An analogue of the Grothendieck-Springer resolution for symmetric spaces
aa r X i v : . [ m a t h . R T ] A p r A PARTIAL ANALOGUE OF THE GROTHENDIECK-SPRINGERRESOLUTION FOR SYMMETRIC SPACES
SPENCER LESLIE
Abstract.
Motivated by questions in the study of relative trace formulae, we construct a gen-eralization of Grothendiecks simultaneous resolution over the regular locus of certain symmetricpairs. We use this space to prove a relative version of results of Donagi-Gaitsgory about theautomorphism sheaf of regular stabilizers. We also obtain partial results toward applicationsin Springer theory for symmetric spaces.
Let G be a connected reductive group over an algebraically closed field k , and let g denoteits Lie algebra. We assume throughout that the characteristic of k is sufficiently large withrespect to G . An important construction in the representation theory of g is the simultaneousresolution of singularities of Grothendieck e g = { ( X, B ) ∈ g × F l G : X ∈ Lie( B ) } , where F l G is the flag variety of Borel subgroups of G . This space plays a central role in Springertheory, where one needs both the property that it simultaneously resolves the singularities ofthe quotient map with respect to the adjoint action χ : g → g //G , and the existence of theCartesian diagram e g reg tg reg t /W, π e χχ (1)where W is the Weyl group acting on a Cartan subalgebra t , π : e g → g is the projection, andwe have made use of the Chevalley isomorphism g //G ∼ = t /W . This diagram may be used toinduce Springer’s W -action on the cohomology of Springer fibers.The variety e g also arises in the theory of G -Higgs bundles as studied by Donagi and Gaitsgory.In [DG02], the authors identify abstract Hitchin fibers as a gerbe over a certain abelian groupscheme which acts on the Hitchin fibration. In their analysis, the restriction of the Grothendieck-Springer resolution to the regular locus of g is used to compare the moduli space of regularcentralizers with the moduli space of regular orbits of g . In his study of the Langlands-Shelstadfundamental lemma, Ngˆo [Ngo06] utilized this connection in an important way. One of the goalsof this present article is to establish an analogous statement in the case of a symmetric space(see Theorem 4.3).More precisely, assume now that G admits an involutive automorphism θ : G → G , and let G be the fixed-point subgroup of θ . The pair ( G, G ) is called a symmetric pair. Passing tothe Lie algebra g = Lie( G ), the differential of θ (which we also denote by θ : g → g ) producesthe decomposition g = g ⊕ g , where g i is the ( − i -eigenspace of θ . Then G acts on the infinitesimal symmetric space g byrestriction of the adjoint action. Studying the G -orbits on g gives a natural generalization ofthe adjoint representation. In fact, the adjoint representation may be recovered by consideringthe involution of g ⊕ g given by swapping the two factors. Date : April 22, 2019.2010
Mathematics Subject Classification.
Primary 20G05 ; Secondary 17B08, 32S45.
Key words and phrases.
Symmetric pair, regular stabilizers, resolution of singularities, Springer theory, relativetrace formulae.
In this paper, we construct a generalization of Grothendieck’s resolution for the quotient of g by the action of G over the regular locus of g under the assumption that θ is quasi-split .This is equivalent to the existence of a Borel subgroup B ⊂ G such that B ∩ θ ( B ) is a torus.In this setting, we define a sub-scheme e g ⊂ g × g e g and, setting e g reg = e g × g g reg , we provethat the induced map π : e g reg → g reg behaves like an analogue of Grothendieck’s resolution: Theorem 0.1.
Let ( g , g ) be a quasi-split symmetric pair with g = g ⊕ g . There is a closedsubscheme e g ⊂ g × g e g with a proper morphism π : e g → g that is an alteration ( [dJ96] ). Wehave a commutative diagram e g ag a /W a , e χ π χ where a is the universal Cartan of the symmetric pair, χ : g → a /W a is the categorical quotientmap, and e χ is the restriction of e χ : e g → g to e g . Furthermore, the restriction e g reg = e g × g g reg is smooth, and the corresponding diagram is Cartesian. See Section 3 for more details. The family of quasi-split symmetric pairs includes the “diag-onal” symmetric space ( g ⊕ g , ∆ g ) as well as the stable (or split) involutions which featurein representation-theoretic approaches to arithmetic invariant theory (see [Tho13]). Remark . Our initial motivation for seeking such a result comes from considering the com-parison of relative trace formulae introduced in [GW14]. In many cases of interest, one needs togeneralize results of Ngˆo on the Langlands-Shelstad fundamental lemma [Ngo06] to the settingof symmetric spaces in order to stabilize these formulae. As noted above, the analogues of theresults of Donagi and Gaitsgory we prove here will play a role for such generalizations.From the perspective of relative trace formulae, the restriction to quasi-split involutionsis very natural. For example, the simple trace formula for symmetric spaces established in[Hah09] degenerates to the identity 0 = 0 if the symmetric space is not geometrically quasi-split. Additionally, Prasad recently showed that generic representations over non-archimedeanfields can be G -distinguished only for such involutions [Pra18].Despite the notation, e g is not a simultaneous resolution of singularities of the categoricalquotient g → g //G . Even for the diagonal symmetric space, the space e g is not isomorphicto the Grothendieck-Springer resolution e g , though their pullbacks to the regular locus areobviously isomorphic. In Section 5, we are identify a (Zariski-dense) interstitial space e g reg ⊂ e g res ⊂ e g which is a family of resolutions of the singularities of g → g //G . In particular, e g res is isomorphic to the Grothendieck-Springer resolution in the diagonal setting. We discuss thisobject in more detail toward the end of the introduction and in Section 5.The proof of Theorem 0.1 occupies Sections 3, using several results from Sections 1 and 2.A key idea is to show (see Proposition 1.12) that the universal Cartan subspace t of g may beequipped with a canonical involution θ can : t → t associated to the symmetric pair. This allowsus to identify the universal Cartan subspace a of the symmetric pair ( g , g ) as a distinguishedsubspace of t . The Grothendieck-Springer resolution is equipped with a smooth map e χ : e g → t ,and we define e g = { ( X, B ) ∈ g × F l G : e χ ( X, B ) ∈ a } , and show that this space has all the desired properties. This relies on a classification of theirreducible components of the fiber product g × t /W t , which in turn relies crucially the existenceof a Kostant-Weierstrass section to the categorical quotient map χ : g → g //G along with G -conjugacy results from [Lev07]. For clarity, we give a concrete description of e g reg as follows: e g reg := { ( X, B ) ∈ g reg × F l G : B ( θ ) = Z B ( X ss ) is a regular θ -stable Borel of Z G ( X ss ) } . (2)To be more precise, we associate to an element ( X, B ) ∈ g × g e g two subgroups of B . Thefirst is the largest θ -stable subgroup contained in B , given by B ( θ ) = B ∩ θ ( B ). For example, if B is θ -split, then B ( θ ) is a maximal torus. Second, if we denote by X ss the semi-simple part of ROTHENDIECK-SPRINGER FOR SYMMETRIC SPACES 3 X , then X ss ∈ g and the centralizer Z B ( X ss ) of X ss in B is a Borel subgroup of the centralizer Z G ( X ss ). Finally, we define a θ -stable Borel subgroup B = θ ( B ) to be regular if its Lie algebracontains a regular nilpotent element n ∈ Lie( B ) that lies in g . This