3) and [15, Corollary 5.6, p.778] we have H ( Z ( w , (2 , , , j )) , T ( w , (2 , , ,j )) ) = H ( Z ( t w , (2 , , , i )) , T ( t w , (2 , , ,i )) ) . By using LES, Corollary 5.4, Corollary 5.10(3), and [15, Corollary 5.6, p.778] we have H ( Z ( t w , (2 , , , i )) , T ( t w , (2 , , ,i )) ) = H ( Z ( t w , (2 , , , i )) , T ( t w , (2 , , ,i )) ) . By using LES and Lemma 6.6(1) we have H ( Z ( t w , (2 , , , i )) , T ( t w , (2 , , ,i )) ) = H ( Z ( t τ , (2 , , , j )) , T ( t τ , (2 , , ,j )) ) . By using LES, Corollary 5.10(3), and [15, Corollary 5.6, p.778] we have H ( Z ( t τ , (2 , , , j )) , T ( t τ , (2 , , ,j )) ) = H ( Z ( t w , (2 , , , i )) , T ( t w , (2 , , ,i )) ) . By using LES, Lemma 6.6(2) we have H ( Z ( t w , (2 , , , i )) , T ( t w , (2 , , ,i )) ) = H ( Z ( t τ , (2 , , , j )) , T ( t τ , (2 , , ,j )) ) . By using LES, Corollary 5.10(3), and [15, Corollary 5.6, p.778] we have H ( Z ( t τ , (2 , , , j )) , T ( t τ , (2 , , ,j )) ) = H ( Z ( t w , (2 , , , i )) , T ( t w , (2 , , ,i )) ) . By using LES, Lemma 6.6(3) we have H ( Z ( t w , (2 , , , i )) , T ( t w , (2 , , ,i )) ) = H ( Z ( t τ , (2 , , , j )) , T ( t τ , (2 , , ,j )) ) . By using LES, Corollary 5.10(3), and [15, Corollary 5.6, p.778] we have H ( Z ( t τ , (2 , , , j )) , T ( t τ , (2 , , ,j )) ) = H ( Z ( t s s , (2 , , , , , T ( t s s , (2 , , , , ) . It is easy to see that H ( t s , α ) = H ( s s , α ) = 0 . We see that H ( s s , α ) =0 , H ( t , α ) = 0 by [15, Corollary 5.6, p.778]. H ( t s s , α ) = 0 by Corollay 5.4.Thus by using LES we have H ( Z ( t s s , (2 , , , , , T ( t s s , (2 , , , , ) = 0 . Thus combining all we have H ( Z ( w , (2 , , , j )) , T ( w , (2 , , ,j )) ) = 0 . Case 5: c = s s s s . In this case we have w = s s s w s s s . Let t = s s s . Since s commutes with s , we have H i ( t w s s , α ) = H i ( t w s s s s s , α ) = 0 for i ≥ H ( Z ( w , (4 , , , i , , , , T ( w , (4 , , ,i , , , ) = H ( Z ( t w , (4 , , , i )) , T ( t w , (4 , , ,i )) ) . By using LES, Corollary 5.5, Corollary 5.10(4), and [15, Corollary 5.6, p.778] we have H ( Z ( t w , (4 , , , i )) , T ( t w , (4 , , ,i )) ) = H ( Z ( t w , (4 , , , i )) , T ( t w , (4 , , ,i )) ) . By using LES, Corollary 5.5, Corollary 5.10(4), and [15, Corollary 5.6, p.778] we have H ( Z ( t w , (4 , , , i )) , T ( t w , (4 , , ,i )) ) = H ( Z ( t w , (4 , , , i )) , T ( t w , (4 , , ,i )) ) . By using LES and Lemma 6.7(1) we have H ( Z ( t w , (4 , , , i )) , T ( t w , (4 , , ,i )) ) = H ( Z ( t τ , (4 , , , j )) , T ( t τ , (4 , , ,j )) ) . By using LES, Corollary 5.10(4), and [15, Corollary 5.6, p.778] we have H ( Z ( t τ , (4 , , , j )) , T ( t τ , (4 , , ,j )) ) = H ( Z ( t w , (4 , , , i )) , T ( t w , (4 , , ,i )) ) . By using LES, Lemma 