aa r X i v : . [ m a t h . AG ] O c t A Kazhdan-Lusztig Atlas on G { P Daoji Huang
Abstract
A stratified variety has a Kazhdan-Lusztig atlas if it can be locally modelledwith Kazhdan-Lusztig varieties stratified by Schubert varieties in some Kac-Moodyflag manifold via stratified isomorphisms. In this paper, we show that the partialflag manifold G { P with the projected Richardson stratification has a Kazhdan-Lusztig atlas, with each chart stratified-isomorphic to a Kazhdan-Lusztig varietyin the affine flag manifold of the formal loop group p G of G . This result generalizesthat of Snider’s on Gr p k, n q with the positroid stratification, and is a geometriccounterpart of the combinatorial correspondence between the poset of projectedRichardson stratification and a certain convex set in the Bruhat order of the Weylgroup of p G given by He and Lam. G { P Let G be a complex, connected, and simply connected reductive group over C . Let B and B ´ be the upper and lower Borel subgroups containing the maximal torus T , and U (resp. U ´ ) denote the unipotent radical of B (resp. B ´ ). Let X ˚ p T q denote the coweightlattice of T . Let W “ N p T q{ T denote the Weyl group of G , and w P W its longestelement. Let S denote the set of simple roots determined by p T, B q . Let Φ denote the setof all roots of G , and Φ ` (resp. Φ ´ ) the set of positive (resp. negative) roots. For any α P Φ, let s α denote the corresponding reflection. When α P S , s α is a simple reflection.The flag variety G { B has a stratification by Schubert cells G { B “ Ů w P W X ˝ w where X ˝ w “ B ´ wB { B and by opposite Schubert cells G { B “ Ů w P W X w ˝ where X w ˝ “ BwB { B .Let X w “ B ´ wB { B denote the Schubert variety and X w “ BwB { B denote the oppositeSchubert variety. The intersections of the Schubert and opposite Schubert cells are calledopen Richardson varieties, denoted by ˚ X wv “ X w ˝ X X ˝ v . These also form a stratificationof G { B , and this stratification is anticanonical. The closure of ˚ X wv is the Richardsonvariety X wv “ X w X X v .We fix a subset J of the simple roots S . Let P denote the parabolic subgroupcorresponding to J , and P ´ the opposite parabolic subgroup of P . Let W P denote thesubgroup of W generated by t s α : α P J u , and W P the set of minimal coset representativesof W { W P . Let w ,P denote the longest element in W P . Let U P (resp. U P ´ ) denote theunipotent radical of P (resp. P ´ ). P admits a Levi decomposition P “ L ˙ U P where L Ą T . 1et π denote the projection G { B Ñ G { P . Denote by ˚Π wv the open projected Richard-son variety π p X w ˝ X X ˝ v q and Π wv the projected Richardson variety π p X w X X v q . G { P admits a projected Richardson stratification G { P “ ğ p w,v qP Q P ˚Π wv , where Q P “ tp w, v q : w P W P , v P W, v ď w u . This is an anticanonical stratificationof G { P . Define an ordering ĺ on W P ˆ W p w , v q ĺ p w, v q if and only if there exists u P W P such that w u ď w and v u ě v . Then p Q P , ĺ q is a subposet of p W P ˆ W, ĺ q , andfor any p w , v q , p w, v q P Q P , p w , v q ĺ p w, v q if and only if Π w v Ď Π wv . For more detailson projected Richardsons, see [KLS14]. Let K denote the local field of Laurent series C pp t ´ qq , and O “ C rr t ´ ss its ring of formalpower series. Also let O ´ be the polynomial ring C r t s . The formal loop group G p K q (alsowritten as p G ) has an extended torus T ˆ C ˚ where C ˚ acts via loop rotation by scaling t . A character on T can be identified with a character on T ˆ C ˚ that is trivial on C ˚ .Furthermore, let δ denote the character on T ˆ C ˚ that is trivial on T and identity on C ˚ . The affine root system of G p K q is given byΦ aff “ t α ` mδ : α P Φ \ t u , m P Z uzt u . and the simple roots S aff “ S \ t α ´ δ u where α P Φ is the lowest root. G p O q is amaximal parabolic subgroup of G p K q . Its unipotent subgroup U G p O q corresponds to theset of roots α ` mδ where m ă