notion arises naturallyfrom studying the action θ induces on the Springer resolution of the nilpotent cone (see Section2).In proving Theorem 0.1, we have two main applications in mind: the study of regular cen-tralizers in g for the action on g (Section 4) and potential applications to Springer theory forsymmetric spaces (Section 5).In Section 4, we introduce the moduli space of regular stabilizers of the action of G on g ,denoted G /N where N is the stabilizer in G of a Cartan subspace of g . As the notationindicates, this space is a partial compactification of the space G /N which parametrizes Cartansubspaces of g [Lev07]. We show that this is naturally a smooth scheme. This space may beequipped with a natural W a -cover G /T → G /N , where W a is the little Weyl group of thesymmetric space. This cover G /T is a partial compactification of the space G /T of pairs( a , b ), with a ⊂ g a Cartan subspace of g and a ⊂ b where b is a θ -split Borel subalgebra of g (Proposition 1.9). In Section 4.1, we prove that there is a Cartesian diagram e g reg G /T g reg G /N , and we show that the horizontal arrows in this diagram are smooth (see Proposition 4.2 andTheorem 4.3). A corollary of this is that the two W a -covers G /T → G /N and a → a /W a are´etale-locally isomorphic in the strong sense that they become isomorphic after a smooth basechange. This implies that one is ´etale-locally a pull-back of the other and vice versa, whencethe terminology. This is the analogue for quasi-split symmetric spaces of the results of [DG02,Section 10].In Section 4.2, we study the tautological sheaf of regular stabilizers C := { ( g, c ) ∈ G × G /N : g | c = Id c } on G /N . We prove that this group scheme is smooth and isomorphic to an abelian groupscheme built out of the fixed point subgroup of the canonical involution on the universal Cartan θ can : T → T . More precisely, let T = T θ can and consider the group scheme T ( S ) := (cid:16) W a -equivariant morphisms e S → T (cid:17) for any G /N -scheme S , where e S = S × G /N G /T . We show (see Theorem 4.6) that thereis a canonical isomorphism C ∼ −→ T . Such a model for the sheaf of regular stabilizers is crucialfor generalizing the approach of Ngˆo to studying fundamental lemmas in the context of relativetrace formulae. Remark . (1) While we assume for simplicity that G der is simply connected for much ofthe article, we address the necessary changes to obtain an isomorphism C ∼ −→ T in thegeneral case in Section 4.3.(2) This group scheme is intimately related to the automorphism group schemes used byKnop [Kno96] in his analysis of collective invariant motion of a G -variety X in char-acteristic zero. Recently, Sakellaridis [Sak18] utilized Knop’s group scheme in a crucialmanner to prove a “beyond endoscopic” transfer statement for rank one spherical vari-eties. It is interesting that it is the “complimentary subgroup” C is central to endoscopicphenomena in the symmetric case.Aside from motivations arising from relative trace formulae, we expect e g to have otherapplications in the representation theory of symmetric pairs. For example, Chen, Grinberg,Vilonen, and Xue (see [CVX15, GVX18, VX18]) have recently studied analogues of Springer SPENCER LESLIE theory for symmetric pairs. While their initial work sought to generalize an approach of Lusztigwhich relies on e g , their most general results rely on a near-by cycles construction in [GVX18].As noted above, the variety e g does not give a simultaneous resolution of singularities for thequotient g → g //G , so it is natural to ask if there is an interstitial space e g reg ⊂ e g res ⊂ e g which generalizes the Grothendieck-Springer resolution in this sense. Toward this question, weconsider in Section 5 such a subspace e g res ⊂ e g , which recovers the classical Grothendieck-Springer resolution in the case of the case of the diagonal symmetric space ( g ⊕ g , ∆ g ) (seeProposition 5.1). Our proposal for e g res is quite natural: we simply extend the construction of e g reg from (5) to all of g .We show in Theorem 5.2 that this definition does indeed form a family of resolutions of thesingularities of the quotient map g → g //G , and give a sufficient criterion in Lemma 5.5 forthis space to be smooth. Thus, there is a precise way in which one may systematically delete G -orbits from e g to obtain a family of resolutions. As we note below, this family can fail to besmooth, or even irreducible, in general.Our argument is similar to the analysis of e g in [Slo80, Chapter 3]. In particular, we needa good understanding of the resolution of singularities of irreducible components of nilpotentcones of symmetric spaces. We review the construction and relevant properties of the resolutiongiven by Sekiguchi and Reeder [Sek84, Ree95] in Section 2, where we introduce the notion of aregular θ -stable Borel subgroup and identify the subset of the fixed-point locus of the Springerresolution which arises in e χ − (0).However, there are very basic cases when the morphism χ : g → g //G does not admita simultaneous resolution. In such cases, our space e g res cannot be smooth and may not evengive rise to an irreducible scheme. We describe a family of such examples using a monodromyargument in Section 5, but for a simple example, consider the case of a quasi-split symmetricpair ( sl (2) , so (2)). Then g ∼ = A k , g //G ∼ = A k , and these isomorphisms may be chosen so that χ corresponds to the map A k → A k ( x, y ) xy. In this case, only the fiber over 0 ∈ A is singular, given by two affine lines meeting transverselyat one point. However, g × a /W a a is a cone, so that there is no way to resolve the singularityof g → g //G at 0 while remaining birational to g × a /W a a . In this case, e g is the blow-upat the cone point and e g res = e g \ G m where G m = SO(2) denotes the open SO (2)-orbit of theexceptional fiber. The two remaining points of the exceptional fiber parametrize the two regular θ -stable Borel subgroups of SL(2), or equivalently the two components of the nilpotent cone of g .This example illustrates both that e g is as close to the Grothendieck-Springer resolution forsymmetric spaces as is possible in general and how one may obtain the resolution of singularitiesof fibers of g → g //G by systematically deleting G -orbits. In this sense, e g is the appro-priate object to study in the case of symmetric spaces and we expect it to have applications torepresentation theory of the symmetric pair ( g , g ) beyond those studied in the present article.In particular, we hope to study the connections between these spaces with the Springer theorydeveloped in [CVX15, GVX18, VX18] in future work.Let us now summarize the paper. We review notation and certain basic properties of sym-metric pairs in Section 1. We then focus on quasi-split involutions, culminating in Proposition1.12. In Section 2, we review the theory of the nilpotent cone N ⊂ g , studying the resolutionsof the components of N . This will be used in the proof of Theorem 5.2. We also introducethe notion of a regular θ -stable Borel subgroup in this section. Section 3 introduces e g , andproves Theorem 0.1. In Section 4, we turn to the primary application of studying the space ofregular stabilizers G /N and the sheaf of regular stabilizers on this space. Finally, with an eyetoward applications in Springer theory, we end by introducing the space e g res ⊂ e g which is a ROTHENDIECK-SPRINGER FOR SYMMETRIC SPACES 5 (potentially non-smooth) family of resolutions of singularities of the quotient map. We give acriterion for when this space is smooth.0.1.
