Trivial Witt groups of flag varieties

TRIVIAL WITT GR OUPS OF FLA G V ARIETIES BAPTISTE CALM ` ES AND JEAN F ASEL Abstract. Let G be a split semi-si mple linear algebraic group o ve r a field k of c haracteristic not 2. Let P be a parab olic subgroup and let L b e a li ne bundle on the pro jectiv e homogeneou s v a riety G/P . W e give a s imple condition on the class of L in Pic( G/P ) / 2 in terms of Dynkin diagrams impl ying that the Witt groups W i ( G/P, L ) are zero for all integ ers i . In particular, if B is a Borel subgroup, then W i ( G/B , L ) i s zero unless L is tr i vial in Pic( G/B ) / 2. The main result of this sho rt note is par t of a more gener al work on Witt gr oups of split pro jectiv e homog eneous v arieties. Howev er, contrary to the re s t o f the work, it has a rather quic k pr o of that do es not require any heavy mac hinery . Since other peo ple in the sub ject ha ve expressed interest and used it in computations, we hav e decided to write it down as a standalone result. The first named author would like to thank Ian Gro jno wski for a fruitful discus- sion. Let G be a split simply connected semi-s imple linear algebra ic gro up ov er a field k , T a maximal split torus o f G and B a B orel s ubg roup containing T . Let ∆ = ∆ G denote the corres po nding Dynkin diagr am of G , in which the vertices are the simple r o ots of G with resp ect to B . Let Θ be a subset of the vertices of the Dynkin diagr am ∆. The standar d parab o lic subg r oup P Θ is defined as the subgroup o f G genera ted by B and the ro ot subgr oups U − α for all α ∈ Θ. An y parab olic subgroup of G is conjuga te to s uch a standard parab olic subg roup, s o a ll pro jective homog eneous v a rieties under G are of the form X Θ = G/P Θ for some subset Θ of ∆. Note that a pro j ective homogeneous v ariet y under a semi-s imple group can alwa ys b e co nsidered as a pro jective ho mogeneous v ariet y under the simply connected co v er of the group, so the assumption that G is s imply connected is harmless for our purp os e s. T o an element λ of the weigh t lattice of the r o ot system of G , w e a s so ciate the line bundle L λ ov er G/B defined as follows. Since G is simply connected, the weigh t λ is a character o f T and extends (uniquely up to isomo r phism) to a one- dimensional repres ent ation V λ of B . W e define L λ as the line bundle who se total space is G × B V λ . The assignment λ 7→ [ L λ ] defines a gr oup isomorphism betw een the weigh t lattice a nd Pic( G/B ). See for example [ MT95 , § 2.1]. T o each simple ro ot α corresp o nds a weigh t ω α characterized b y the re la tion h ω α , β ∨ i = δ α,β for all simple ro o ts β . These weights ar e called fundamen tal weigh ts, and they fo rm a basis of the weight lattice. The Picard group o f G/B is thus the free abe lian g roup gener ated by the classes [ L ω ] with ω a fundamen tal weigh t. The Picard group of X Θ injects in the o ne of G/B = X ∅ by pull-back, and its image is the free a be lia n group g enerated by the classes of line bundles L ω α where α is a simple ro ot no t in Θ (see [ MT95 , Prop. 2.3]). W e subsequently ident ify Pic( X Θ ) with its image in Pic( G/B ). This induces a bijection betw een the subsets of ∆ − Θ and the Picard group mo dulo 2 of X Θ , sending a subset Λ to the Date : Septem ber 25, 2018. 2010 Mathematics Subje ct Classific ation. 19G99, 11E81, 20G10, 20G15 , 14M15, 14M17. Key wor ds and phr ases. Witt group, Flag v ari et y , Pro jectiv e homogeneo us v ariety . 1 2 BAPTISTE CALM ` ES AND JEAN F ASEL class [ L P α ∈ Λ ω α ] in Pic( X Θ ) / 2. F or a line bundle L on X Θ , w e use the notatio n Λ( L ) for the in verse bijection applied to its cla ss in Pic( X Θ ) / 2. In other w ords, a simple ro ot α is in Λ ( L ) if a nd o nly if [ L ω α ] app ea rs with an o dd m ultiplicit y in the decomp osition of [ L ] ∈ Pic( G/B ). W e say that a simple ro o t α ∈ ∆ − Θ is not adjac ent to Θ if no edge of ∆ connects α and a vertex of Θ, i.e. if α is or thogonal to all simple ro ots β ∈ Θ. Theorem. Assume t he char acteristic of k is not 2 . L e t L b e a line bund le on X Θ and let Λ = Λ( L ) b e the asso ciate d subset of ∆ − Θ . If ther e is a vertex α ∈ Λ that is not adjac ent to Θ , then W i ( X Θ , L ) = 0 for al l inte ge rs i . Pr o of. Let Θ ′ be the subset Θ ∪ { α } . Then, there is a vector bundle V o f rank 2 ov er X Θ ′ and an isomo rphism X Θ ≃ P X Θ ′ ( V ) of schemes ov er X Θ ′ . See the lemma below for a pro o f. The theorem is then implied by the following fact: for any vector bundle V of rank 2 ov er a regular ba se X and any line bundle L on P X ( V ) such that [ L ] ∈ Pic( P X ( V )) / 2 is not in the image of the pull-back from Pic( X ) / 2, the Witt groups W i ( P X ( V ) , L ) are zero for all v alues of i [ W al03 , Thm. 1.3]. Obviously , in our case, [ L ] is not in the image o f the pull-back from P ic( X Θ ′ ) / 2, since P ic( X Θ ) / 2 = Pic( X Θ ′ ) / 2 ⊕ Z / 2 [ L ω α ] and α ∈ Λ precise ly means that the comp onent of L on [ L ω α ] is nonzer o.  