0. Similarly, U G p O ´ q corresponds to the set of roots α ` mδ where m ą x W “ W ˙ X ˚ p T q acts on Φ aff , where W acts on δ trivially and α P Φ the usual way. Also for any t µ P X ˚ p T q , t µ ¨ p α ` mδ q “ α ` p m ` x µ, α yq δ. Let I “ t f P G p O q : f p8q P B Ă G u be the standard Iwahori subgroup and I ´ “ t f P G p O ´ q : f p q P B ´ Ă G u the opposite Iwahori subgroup of I . The affine flag varietyis G p K q{ I . The Schubert and opposite Schubert varieties are defined as X w “ I ´ w I { I and X w “ I w I { I where w P x W . Let p H, B ˘ H , T H , W H q be a pinning of a Kac-Moody Lie group. A Kazhdan-Lusztigvariety in the Kac-Moody flag variety H { B H is the intersection of an opposite Schubertcell X w ˝ “ B H wB H { B H and a Schubert variety X v “ B H vB H { B H , where w, v P W H .Let M be a smooth variety bearing a stratification Y , whose minimal strata Y min arepoints. To be precise, M “ Ů y P Y M ˝ y where Y is a ranked poset, each M ˝ y is a locallyclosed subvariety with closure M y , and M y “ Ů y ď y M ˝ y . We say that p M, Y q has a Kazhdan-Lusztig atlas with the modelling Kac-Moody H { B H if2a) there is a ranked poset injection v : Y opp Ñ W H whose image is Ť f P Y min r v p M q , v p f qs ,(b) M can be covered by charts U f centered around the minimal strata f P Y min suchthat for each f , there is a stratification-preserving isomorphism c f : U f Ñ X v p f q˝ X X v p M q Ă H { B H . Our goal is to show that G { P equipped with the projected Richardson stratification hasa Kazhdan-Lusztig atlas with the modelling flag variety G p K q{ I . The combinatorial partof the result concerning the poset structures has been established by He and Lam [HL15].Our result is a generalization of Snider’s result on the Grassmannian variety Gr p k, n q of k -planes in n -space with the positroid stratification [Sni10]. In that case, Gr p k, n q admitsa Bruhat atlas [HKL], where each chart is modelled by an opposite Schubert cell (i.e., v p M q “
1) of the corresponding type A affine flag variety.
Theorem 1. G { P with the projected Richardson stratification has a Kazhdan-Lusztigatlas w U P ´ P { P – X w t λ w ,P w w ˝ X X t λ w ,P w where w P W P , w w “ w w ,P , and λ is a dominant coweight such that for any α P S , x α, λ y “ iff α P J .Under the isomorphism, the strata correspond as follows w U P ´ P { P X Π wv – X w t λ w ,P w w ˝ X X vt λ w ´ p for all v P W, w P W P , v ď w q Remark.
This result has also been established independently recently by Galashin, Lam,and Karp [GKL19] through a slightly different map.
To describe the chart isomorphisms, we use Bott-Samelson maps. For a detailed de-scription of Bott-Samelson variety in full Kac-Moody generality, see [Kum12]. The casesrelevant to us are finite or affine type. For any w P W (resp. x W ) and Q “ p s α , ¨ ¨ ¨ , s α l p w q q a reduced word for w , we have a Bott-Samelson map m Q : A l p w q Ñ G (resp. p G ) suchthat m Q p z , ¨ ¨ ¨ , z l p w q q “ ś l p w q i “ exp p z i R α i q Ă s α i where R α i is a root vector that spans theroot space indexed by α i . First we establish some lemmas. (The dot notation “ ¨ ” in thispaper denotes the conjugation action on groups and the induced action Lie algebras. Ittakes higher precedence than binary set operations.) Lemma 1.
For any w P W P , w ¨ Φ J Ă Φ ` .Proof. We have l p w q “ | w ¨ p Φ ´ z Φ ´ J \ Φ ´ J q X Φ ` | “ | w ¨ p Φ ´ z Φ ´ J q X Φ ` q| ` | w ¨ Φ ´ J X Φ ` | and w ,P ¨ Φ ´ “ p Φ ´ z Φ ´ J q \ Φ J . Then l p ww ,P q “ | ww ,P ¨ Φ ´ X Φ ` | “ |p w ¨ p Φ ´ z Φ ´ J q X Φ ` q\p w ¨ Φ J X Φ ` q| “ | w ¨p Φ ´ z Φ ´ J qX Φ ` |`| w ¨ Φ J X Φ ` | . Since l p ww ,P q “ l p w q` l p w ,P q ,we must have | w ¨ Φ ´ J X Φ ` | ` | Φ J | “ | w ¨ Φ J X Φ ` | . Therefore, | w ¨ Φ ´ J X Φ ` | “ | w ¨ Φ J X Φ ` | “ | Φ J | , and the claim follows.3 emma 2. Let Q be a reduced word for w where w P W P , then Im p m Q q w ´ “ w ¨ U P ´ X U. Proof.