Notation.
Algebraic groups will be denoted in Roman font, while Lie algebras will be infraktur font.For any G -variety V on which an endomorphism θ acts, we denote by V θ the fixed pointsubvariety of V . For any subspace U ⊂ g , we denote its centralizer in a subgroup H ⊂ G by Z H ( U ). In particular, for X ∈ g we have Z G ( X ) = Z G ( X ) θ . We set Z ( G ) to be the center of G . Similarly, we denote the centralizer of U in the Lie algebra h = Lie( H ) by z h ( U ). For any group H on which θ acts, we denote τ ( g ) = g − θ ( g ).For any group H , we use H ◦ to denote the connected component of the identity.0.2. Acknowledgements.
I want to thank Jayce Getz for introducing me to questions whichled directly to this project, as well as for many helpful conversations. I also thank Ngˆo BaoChau, Aaron Pollack, and David Treumann for helpful discussions. Finally, I want to thankJack Thorne for comments that led to the discovery of an error in an earlier version of thisarticle.
Contents
1. Preliminaries 52. Nilpotent cones of symmetric spaces 103. A simultaneous resolution over the regular locus 114. Moduli space of regular stabilizers 145. Smoothness and resolution of singularities 21References 251.
Preliminaries
Let k , G , g , and θ be as above. We assume that char( k ) = 2 is either 0 or greater than 2 κ ,where κ is the supremum of the Coxeter numbers of the simple components of G . Remark . Much of this article works for char( k ) = 2 good for G , which is a much weakerassumption. The only aspect relying on the restriction to char( k ) > κ is the theory of theresolutions of singularities of the nilpotent cone from [Ree95]. We expect that appropriateapplication of the techniques used in [Lev07] should allow for Reeder’s results to be extendedto good characteristic.For simplicity, we assume that the derived subgroup G (1) of G is simply connected, except inSection 4.3. This is not a serious restriction since for any isogenous group G ′ with involution θ ′ there exists a unique involution θ sc of G such that, if p : G → G ′ is the surjective isogeny, thediagram G GG ′ G ′ θ sc p pθ ′ commutes; see [Ste68, 9.16] and [Lev07, Lemma 1.3]. In particular, θ ′ and θ sc induce the sameinvolution on g . We abuse notation and also denote by θ : g → g the associated linear involutionof g .There is a direct-sum decomposition g = g ⊕ g , where g i is the ( − i -eigenspace of θ in g .Let G = { g ∈ G : θ ( g ) = g } be the fixed point subgroup of θ in G . The assumption that G der is simply connected means that the connected components of G is controlled by its image inthe abelianization map ν : G/G der → G ab ∼ = G km . The restriction of the adjoint action to G normalizes g , and g = Lie( G ). We will often use i ∈ { , } as a subscript to indicate objectsassociated to the corresponding ( − i -eigenspace; for example, we denote by N the cone ofnilpotent elements in g (see Section 2).1.1. Basics of symmetric pairs.
Let ( g , g ) be a symmetric pair with associated involution θ . We record here some structural facts about ( g , g ) and point the reader to [Lev07] for moredetail. We begin by noting that the Jordan decomposition behaves well with respected to thedecomposition of g = g ⊕ g . Lemma 1.2.
For X ∈ g and for i = 0 , , X ∈ g i if and only if X ss , X nil ∈ g i where X = X ss + X nil is the Jordan decomposition of X ∈ g . In particular, there is a well-defined notion of the semi-simple locus g ss of g , namely g ∩ g ss .A toral subalgebra a ⊂ g is a Cartan subspace of g if it is maximal in the collection of toralsubalgebras of g . Such a subalgebra lies in the semi-simple locus of g . Define the rank ofthe symmetric space r = rank( g ) to be dim( a ) for a Cartan subspace a (see [Lev07, Theorem2.11]). A torus A in G is θ -split if θ ( a ) = a − for all a ∈ A . A maximal such torus is called amaximal θ -split torus. Any two maximal θ -split tori of G are conjugate by an element of G [Lev07, Section 2].We say an element X ∈ g is regular if its centralizer Z G ( X ) ⊂ G has the smallest possibledimension, and denote g reg as the set of regular elements. We refer to [KR71] for properties ofregular elements. An element is regular semi-simple if it is both regular and semi-simple, andset g rss = g reg ∩ g ss to be the regular semi-simple locus.1.2. Quasi-split symmetric pairs.
Define a parabolic subgroup P ⊂ G to be θ -split if P ∩ θ ( P ) is a Levi subgroup of P . Fix a maximal θ -split torus A . Proposition 1.3. [Vus74, Section 1]
Let P ⊃ A be a θ -split parabolic subgroup. Then P isminimal among θ -split parabolic subgroup if and only if P ∩ θ ( P ) = Z G ( A ) . Any two minimal θ -split parabolic subgroups of G are conjugate by an element of G . Definition 1.4.
A symmetric pair ( g , g ) with associated involution θ is called quasi-split ifthere exists a Borel subgroup B that is θ -split. This is equivalent to B ∩ θ ( B ) being a torus.The pair ( g , g ) (resp., θ ) is split if it is quasi-split and the torus B ∩ θ ( B ) is θ -split.We will be exclusively interested in quasi-split symmetric pairs in the sequel. The followingcharacterizations is well known. Proposition 1.5.
A symmetric pair ( g , g ) is quasi-split if and only if the following equivalentstatements hold:(1) There exists a θ -split Borel subgroup of G .(2) The centralizer of a maximal θ -split torus is abelian.(3) There exists a regular element of g contained in g ; that is, g ∩ g reg = ∅ . Furthermore, a quasi-split θ is split if and only if g contains a Cartan subalgebra of g . We assume now and for the remainder of the paper that ( g , g ) is quasi-split. Let A ⊂ G bea maximal θ -split torus. By Proposition 1.5, T := Z G ( A ) is a maximal torus.1.3. The little Weyl group and θ -split Borel subgroups. Associated to the tori A ⊂ T ,we have the absolute Weyl group W T and the little Weyl group W A = N G ( A ) /Z G ( A ). For ageneral symmetric pair, the little Weyl group W A is not naturally a subgroup of W T , but asubquotient. When the symmetric pair is quasi-split, W A may be identified with the fixed-pointsubgroup ( W T ) θ : Lemma 1.6.