Example. Let B b e a B orel subg roup o f G . Then W i ( G/B , L ) = 0 for any i unless [ L ] = [ O ] ∈ Pic( G/B ) / 2. Indeed, G/B = X ∅ so any L suc h that Λ( L ) 6 = ∅ satisfies the as sumption of the theorem. Here a re mor e examples with a g r oup G of type D 4 . α (a) (b) α (c) (d) (e) The vertices in Θ are marked , while the vertices in Λ ( L ) a re marked . Cases (a) and (c) s atisfy the as sumption of the theo rem (with the g iven α ), while cas es (b) a nd (d) do not. With Θ as in case (e), no Λ ( L ) c an sa tisfy the assumption o f the theorem. Remark. The co nv erse of the theorem is no t true: it ca n happ en that a line bundle L is such that W i ( X Θ , L ) = 0 for all v alues of i , while all α ∈ Λ( L ) are adjacent to Θ. Examples ar e provided by quotient s of a group of t ype A n by a maximal prop er parab olic subgroup: all v ertices are in Θ but the d -th o ne. This yields a Gra ssmann v a riety G r ( d, d + e ) with n = d + e − 1. When d and e ar e o dd, and when L is a generator of the Picard gr o up of G r ( d, d + e ), then W i ( G r ( d, d + e ) , L ) = 0 for all v a lues of i (see [ BC07 ]). Since we could not find a reference in the literature for the following well-kno wn fact, we include a pro of (actually v a lid ov er an y base, not necessarily a field). Lemma. If α is a simple r o ot ortho go nal to al l simple r o ots in Θ , then the natura l pr o je ction X Θ → X Θ ∪{ α } is isomorphic to the pr oje ctive bund le asso ciate d to a ve ct or bund le of r ank 2 over X Θ ∪{ α } . Pr o of. Recall that G is simply connected. Let P = P Θ and Q = P Θ ∪{ α } . Let L b e the Levi subgroup of Q cont aining the maximal torus T . Its der ived subgr oup [ L, L ] is semi-simple, simply connected and has Dynkin type given by the full subgra ph of ∆ w ho se vertices are in Θ ∪ { α } [ Dem64 , Prop. 6.2.7]. Thus, [ L, L ] ≃ SL 2 × H with H a semi-simple simply connected s ubgroup of P . The r adical, Rad( L ), is a TRIVIAL WITT GR OUPS OF FLAG V ARIETIES 3 central tor us in L and the multiplication map [ L, L ] × Rad( L ) → L is surjective [ Dem64 , 6.2.3]. Its k ernel ν is is omorphic to [ L, L ] ∩ Rad( L ), and is therefore a finite subgroup o f the center µ 2 × µ of [ L, L ], where µ is a finite group of multiplicative t yp e. There are tw o cases: (1) the kernel ν is included in µ or (2) it surjects to µ 2 through the pro jection map µ 2 × µ → µ 2 . In the first cas e , the pro jection SL 2 × H × Ra d( L ) → SL 2 factors as a (surjective) ma p L → SL 2 . In the second ca se, let us show that Rad( L ) ha s a surjection to G m such tha t the following dia gram on the left commutes. ν   / /     Rad( L )     µ 2   / / G m N Z r o o o o Z / 2 ?  O O Z o o o o O O    T aking Cartier duals, it amounts to chec king the existence of a dashed map s uch that the right dia gram commutes: ta ke the image of 1 ∈ Z / 2 in N , lift it to Z r and send 1 ∈ Z to this lift. This implies that the induced surjection SL 2 × H × Ra d( L ) → SL 2 × G m factors as a surjectio n L → (SL 2 × G m ) /µ 2 = GL 2 . So, in cases (1) and (2), taking the standa rd representation V o f SL 2 (resp. GL 2 ), and using that L is the quotient o f Q b y its unipotent radical, w e obtain a surjectiv e map Q → SL( V ) (resp. GL( V )); let K be its k ernel. Both Rad( L ) ⊂ T ⊂ P and H ⊂ P , so K ⊂ P . F urthermore, P / K is a Borel subgroup of Q/K since Q/ P = L/ ( P ∩ L ) ∼ = P 1 . Let us consider the vector bundle G × Q V → G/Q . Its asso ciated pro jectiv e spac e is G × Q ( Q/K ) / ( P /K ) = G × Q Q/P = G/P , as exp ected.  References [BC07] P . Balm er and B. Calm ` es. Witt gr oups of Grassmann v arieties. arXiv:0807.3296 , 2007. [Dem64] M. Demaz ure. Sch ´ emas en gr oup es. III: St ructur e des sch´ emas e n gr oup es r´ eductifs, Exp. XXII . S ´ eminaire de G ´ eom´ etrie Alg´ ebrique du Bois Marie 1962/ 64 (SGA 3). Dirig´ e par M. Demazure et A. Grothendiec k. Lecture Notes in Mathematics, V ol. 153. Springer-V erl ag, Berlin, 1962/1 964. [MT95] A . S. Merkurj ev and J.-P . Tignol. The multipliers of simili tudes and the Brauer group of homogene ous v arieties. J. R eine A ngew. Math. , 461:13–47, 1995. [W al03] C. W alter. Grothend iec k-Witt groups of pro j ectiv e bundles. preprint, http://w ww.math.u iuc.edu/K- theory/0644/ , 2003. Baptiste Calm ` es, Universit ´ e d’Ar tois, Labora toire de Ma th ´ ema tiques de Lens, France URL : http: //www.mat h.uni-bi elefeld.de/~bcalmes Jean F asel, Ludwig Maximilian Universit ¨ at, M unich, Germany