6.7(2) we have H ( Z ( t w , (4 , , , i )) , T ( t w , (4 , , ,i )) ) = H ( Z ( t τ , (4 , , , j )) , T ( t τ , (4 , , ,j )) ) . By using LES, Corollary 5.10(4) and [15, Corollary 5.6, p.778] we have H ( Z ( t τ , (4 , , , j )) , T ( t τ , (4 , , ,j )) ) = H ( Z ( t s s , (4 , , , , , T ( t s s , (4 , , , , ) . We see that H ( s s , α ) = 0 , H ( t , α ) = 0 by [15, Corollary 5.6, p.778]. Since s , s commutes with s we have H ( t s , α ) = H ( s , α ) = 0 . By Corollary 5.5 we have H ( t s s , α ) = 0 . Thus by using LES we have H ( Z ( t s s , (4 , , , , , T ( t s s , (4 , , , , ) = 0 . Thus com-bining all we have H ( Z ( w , (4 , , , i , , , , T ( w , (4 , , ,i , , , ) = 0 . Case 6: c = s s s s . In this case we have w = s s s s w s s . Let t = s s s s . Byusing LES and [15, Corollary 5.6, p.778] we have H ( Z ( w , (4 , , , , i , , , T ( w , (4 , , , ,i , , ) = H ( Z ( t w , (4 , , , , i )) , T ( t w , (4 , , , ,i )) ) . By using LES, Corollary 5.6, Corollary 5.10(5), and [15, Corollary 5.6, p.778] we have H ( Z ( t w , (4 , , , , i )) , T ( t w , (4 , , , ,i )) ) = H ( Z ( t w , (4 , , , , i )) , T ( t w , (4 , , , ,i )) ) . By using LES, Corollary 5.6, Corollary 5.10(5), and [15, Corollary 5.6, p.778] we have H ( Z ( t w , (4 , , , , i )) , T ( t w , (4 , , , ,i )) = H ( Z ( t w , (4 , , , , i )) , T ( t w , (4 , , , ,i )) ) . By using LES and Lemma 6.8(1) we have H ( Z ( t w , (4 , , , , i )) , T ( t w , (4 , , , ,i )) ) = H ( Z ( t τ , (4 , , , , j )) , T ( t τ , (4 , , , ,j )) ) . By using Corollary 5.10(5) and [15, Corollary 5.6, p.778] we have H ( Z ( t τ , (4 , , , , j )) , T ( t τ , (4 , , , ,j )) ) = H ( Z ( t w , (4 , , , , i )) , T ( t w , (4 , , , ,i )) ) . By using LES, Lemma 6.8(2) we have H ( Z ( t w , (4 , , , , i )) , T ( t w , (4 , , , ,i )) ) = H ( Z ( t τ , (4 , , , , j )) , T ( t τ , (4 , , , ,j )) ) . IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G By using LES, Corollary 5.10(5), and [15, Corollary 5.6, p.778] we have H ( Z ( t τ , (4 , , , , j )) , T ( t τ , (4 , , , ,j )) ) = H ( Z ( t s s , (4 , , , , , , T ( t s s , (4 , , , , , ) . We see that H ( s s , α ) = 0 , H ( t s , α ) = 0 . Further, by using [15, Corollary 5.6,p.778] we have H ( s s s , α ) = 0 , H ( t , α ) = 0 . By Corollary 5.6 we have H ( t s s , α ) =0 . Therefore by using LES we have H ( Z ( t s s , (4 , , , , , , T ( t s s , (4 , , , , , ) = 0 . Thus combining all we have H ( Z ( w , (4 , , , , i , , , T ( w , (4 , , , ,i , , ) = 0 . Case 7: c = s s s s . In this case we have w = s s s s s s w s s s s . Let t = s s s s s s . Let i ′ = (4 , , , , , , l , , , . Recall that l r = ( i r , . Let i ′ r = (4 , , , , , , i r )be the reduced expressions of t w r for r = 1 , , . Let j ′ r = (4 , , , , , , j r ) be the reducedexpressions of t τ r for r = 1 , , . Let j ′ = (4 , , , , , ,