When θ is quasi-split, there is a natural embedding W A ֒ → W T , where S = Z G ( A ) is the θ -stable maximal torus containing A , where under this inclusion W A =( W S ) θ . ROTHENDIECK-SPRINGER FOR SYMMETRIC SPACES 7
Proof.
By definition W A = N G ( A ) /Z G ( A ), and in this case Z G ( A ) = Z G ( T ) = T . This impliesthat N G ( A ) ⊂ N G ( T ), giving the first claim.Let w ∈ ( W T ) θ and suppose n w represents w . Then θ ( n w ) = n w s for some t ∈ T . We needto show that n w ∈ N G ( A ). Indeed, for any a ∈ A , n w an − w ∈ T and θ ( n w an − w ) = θ ( n w ) θ ( a ) θ ( n w ) − = n w ( ta − t − ) n − w = n w a − n − w = ( n w an − w ) − , so that n w an − w ∈ A giving the inclusion. Then the second claim now follows easily. (cid:3) Remark . The above proposition gives an inclusion W A ⊂ W T when A is a maximal θ -splittorus and T is its centralizer. If we instead consider a θ -fixed Borel subgroup B and θ -stablemaximal torus T ′ ⊂ B and set ( W T ′ ) = N G ( T ′ ) θ /Z G ( T ′ ) θ , then we have the inclusions( W T ′ ) ⊂ ( W T ′ ) θ ⊂ W T ′ . (3)The subscript 0 is motivated by the fact that it is possible to choose T ′ ⊂ B such that T ′ is amaximal torus in G and ( W T ′ ) = W ( G , T ′ ) is the Weyl group of ( G , T ′ ). This distinctionwill be relevant in our discussion of resolutions of singularities of nilpotent cones in Section 2. Example . Consider the simply connected form of E , and the following involution: let ρ be the automorphism induced by the non-trivial diagram automorphism, and let s = ˇ α ( − α is the highest root, and ˇ α ( t ) is the corresponding cocharacter of T . Set θ = i s ◦ ρ ,where i s is conjugation by s . Setting W = W T ′ , we have that W θ is a Weyl group of type F .On the other hand, W is the Weyl group of G (which is type C ). Thus, [ W θ : W ] = 3, and[ W : W θ ] = 45.On the other hand, this corresponds to the split involution of type E listed in [Lev07, pg.549]. It follows that W A = W T . (cid:3) Returning to our maximal θ -split torus A and centralizer T , note that there are | W | Borelsubgroups containing A . By [S +
85, Proposition 2.9], we know that there exists a θ -split Borelsubgroup B ⊃ T . The following proposition enumerates says that the θ -split Borel subgroupscontaining T is a W A -torsor. Proposition 1.9.
Fix a θ -split Borel B ⊃ T . Then any other θ -split Borel B ′ is of the form wBw − for some w ∈ W A ⊂ W T . In particular, for any maximal θ -split torus A , the set of θ -split Borel subgroups containing it form a W A -torsor.Remark . A slight variation of this argument shows that there is a W A -torsor of minimal θ -split parabolic subgroups P containing a maximal θ -split torus A for arbitrary symmetricpairs. We leave the details to the reader. Proof.
Recall W A is the fixed-point subgroup of the induced action on W = W T . Any w ∈ W θ takes B to another θ -split Borel subgroup. Indeed, wBw − ∩ θ ( wBw − ) = wBw − ∩ θ ( w ) θ ( B ) θ ( w ) − = wBw − ∩ wθ ( B ) w − = w ( B ∩ θ ( B )) w − = wT w − = T. To finish, for any other Borel vBv − where θ ( v ) = v , we claim that T ( vBv − ∩ θ ( v ) B op θ ( v ) − . Conjugating by v , the claim is equivalent to T ( B ∩ wB op w − for some w = 1 ∈ W T . Thislast claim is obvious by general theory, so we conclude that vBv − is not θ -split. (cid:3) SPENCER LESLIE
Canonical involution on the universal Cartan.
We end this section by recalling theuniversal Cartan subspace a of a quasi-split symmetric pair ( g , g ), and showing that the uni-versal Cartan t of g inherits a universal involution θ can : t → t such that a may be identified asthe ( − a ⊂ t in Section 3.For the moment, let X = G/H be a homogeneous variety of G admitting an open orbit forsome Borel subgroup B . Such varieties are called spherical, and symmetric varieties are specialcases. To any such variety, one may attach a conjugacy of parabolic subgroups characterizedas follows: let B ⊂ G be a Borel subgroup, and let X open be the open B -orbit on X . We set P ( X ) ⊃ B to be the maximal standard parabolic subgroup stabilizing X open : P ( X ) = { g ∈ G : gX open = X open } . Define the universal Cartan subgroup of G as the quotient T = B/ [ B, B ]. Note that for anyother Borel subgroup B ′ , there is a canonical isomorphism T = B/ [ B, B ] ∼ = B ′ / [ B ′ , B ′ ] , justifying the name. This quotient inherits an action of the Weyl group W of G , and therestriction of the quotient B → T to any maximal torus H ⊂ B induces a W -equivariantisomorphism H ∼ −→ T . We also have the Lie algebra version s = b / [ b , b ]; this is the universalCartan subalgebra, which also inherits a W -action.There is a canonical torus A X associated to the variety X , known as the universal Cartanof X . One may realize A X as the quotient of P ( X ) through which P ( X ) acts on the quotient U \\ X ◦ where U ⊂ B is the unipotent radical of B . In particular, we have quotient homo-morphism of universal Cartans T → A X , and a corresponding map of Lie algebras t → a X . Moreover, there is a finite Coxeter group W X associated to X , called the little Weyl group of X , which may be realized as a subquotient of W and so that the quotient t → a X is equivariantwith respect to the appropriate subgroup of W . The rank of X is defined to be the rank of A X .Returning to the case of a symmetric space X θ = G/G , P ( X θ ) is conjugate to a mini-mal θ -split parabolic subgroup. To see this P ( X θ ) is a Borel subgroup under the quasi-splitassumption. Lemma 1.11.