Since U P ´ “ w ,P ¨ U ´ X U ´ , we have w ¨ U P ´ X U “ ww ,P ¨ U ´ X w ¨ U ´ X U . Let J bethe set of simple roots that generate P , Φ J be the set of positive roots generated by J , andΦ ´ J be the set of negative roots generated by ´ J . Since w P W P , we have w ¨ Φ ´ J Ă Φ ´ by Lemma 1. Therefore, w ¨ Φ ´ X Φ ` “ w ¨ p Φ ´ J \ p Φ ´ z Φ ´ J qq X Φ ` “ w ¨ p Φ ´ z Φ ´ J q X Φ ` .Moreover, w ,P ¨ Φ ´ “ Φ J \ p Φ ´ z Φ ´ J q , so ww ,P ¨ Φ ´ X Φ ` “ w ¨ p Φ J \ p Φ ´ z Φ ´ J qq X Φ ` Ą w ¨ Φ ´ X Φ ` , which means w ¨ U ´ X U Ă ww ,P ¨ U ´ X U . Therefore w ¨ U P ´ X U “ w ¨ U ´ X U ,which has dimension l p w q . The dimensions of both sides match up, so it suffices to showthat for any z , m Q p z q w ´ P w ¨ U ´ X U .We proceed by induction on the length of Q . Let q ρ be the sum of fundamentalcoweights and we use the fact that for any x P G , lim t Ñ8 q ρ p t q ¨ x “ x P U ´ . When w “ s α , we need exp p zR α q P s α ¨ U ´ which is given by lim t Ñ8 p s α ¨ q ρ p t qq ¨ exp p zR α q “ s α exp p zR α q s α P U ´ α . Clearly exp p zR α q P U , so the statement is true when w is asimple reflection.Now suppose s α w ą w . Notice that s α preserves the set Φ ´ zt´ α u . It follows that s α ¨ p w ¨ Φ ´ X Φ ´ zt´ α uq Ă Φ ´ . In other words, all negative roots in w ¨ Φ ´ that are not ´ α remain negative under the action of s α . Since s α w ą w , s α w ¨ Φ ´ contains more positiveroots than w ¨ Φ ´ , it must be the case that α P p s α w q ¨ Φ ´ , and thus U α Ă s α w ¨ U ´ .Assume by induction m Q p z q w ´ P w ¨ U ´ X U , we show that exp p zR α q s α m Q p z q w ´ s α Pp s α w q ¨ U ´ X U . Nowlim t Ñ8 p s α w ¨ q ρ p t qq ¨ p exp p zR α q s α m Q p z q w ´ s α q“ lim t Ñ8 p s α w ¨ q ρ p t qq ¨ p exp p zR α qq lim t Ñ8 p s α w ¨ q ρ p t qq ¨ p s α m Q p z q w ´ s α q .The first factor is 1 by U α Ă s α w ¨ U ´ and the second factor is 1 by lim t Ñ8 p w ¨ q ρ p t qq ¨ m Q p z q w ´ “ p zR α q s α m Q p z q w ´ s α Pp s α w q ¨ U ´ .We are left to show exp p zR α q s α m Q p z q w ´ s α P U . Since m Q p z q w ´ P w ¨ U ´ X U , s α m Q p z q w ´ s α P s α w ¨ U ´ X s α ¨ U . Since α P s α w ¨ Φ ´ but s α ¨ Φ ` “ Φ ` zt α u Y t´ α u , s α w ¨ Φ ´ X s α ¨ Φ ` Ă Φ ` zt α u . It follows that s α m Q p z q w ´ s α P U and exp p zR α q s α m Q p z q w ´ s α P U . Let w , w P W P such that w w “ w w ,P and l p w q ` l p w q “ l p w w ,P q . Choose Q a reduced word for w and Q a reduced word for w . Let m , m denote the