1) be the reduced expression of t s . By Lemma 4.1(2) and Corollary 5.2(2) we have H i ( s w , α ) = 0 for i ≥ . Since s commutes with s , s , we have H i ( s w , α )= H i ( s w s s s s , α )= H i ( s w s s s , α ) for i ≥ . Thus we have H i ( s w s s s , α ) = 0 for i ≥ . Therefore by using SES we have H i ( t w s s s , α ) = 0 for i ≥ . Since s commutes with s we have H i ( t w s s , α ) = H i ( t w s , α ) for i ≥ . H i ( t w s , α ) = H i ( t [1 , s s s , α ) = 0 for i ≥ H i ( t w s s , α ) = 0 for i ≥ . Thus by using LES, above discussion,and [15, Corollary 5.6, p.778] we have H ( Z ( w , i ′ ) , T ( w ,i ′ ) ) = H ( Z ( t w , i ′ ) , T ( t w ,i ′ ) ) . By using LES, Lemma 5.8(4), Corollary 5.10(6), and [15, Corollary 5.6, p.778] we have H ( Z ( t w , i ′ ) , T ( t w ,i ′ ) ) = H ( Z ( t w , i ′ ) , T ( t w ,i ′ ) ) . By using LES, Lemma 5.8(4), Corollary 5.10(6), and [15, Corollary 5.6, p.778] we have H ( Z ( t w , i ′ ) , T ( t w ,i ′ ) ) = H ( Z ( t w , i ′ ) , T ( t w ,i ′ ) ) . By using LES and Lemma 6.9(1) we have H ( Z ( t w , i ′ ) , T ( t w ,i ′ ) ) = H ( Z ( t τ , j ′ ) , T ( t τ ,j ′ ) ) . By using LES, Corollary 5.10(6), and [15, Corollary 5.6, p.778] we have H ( Z ( t τ , j ′ ) , T ( t τ ,j ′ ) ) = H ( Z ( t s s , ( j ′ , , T ( t s s , ( j ′ , ) . By using LES and Lemma 6.9(2) we have H ( Z ( t s s , ( j ′ , , T ( t s s , ( j ′ , ) = H ( Z ( t s , j ′ ) , T ( t s ,j ′ ) ) . By [15, Corollary 5.6, p.778] we see that H ( s s , α ) = 0 , H ( s s s , α ) = 0 ,H ( s s s s s , α ) = 0 , and H ( t , α ) = 0 . By Lemma 5.8(1) we have H ( s s s s , α ) =0 . Since s , s commute with s , we have H ( t s , α ) = H ( s s s s , α ) . It is easy tosee by using SES that H ( s s s s , α ) = 0 . Thus we have H ( t s , α ) = 0 . There-fore by using LES we have H ( Z ( t s , j ′ ) , T ( t s ,j ′ ) ) = 0 . Thus combining all we have H ( Z ( w , i ′ ) , T ( w ,i ′ ) ) = 0 . (cid:3) Corollary 7.2.
Let c be a Coxeter element such that c is of the form [ a , a , a − · · · [ a k , a k − − with a = 3 or a = 2 and a k = 1 . Let ( w , i ) be a reduced expres-sion of w in terms of c as in Theorem 7.1. Then, Z ( w , i ) has no deformations.Proof. By Theorem 7.1 and by [5, Proposition 3.1, p.673], we have H i ( Z ( w , i ) , T ( w ,i ) ) =0 for all i > . Hence, by [13, Proposition 6.2.10, p.272], we see that Z ( w , i ) has nodeformations. (cid:3) non rigidity for G Now onwards we will assume that G is of type G . Note that the longest element w ofthe Weyl group W of G is equal to − identity. We recall the following proposition from[17, Proposition 1.3, p.858]. We use Proposition 3.1 and the notation as in [17] to deducethe following:
Lemma 8.1.