For any maximal θ -split torus A , there is an isogeny of tori A → A X . In par-ticular, the two Lie algebras
Lie( A ) and a are (non-canonically) isomorphic by an isomorphismwhich intertwines the actions of W A ∼ = W a .Proof. Let T = Z G ( A ). For any θ -split Borel subgroup B ⊃ T , consider the canonical isomor-phism b / [ b , b ] ∼ = t . Restricting the quotient to Lie( T ) induces an isomorphism Lie( T ) ∼ = t .The θ -split condition implies that if x = eG ∈ ( X θ ) open , then B x = B ∩ G = B θ = T θ = T ∩ G . It follows from [Lev07, Lemma 1.3] that A ֒ → T → T /T θ ∼ = U \ B/B x ∼ = A is an isogeny.Passing to Lie algebras gives the second statement, and the statement about Weyl group actionsfollows from the W -equivariance of t → a . (cid:3) Hereafter, we will denote the universal Cartan of
G/G simply by A , its Lie algebra by a , andthe little Weyl group by W a . Additionally, the notation t will always denote the the Lie algebraof the universal Cartan of G . The previous lemma and discussion imply that this is consistentwith our previous notation, at least up to a non-canonical isomorphism. In particular, we havean inclusion of the Weyl groups W a ⊂ W .As is visible in the proof of the lemma, the isomorphism t ∼ = Lie( T ) induced by any θ -splitBorel descends through the quotient q : Lie( T ) → Lie( T ) / Lie(( T ∩ G ) ◦ ) ∼ = Lie( A ) to give acommutative diagram t a Lie( T ) Lie( A ) . ∼ = ∼ = q (4) ROTHENDIECK-SPRINGER FOR SYMMETRIC SPACES 9
There is a natural splitting of Lie( T ) → Lie( A ) induced by the involution θ acting on Lie( T )Lie( T ) = Lie( A ) ⊕ Lie( A ) ⊥ , where Lie( A ) ⊥ = { X ∈ Lie( T ) : θ ( X ) = X } , and q corresponds to the projection onto the firstfactor. The commutativity of the diagram induces a splitting a ֒ → t for any choice of θ -stableBorel subgroup. We claim that the image of this splitting is in fact independent of A , T , and B . Proposition 1.12.
There exists a canonical involution θ can : t → t inducing a decomposition t = t ⊕ t . Moreover t ∼ = a and the image of the splitting a is t .Proof. For any Borel subgroup B , there exists gG ∈ G/G such that B is θ g -split, where θ g ( h ) = gθ ( g − hg ) g − is the conjugate involution. Moreover, any other such involution is ofthe form θ bg for some b ∈ B . Note that if S = B ∩ θ g ( B ) is the θ g -stable maximal torus of B determined by g , then bSb − = B ∩ θ bg ( B ) for any b ∈ B . Thus for any pair ( B, S ), thereexists a conjugate involution θ g such that B is θ g -split and S = B ∩ θ g ( B ) is the distinguished θ g -stable Cartan subgroup.Fix a Borel B with an involution θ g as above, and denote by θ g be the induced involution onLie( S ). We have the induced isomorphism ϕ B : Lie( S ) ֒ → b → b / [ b , b ] ∼ = t , and consider the involution θ ∗ on t induced by this isomorphism. For any other Borel subgroup P and involution θ h such that P is θ h -split with corresponding stable torus T , there is a g ∈ G such that ( P, T ) = ( g Bg − , g Sg − ). The choice of g is determined up to the T × S -action( t, s ) · g tg s , so that the induced map ad( g ) : Lie( S ) ∼ −→ Lie( T ) is independent of all choicesand is equivariant with respect to the involutions: θ h (ad( g )( X )) = ad( g )( θ g ( X )) . Let θ ∗∗ denote the involution on t induced by ϕ P : Lie( T ) → t . We have the commutativediagram Lie( S ) Lie( T ) t t . ϕ B ad( g ) ϕ P = Indeed, the unique isomorphism b / [ b , b ] ∼ = p / [ p , p ] is induced by ad( g ). Note that for t ∈ t there exists a unique X ∈ Lie( S ) such that t ∼ = X (mod [ b , b ]) and a unique Y ∈ Lie( T ) suchthat t ∼ = Y (mod [ p , p ]). Then the above diagram implies ad( g )( X ) = Y , so that θ ∗∗ ( t ) ∼ = θ h ( Y ) (mod [ p , p ])= ad( g )( θ g ( X )) (mod [ p , p ]) ∼ = θ g ( X ) (mod [ b , b ]) ∼ = θ ∗ ( t ) , where every isomorphism used is the canonical one. In particular, the two involutions areidentified. We conclude that induced involution θ can : t → t is independent of the choicesinvolved. Let t = t ⊕ t be the induced decomposition, where t i is the ( − i -eigenspace of θ can .Now consider the case of a θ -split Borel B with maximal θ -split torus A ⊂ S = B ∩ θ ( B ).Then the construction of θ can implies we have an ( θ, θ can )-equivariant isomorphism ϕ B : Lie( S ) ∼ −→ t . In particular, we obtain an isomorphism Lie( A ) ∼ = t of ( − a ֒ → t is an isomorphism onto t . (cid:3) We remark that a similar argument produces a universal involution θ can : T → T whichdifferentiates to the involution discussed in the proposition.
Corollary 1.13.
Let T be the universal Cartan of G . There is a canonical involution θ can : T → T . In particular, there is a universal regular fixed-point subgroup T = T θ can . We will use this corollary in Section 4 when the universal fixed-point torus T is used to studythe universal stabilizer group scheme.2. Nilpotent cones of symmetric spaces
In this section, we discuss the nilpotent cone N = N ∩ g and desingularizations of nilpotent G -orbits. We introduce the notion of a regular θ -stable Borel subgroup. Aside from thisdefiniton, this section will be used in Section 5 to study the generalization of the Grothendieck-Springer resolution over the entire space g .The variety N need not be irreducible. In fact, there is a bijection between connectedcomponents of N reg = N ∩ g reg and irreducible components of N . Motivated by this, weadopt the notation π ( N ) to denote the set of irreducible components of N . We refer thereader to [KR71] in characteristic zero and [Lev07] in good characteristic for further details.There is a general construction of resolutions of singularities for nilpotent orbit closures dueto [Ree95, Sek84] which generalizes the Springer resolution of the nilpotent cone. As we areworking in the special case of quasi-split symmetric spaces and only consider resolutions ofregular nilpotent orbits, we describe the resolution in a simpler, albeit less general fashion.Fix a regular nilpotent e ∈ N lies in the Lie algebra Lie( B ) of a unique Borel subgroup B ,which is necessarily θ -stable. Recall the Springer resolution of the nilpotent cone of g : e N = { ( X, B ) ∈ N × F l G : X ∈ Lie( B ) } ∼ = G × B n , where we may choose B to be our θ -stable Borel subgroup. The map ˇ π : e N −→ N given by(
X, B ) X is the Springer resolution of singularities. Consider the involution θ ∗ on e N definedby θ ∗ ( X, B ) = ( − θ ( X ) , θ ( B )) . Fixing a θ -stable torus T ⊂ B , we denote for the remainder of this section W = W T . In theprevious section (3), we introduced two subgroups W ⊂ W θ ⊂ W . Reeder shows in [Ree95]that the fixed-point variety e N θ ∗ may be decomposed as a disjoint union of vector bundles over F l G indexed by W \ W θ : e N θ ∗ = G w ∈ W \ W θ E w . The restriction of ˇ π naturally maps to N , and we have the following: Proposition 2.1.