Bott-Samelson maps m Q , m Q , respectively. Lemma 3. w ´ Im p m Q q “ w ¨ U P ´ X U ´ .Proof. The dimensions of both sides are l p w q “ l p w q ´ l p w ,P q ´ l p w q , so it suffices toshow that for any z , w ´ m Q p z q P w ¨ U P ´ X U ´ , or m Q p z q ´ w P w ¨ U P ´ X U ´ . Thereexists a reduced word Q of w w ´ w such that w m Q p z q w “ m Q p z q ´ . Our statementreduces to w m Q p z q w w P w ¨ U P ´ X U ´ . Conjugating by w , we get m Q p z q w w w P w w ¨ U P ´ X U . Notice that w w ´ w “ w w w ,P P W P , since l p w w ´ w q “ l p w q “ l p w q ´ l p w ,P q ´ l p w q and l p w w q “ l p w q ´ l p w q . The claim then follows from theprevious lemma, since w w ¨ U P ´ “ w w w ,P ¨ U P ´ . emma 4. t λ ¨ U P Ă U G p O ´ q , and t λ ¨ U P ´ Ă U G p O q .Proof. The roots that correspond to U P are Φ ` z Φ J . For any β in this set, x λ, β y ą t λ ¨ β “ β ` x λ, β y δ , which is in the roots of U G p O ´ q . The proof for the second partis similar.We will now give parametrizations of both sides of the isomophisms separately. Let p z , ¨ ¨ ¨ , z | Q | , z | Q |` , z | Q |`| Q | q P A l p w w ,P q be parameters. For ease of notation, let z “ p z , ¨ ¨ ¨ , z | Q | q , z “ p z | Q |` , ¨¨¨ , | Q |`| Q | q , and z “ p z , ¨ ¨ ¨ , z | Q | , z | Q |` , z | Q |`| Q | q .To parametrize X w t λ w ´ ˝ X X t λ w ,P w “ X w t λ w ,P w w ˝ X X t λ w ,P w , use the map ψ w : z ÞÑ m p z q t λ w ,P w m p z q I { I . We first see the image of this map lies inside X w t λ w ,P w w ˝ . Since t λ w ,P w is the smallestelement in the set W t λ W [HL15], l p w t λ w ,P w w q “ l p w q ` l p t λ w ,P w q ` l p w q . Choosea reduced word Q for t λ w ,P w , then the Bott-Samelson map m Q Q Q ˝ π where π isthe projection G p K q Ñ G p K q{ I parametrizes X w t λ w ,P w w ˝ . Setting the coordinates thatcorrespond to Q to 0 we get ψ w . The fact that that the image of ψ w lies in X t λ w ,P w will follow from the proof of Proposition 1 below.Now we describe the parametrization of w U P ´ P { P Ă G { P . First we consider theparametrization of w ¨ U P ´ p : A l p w ,P w q Ñ w ¨ U P ´ ,p p z q “ m p z q w ,P w m p z q . The fact that this is indeed a parametrization follows from Lemma 2 and 3. Since p p z q P w U P ´ w ´ , there exists a unique u ´ p z q P U ´ X w U P ´ w ´ such that u ´ p z q p p z q P U X w U P ´ w ´ by Lemma 2.2 in [KWY13]. The parametrization of wU P ´ P { P Ă G { P isgiven by φ w : z ÞÑ u ´ p z q m p z q P { P. Proposition 1.