Let c ∈ W be a Coxeter element. Then, we have (1) w = c . (2) For any sequence i = ( i , i , i ) of reduced expressions of c ; the sequence i =( i , i , i ) is a reduced expression of w . Proof.
Proof of (1): Let η : S −→ S be the involution of S defined by i → i ∗ , where i ∗ is given by ω i ∗ = − w ( ω i ) . Since G is of type G , w = − identity. Therefore, wehave i = i ∗ for every i. Let h be the Coxeter number. By [17, Proposition 1.7], we have h ( i, c ) + h ( i ∗ , c ) = h. Since h = 2 | R + | / i = i ∗ , we have h ( i, c ) = h/ , as | R + | = 6 . By Proposition 3.1, we have c ( ω i ) = − ω i for all i = 1 , . Since { ω i : i = 1 , } forms an R -basis of X ( T ) ⊗ R , it follows that c = − identity. Hence,we have w = c . The assertion (2) follows from the fact that l ( c ) = 2 and l ( w ) = | R + | = 6 . (see [9, p.66, Table 1]). (cid:3) Let c be a coxeter element of W. Then c = s s or c = s s . Then from Lemma 8.1 wehave w = s s s s s s , or w = s s s s s s according as c = s s or c = s s . Let i (repectively, i ) be the the reduced expression of w = s s s s s s (respectively, w = s s s s s s ). Then we have Theorem 8.2. H ( Z ( w , i r ) , T ( w ,i r ) ) = 0 for r = 1 , . Proof.
Let c = s s . Let i = (1 ,
2) be the sequence corresponding to c. Then using LES,we have: 0 −→ H ( c, α ) −→ H ( Z ( c, i ) , T ( c,i ) ) −→ H ( s , α ) −→ H ( c, α ) g −→ H ( Z ( c, i ) , T ( c,i ) ) −→ . IGIDITY OF BOTT-SAMELSON-DEMAZURE-HANSEN VARIETY FOR F AND G By using SES, we see that H ( s s , α ) = C α + α ⊕ C α +2 α . Now H ( s , α ) α + α = 0 . Hence g is a non zero homomorphism. Hence H ( Z ( c, i ) , T ( c,i ) )) = 0 . By Lemma 6.2, thenatural homomorphism H ( Z ( w , i )) −→ H ( Z ( c, i ) , T ( c,i ) )is surjective.Hence we have H ( Z ( w , i ) , T ( w ,i ) ) = 0 . Let c = s s , u = s s s . Let j = (2 , ,
2) be the sequence corresponding to u. Thenusing LES, we have:0 −→ H ( u, α ) −→ H ( Z ( u, j ) , T ( u,j ) ) −→ H ( Z ( s s , (2 , , T ( s s , (2 , ) −→ H ( u, α ) h −→ H ( Z ( u, j ) , T ( u,j ) ) −→ H ( Z ( s s , (2 , , T ( s s , (2 , ) → . We see that H ( u, α ) = C α ⊕ C α + α ⊕ C α +2 α , H ( s , α ) α + α = 0 , and H ( s s , α ) α + α =0 . Therefore by LES we have H ( Z ( s s , (2 , , T ( s s , (2 , ) α + α = 0 . Hence h is a non zerohomomorphism. Hence H ( Z ( u, j ) , T ( u,j ) )) = 0 . By Lemma 6.2, the natural homomorphism H ( Z ( w , i )) −→ H ( Z ( u, j ) , T ( u,j ) )is surjective.Hence we have H ( Z ( w , i ) , T ( w ,i ) ) = 0 . (cid:3) Acknowledgements
The authors would like to thank to the Infosys Foundation for thepartial financial support.
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