Assume that θ is quasi-split. Then for each component N i , there existsexactly one w = w ( i ) ∈ W \ W θ such that the restriction of ˇ π to E w ( i ) is a resolution ofsingularities ˇ π : E w ( i ) −→ N i . Proof.
This follows from [Ree95, Proposition 3.2], the proof of [Ree95, Proposition 4.1], and ourassumption that ( g , g ) is quasi-split. (cid:3) In general, the number [ W θ : W ] is greater than π ( N ). In particular, there may exist θ -stable Borel subgroups B ⊂ G such that Lie( B ) ∩ N reg = ∅ . For example, consider the splitinvolution for the exceptional Lie algebra g . Then g ∼ = sl (2) ⊕ sl (2) and g ∼ = V ⊠ Sym ( V ),where V is the standard representation. In this case, the nilpotent cone N is irreducible [Lev07,Lemma 6.19 (c)], but [ W θ : W ] = 3 since this is an inner involution. Thus, there is only asingle orbit of θ -stable Borel subgroups meeting N reg , and there are two orbits which do not. Definition 2.2.
Suppose that B ∈ F l θG is a θ -fixed Borel subgroup. If this intersection B ∩N reg is non-empty, we say that B is a regular θ -stable Borel subgroup. ROTHENDIECK-SPRINGER FOR SYMMETRIC SPACES 11
It is only regular θ -stable Borel subgroups that contribute to the fibers of the resolutions inProposition 2.1. Denote the set of regular θ -stable Borel subgroup of G by ( F l θG ) reg , so that( F l θG ) reg = G i ∈ π ( N ) C w ( i ) , where C w ( i ) = { B ∈ F l G : Lie( B ) ∩ ( N i ) reg = ∅} is the closed G -orbit of regular θ -stable Borelsubgroups whose Lie algebras meet the regular locus of the component N i ⊂ N .3. A simultaneous resolution over the regular locus
In this section, we define and study a subscheme e g ⊂ g × g e g which fits into a diagramanalogous to the Grothendieck-Springer resolution where t is replaced by the universal Cartansubspace of g and prove Theorem 0.1, which we now recall. Theorem 3.1.
Let ( g , g ) be a quasi-split symmetric pair with g = g ⊕ g . There is a subscheme e g ⊂ g × g e g with a proper morphism π : e g → g that is an alteration in the sense of de Jong.We have a commutative diagram e g ag a /W a , e χ π χ where a is the universal Cartan of the symmetric pair, χ : g → a /W a is the categorical quotientmap, and e χ is the restriction of e χ : e g → g to e g . Furthermore, the restriction e g reg = e g × g g reg is smooth, and the corresponding diagram is Cartesian. We prove this theorem in the next section by defining e g to be a distinguished irreducible com-ponent of the fiber product g × g e g , proving several desirable properties including the statementabout the restriction to the regular locus. Remark . After completing this article, we became aware that this component is intimatelyrelated to a construction appearing in the work of Knop in the context of spherical varieties[Kno94]. The relation is that for a quasi-split symmetric space
G/G , e g reg may be identifiedas the fiber over a point x ∈ G/G in a component of a certain cover of the cotangent bundle T ∗ ( G/G ) Knop uses in his analysis of automorphisms of spherical varieties. However, whileKnop identifies this component using a section of the invariant moment map over the semisimplelocus, we use of a Kostant-Weierstrauss section to study this space over the regular locus. Thisapproach allows us to study non-semisimple elements and enables us to see that the object isindeed smooth over g reg , which is crucial to the applications in Section 4.3.1. Components of the fiber product.
Consider the Cartesian diagram g × t /W t tg t /W. πχ The fiber product is not irreducible, and we must study the various irreducible components.
Proposition 3.3.
The irreducible components of g × t /W t all have the same dimension. Theyeach surject onto g , and are permuted transitively by the Weyl group action on the secondfactor. Finally, each component is stable under the G ◦ -action on the left.Proof. We claim that g × t /W t is a complete intersection in g × t . To see this, note that t /W ∼ −→ A r k is an affine space and the morphism t → t /W is flat of relative dimension 0 with t smooth. This implies that g × t /W t → g is also flat of relative dimension 0, so thatdim( g × t /W t ) = dim( g × t ) − r . Note that g × t /W t ⊂ g × a is the zero set of the r equations induced by the coordinatesof χ ( g ) = π ( t ). Thus, g × t /W t is a complete intersection in g × t . This implies that allthe components have the same dimension. Note that all the fibers of g × t /W t → g are W -orbits, so that each component maps finite-to-one onto g , and W acts transitively on thecomponents. The final statement follows from the fact that G ◦ is connected, and that the fibersof g × t /W t → t /W are G -stable. (cid:3) We now make these components more explicit. Set ˆ g := g × a /W a a , so that there is aCartesian diagram ˆ g ag a /W a . π ˆ χ χ (5)Fix a set of coset representatives v ∈ W/W a . Then for each v ∈ W/W a , define v : a → t by v ( a ) = v · a . Then the composition a v −→ a → t /W is W a -invariant, so that it factors uniquely to give a diagram a ta /W a t /W, vv/W A (6)by the universal property of the categorical quotient. Composing (5) with (6), we obtain aclosed embedding, also denote by v , v : ˆ g → g × t /W t . Denoting the image by C v ⊂ g × t /W t , then C v → g is surjective for each v . Lemma 3.4. C v is irreducible for each v ∈ W/W a .Proof. It suffices to prove ˆ g is irreducible. Recalling that since ( g , g ) is quasisplit, the inter-section of g with the regular semi-simple locus of g is non-empty (if fact, it is dense). Set g rss = g ∩ g rss . Since W a permutes the irreducible components, it suffices to show that ˆ g rss is irreducible. This will follow from the existence and properties of the Kostant-Weierstrasssection, as we now explain.Fix a regular nilpotent element e ∈ g . Then there exists an r = rank( g ) dimensionalaffine subspace e + v ⊂ g such that (see [KR71, Section II.3] for characteristic zero and [Lev07,Lemma 6.30] for good characteristics):(1) The restriction χ | e + v : e + v −→ a /W a is an isomorphism,(2) every element X ∈ e + v is regular in g , and(3) each regular G ∗ -orbit in g meets e + v in exactly one point.Here, G ∗ = { g ∈ G : g − θ ( g ) ∈ Z ( G ) } = F G ◦ , where F = { a ∈ A : a ∈ Z ( G ) } . Let κ denotethe inverse isomorphism κ : a /W a → e + v , known as a Kostant-Weierstrass section. Considerthe morphism σ : a → ˆ g defined by σ ( a ) = ( κ ( a ) , a ) . This is a section of ˆ χ : ˆ g → a , so that the image is an irreducible closed subscheme of ˆ g withan open dense subscheme σ ( a reg ). This implies that G ◦ · σ ( a reg ) is irreducible, as G ◦ is smoothand connected. Applying [Lev07, Lemma 6.29], we see that G ◦ · σ ( a reg ) = ˆ g rss , implying that ˆ g is irreducible. (cid:3) Let I denote the set of irreducible components of g × t /W t . ROTHENDIECK-SPRINGER FOR SYMMETRIC SPACES 13
Corollary 3.5.