The map ψ w ˝ φ ´ w is a stratified isomorphism on the w -charts.Proof. Assume m p z q “ u ´ p z q m p z q P { P P ˚Π wv where p w, v q P W J ˆ W , v ď w . Thismeans there exists p P P such that u ´ p z q m p z q p P BwB X B ´ vB . We may take p P L X U ´ . Let u ` p z q “ u ´ p z q p p z q , then u ´ p z q m p z q “ u ` p z q m p z q ´ w w ,P . Since u ´ p z q P U ´ and u ` p z q P U , we have m p z q p P B ´ vB and p ´ w ,P w m p z q P Bw ´ B .We want to show that m p z q t λ w ,P w m p z q I { I P X vt λ w ´ . Since p P L X U ´ , p com-mutes with t λ , so m p z q t λ w ,P w m p z q “ m p z q pt λ p ´ w ,P w m p z q P B ´ vBt λ Bw ´ B .We now argue that B ´ vBt λ Bw ´ B Ă I ´ vt λ w ´ I . It suffices to show that for all b , b P B , vb t λ b w ´ P I ´ vt λ w ´ I .Write b “ l u , where l P L X B and u P U P . By Lemma 4, u t λ “ t λ u , where u P U G p O q . Since b w ´ P G normalizes U G p O q , there exists u P U G p O q Ă I such that u b w ´ “ b w ´ u .So far we have shown that vb t λ b w ´ “ vl t λ b w ´ u . Since l P L X B , l commuteswith t λ . Therefore, vl t λ b w ´ u “ vt λ b w ´ u for some b P B . Now write b “ u l where u P U P and l P L X B . Again by Lemma 4, we have t λ u “ u t λ where u P G p O ´ q . Since v P G normalizes U G p O ´ q , we have vu “ u v for some u P U G p O ´ q Ă I ´ . Furthermore, by Lemma 1, since w P W J , w p L X B q w ´ Ă B . Therefore, there exists some b P B such that l w ´ “ w ´ b . Summing up, we have vt λ u l w ´ u “ u vt λ w ´ bu P I ´ vt λ w ´ I as desired. Let G “ SL p C q , S “ t α , α , α u , J “ t α , α u , and λ “ p , , , ´ q . Let s , s , s bethe simple reflections associated to the simple roots α , α , α . In this case w w ,P “ p zR α q “ „ z ´ , exp p zR α q “ „ z ´ , exp p zR α q “ „ z ´ . We show an example of computations on the chart w U P ´ P { P where w “ “ s s s .Then w “ w w ,P w ´ “ “ s s . We demonstrate that the chart isomorphism isstratification preserving by showing explicitly that the equations of the projected Richard-son divisors that meet w U P ´ P { P match the corresponding equations of the Schubert di-visors on the Kazhdan-Lusztig variety X w t λ w ´ ˝ X X t λ w w ,P Ă y SL p C q{ I . This is enough,because the stratifications on both sides are uniquely determined by divisors.The chart w U P ´ { P is parametrized as follows φ w : p z , z , z , z , z q ÞÑ »——– z z z ` z z ` z z z z ´ fiffiffifl P { P. The codimension one projected Richardson divisors on this G { P are Π , Π , Π ,Π , and Π . Among these, the first four have nonempty intersections with our chartunder consideration. Their equations on this chart are z , z z ` z z , z , z , respectively.Meanwhile, the Kazhdan-Lusztig variety X w t λ w ´ ˝ X X t λ w w ,P is parametrized as fol-lows: ψ w : p z , z , z , z , z q ÞÑ »——– z z z ´ fiffiffifl »——– t ´ ´ { t fiffiffifl »——– z z ´ ´ fiffiffifl I { I “ »——– tz tz ´ z t ´ ´ z t ´ ´ t ´ tz ´ t fiffiffifl I { I Following the rules given in [Hua], we compute that the equations for X w t λ w ´ ˝ X X s t λ w w ,P , X w t λ w ´ ˝ X X s t λ w w ,P , X w t λ w ´ ˝ X X s t λ w w ,P , and X w t λ w ´ ˝ X X t λ w w ,P s arealso z , z z ` z z , z , z , respectively. 6 eferences [GKL19] Pavel Galashin, Steven N. Karp, and Thomas Lam. Regularity theorem fortotally nonnegative flag varieties. https://arxiv.org/abs/1904.00527, 2019.[HKL] Xuhua He, Allen Knutson, and Jiang-Hua Lu. Bruhat atlases. preprint.[HL15] Xuhua He and Thomas Lam. Projected richardson varieties and affine schubertvarieties. Annales de linstitut Fourier , 65(6):23852412, 2015.[Hua] Daoji Huang. An analogue of fultons ideals for matrix schubert varieties inthe affine case. in preparation.[KLS14] Allen Knutson, Thomas Lam, and David E Speyer. Projections of richardsonvarieties.
Journal fr die reine und angewandte Mathematik (Crelles Journal) ,2014(687), Jan 2014.[Kum12] Shrawan Kumar.
Kac-Moody groups, their flag varieties and representationtheory , volume 204. Springer Science & Business Media, 2012.[KWY13] Allen Knutson, Alexander Woo, and Alexander Yong. Singularities of richard-son varieties.