The map v C v is a bijection between W/W a −→ I . Proof.
By the previous lemma, the map is well defined. Noting that g rss × t /W t = [ v C rssv , and C rssv ∩ C rssw = ∅ if v = w ∈ W/W A , the corollary now follows. (cid:3) Over the regular locus.
Set e g reg := C reg for the restriction of C to the regular locus.Since C is isomorphic to the fiber product g × a /W a a , we have a Cartesian diagram e g reg ag reg a /W a . e χ π χ (7)For our applications, we need another description of e g reg . There is a natural proper map g × g e g → g × t /W t induced by the map e g → g × t /W t . Moreover, if we restrict to the regularlocus, we obtain an isomorphism g reg × g e g ∼ −→ g reg × t /W t , where we use the fact that g reg ⊂ g reg and that g reg × g reg (cid:0) g reg × t /W t (cid:1) ∼ = g reg × t /W t . Corollary3.5 thus enumerates those irreducible components of g × g e g that map onto the regular locusof g . In particular, there is a unique irreducible component, e g := C , of g × g e g such that g reg ∼ −→ C reg .We denote π : e g → g for the induced proper morphism. By our previous considerations, e g × g g reg ∼ = e g reg . Our goal is to give a description of this scheme in terms of Borel subgroupsof G .For an element ( X, B ) ∈ g × g e g , we define the following two subgroups. Firstly, let B ( θ ) = B ∩ θ ( B ) denote the largest θ -stable subgroup of B ; it has the Lie algebra b ( θ ) = b ∩ θ ( b ).Secondly, let Z B ( X ss ) = B ∩ Z G ( X ss ) be the corresponding Borel subgroup of Z G ( X ss ), where X = X ss + X nil is the Jordan decomposition. Denote by z b ( X ss ) the Lie algebra of Z B ( X ss ). Proposition 3.6.
With the definitions as above, we have that e g reg := { ( X, B ) ∈ g reg × F l G : B ( θ ) = Z B ( X ss ) is a regular θ -stable Borel of Z G ( X ss ) } . Moreover, e g reg → a is smooth. With the definition e g := C , this proposition proves Theorem 3.1. For ease of language, werefer to such Borel subgroups as maximally split regular Borel subgroups. This terminology isjustified as any Borel subgroup in the fiber of X ∈ g reg satisfies Z B ( X ss ) ⊂ B ( θ ). Proof.
By definition of C , we know that the map χ | e g reg lands in a . Moreover, diagram (7)and [Lev07, Corollary 6.31] implies that this map is smooth.Let S ⊂ g reg × g g denote the right-hand side. To complete the proof, we first need to showthat the map S → t factors through e g reg .Let g ∈ Z G ( X ss ) be such that g − B ( θ ) g is split for the restriction of θ to Z G ( X ss ). Note that X ≡ X ss (mod [ b , b ]) , so we are free to assume X = X ss . Note that Z G ( X ss ) B = P ( X ss ) is a parabolic subgroupof G with Levi subgroup Z G ( X ss ). If P ( X ss ) = Z G ( X ss ) U P , set U θ = B ∩ U P . Then U θ isthe largest unipotent subgroup of B such that θ ( U θ ) ∩ U θ = 1, and we have the decomposition B = B ( θ ) · U θ . In particular, for g ∈ Z G ( X ss ) we have g − U θ g ⊂ U θ . We claim that g − Bg is θ -split. Indeed, θ ( g − Bg ) = θ ( g − B ( θ ) g ) θ ( U θ ) , so that by the Levi decomposition for P ( X ss ), θ ( g − Bg ) ∩ g − Bg = θ ( g − B ( θ ) g ) ∩ g − B ( θ ) g is a maximal torus in Z G ( X ss ). Thus, B is θ g -split.Since g ∈ Z G ( X ss ), θ g ( X ss ) = Ad( g − ) ◦ θ ◦ Ad( g )( X ss ) = − X ss , so that in t θ can [ X ss (mod [ b , b ])] = θ g ( X ss ) (mod [ b , b ]) = − X ss (mod [ b , b ]) . Thus, the map S → t factors through a , so that we have a map S → e g reg . Since W a actstransitively on the fibers of e g reg , the argument above and Proposition 1.9 combine to show thatthis map is an isomorphism on geometric points. As e g reg is smooth, this is sufficient. (cid:3) We explicate the fibers of π : e g reg → g reg on geometric points. Suppose that X = X ss + X nil ∈ g reg , and let a be a Cartan subspace of g containing X ss . Then A ⊂ Z G ( X ss ), whereLie( A ) = a . Let B split be a θ -split Borel subgroup containing A . Then P ( X ) = Z G ( X ss ) B split is a θ -split parabolic subgroup with Levi subgroup Z G ( X ss ). The assumption that X is regularis equivalent to X nil ∈ z g ( X ss ) reg [KR71, Theorem 7]. Therefore, there is a unique Borelsubgroup B ⊂ Z G ( X ss ) such that X nil lies in the nilradical of Lie( B ). Setting B ′ = BU P ,where U P is the unipotent radical of P ( X ) = Z G ( X ss ) U P , then B ′ is a Borel subgroup of G and ( X, B ′ ) ∈ π − ( X ). This sets up a bijection { θ -split parabolic subgroups with Levi Z G ( X ss ) } ←→ π − ( X ) P ( X ) = Z G ( X ss ) B split ←→ ( X, B ′ )Since any two θ -split Borel subgroups B , B ⊃ A give the same parabolic subgroup P ( X ) ifand only if B = wB w − for some w ∈ Stab W a ( X ss ), the left-hand side is in bijection with W a / Stab W a ( X ss ) ∼ = W a · X ss . Thus, this gives the entire fiber. Noting that since W a permutesthe Borel subgroups in the fiber over a given regular element X ∈ g reg through the action of W A for some maximal θ -split torus A contained in Z G ( X ss ) and W A = N G ( A ) /Z G ( A ), we havethe following corollary. Corollary 3.7.
For a regular element X ∈ g reg , π ( π − ( X )) ⊂ F l G lies in a single G -orbit, where π : e g → F l G . Furthermore, for any X ∈ g if ( X, B ) , ( X, B ) ∈ π − ( X ) and B ( θ ) = B ( θ ) , then B is G -conjugate to B .Proof. The first claim follows from the discussion above. For the second claim, we first assumethat X ∈ g ss . Fixing e ∈ Lie( B ( θ )) ∩ N ( z g ( X )) reg , then ( X + e, B i ) ∈ e g reg for i = 1 , X + e ∈ g reg . The second claim now follows from the first claim for X ∈ g ss . For general X ∈ g , note that ( X, B ) , ( X, B ) ∈ π − ( X ) implies that ( X ss , B ) , ( X ss , B ) ∈ π − ( X ss ),where X = X ss + X nil is the Jordan decomposition of X . (cid:3) Remark . It is natural to ask for an explicit description of e g . By the construction of C , wehave that ( X, B ) ∈ e g if and only if X (mod [ b , b ]) ∈ a ⊂ t . For example, (0 , B ) ∈ e g for any B ∈ F l G . Since dim F l G = dim( N ), the diagram e g ag a /W a , e χ π χ does not give a simultaneous resolution of singularities and the map e g → g is not small. Wediscuss the question of whether there is an intermediate space e g reg ⊂ e g res ⊂ e g in Section 5below. 4. Moduli space of regular stabilizers
In this section, we generalize to the case of quasi-split symmetric spaces several results ofDonagi and Gaitsgory [DG02, Section 10]. These fundamentally rely on Theorem 3.1 over theregular locus.
ROTHENDIECK-SPRINGER FOR SYMMETRIC SPACES 15
Regular stabilizers.
With our set up as before, we have a Cartesian diagram e g reg ag reg a /W a . e χ π χ The space a /W a is the moduli space of regular G -orbits. We shall introduce a new space whichparametrizes regular stabilizers .In their study of the moduli of G -Higgs bundles [DG02], Donagi and Gaitsgory introduce themoduli space of regular centralizers G/N , where N is the normalizer of a fixed maximal torus T . This is a partial compactification of the space of Cartan subalgebras of g and is a smoothsubscheme of the Grassmanian of r -planes in g , Gr r ( g ). It comes equipped with a naturalsmooth morphism ϕ : g reg → G/N which sends X ∈ g reg to its centralizer. There is a ramified W = T \ N -cover G/T → G/N ,where
G/T := { ( c , b ) ∈ G/N × F l G : c ⊂ b } . This is a partial compactification of the quotient
G/T , which corresponds to c being a Cartansubalgebra. We refer the reader to [DG02, Section 2] for the definition of a W -cover. This isa partial compactification of the quotient map G/T → G/N , which corresponds to restrictingto the regular semi-simple locus. By the proof of [DG02, Prop. 1.5], there exists a Cartesiansquare e g reg G/T g reg G/N . ϕ This has the consequence that the W -cover G/T → G/N is ´etale-locally isomorphic to the W -cover t → t /W . In the next section, we prove a relative version of Theorem 11.6 in [DG02], whichgives an isomorphism between two commutative group schemes over G/N . This isomorphismwas used in a fundamental way in [Ngo06], who worked over the base t /W rather than G/N .The ´etale-local isomorphism [DG02, Proposition] between these two W -covers allows for passagebetween these two bases. The goal of this section is to prove an analogue of this statement inthe case of a quasi-split symmetric pair ( g , g ).To this end, we assume that the torus T = Z G ( A ) is the centralizer of a maximal θ -splittorus A . Using the pairing [ · , · ] : g × g → g , we let Ab r ( g ) ⊂ Gr r ( g ) denote the closedsubscheme of the Grassamanian of r -planes in g on which the restriction of [ · , · ] vanishesidentically. Consider the map ϕ : g reg → Ab r ( g ) X z g ( X ) . Essentially the same argument of [DG02, Section 10.1] applies to show that this is a well definedmorphism of schemes. We define the image of this map to be G /N , where N = N G ( A ) ⊂ G is the normalizer of A in G . The following lemma tells us that notation G /N is reasonable. Lemma 4.1.
The k -points of G /N parametrizes maximal abelian subalgebras of g ( k ) thatmeet g reg ( k ) . Moreover, the quotient G /N embeds as an open subvariety parameterizing Car-tan subspaces of g .Proof. Let X ∈ g reg have centralizer c = z g ( X ), which is a maximal abelian subalgebra of g . Asthis is θ -stable, it decomposes c = c ⊕ c , where c ∼ = Lie( Z G ( X )) [Lev07, Lemma 4.2]. Then c c gives ϕ ( X ). The maximality follows from the regularity of X . Moreover, if we are given such an abelian subalgebra c ′ ⊂ g , then it is contained in the centralizer of any regular element X ∈ c ′ . Therefore, c ′ ⊂ z g ( X ) and maximality forces equality.It is known that the quotient G /N parametrizes Cartan subspaces [Lev07, Theorem 2.11],and the embedding is obvious. (cid:3) Proposition 4.2.
The map ϕ : g reg → G /N is smooth.Proof. An argument mirroring the one in [DG02, Section 10.1] works in our setting. We includethe argument for completeness.Set c = ϕ ( x ) ∈ Ab r ( g ). Using the definition of Ab r ( g ), we may express the tangent space T c ( Ab r ( g )) as the space of maps T : c → g / c such that[ T ( y ) , y ] + [ y , T ( y )] = 0 (8)for all y , y ∈ c . To see this, we have by definition that T c ( Ab r ( g )) = { c ′ ∈ Ab r ( g h ǫ i ) : p ( c ′ ) = a } , where ǫ = 0 and where p : g h ǫ i → g is the projection onto the first factor. Any linear map T : c → g satisfying (8) gives rise to such an algebra by setting for any k -algebra R c ′ T ( R ) = span R h ǫ i { a + ǫT ( a ) : a ∈ c ( R ) } . It is easy to see that c ′ T = c ′ T if and only if T ( y ) − T ( y ) ∈ c for all y ∈ c and that any c ′ arisesin this way. This gives the claimed description.In terms of this description, the differential dϕ : T x ( g reg ) ∼ = g → T c ( Ab r ( g )) ∼ = g / c sends v ∈ g to the unique map T : c → g / c such that[ T ( x ) , y ] + [ y, u ] = 0 for all y ∈ c . This identify implies that [ T ( x ) − v, y ] = 0 for all y so that we may identify T ( x ) ≡ v (mod c ).Therefore, letting ev : T c ( Ab r ( g )) → g / c be the map T T ( x ), we see that the composition g ∼ = T x ( g reg ) dϕ −−→ T c ( Ab r ( g )) ev −→ g / c coincides with the tautological quotient map. Finally, the identity [ T ( x ) , y ] = − [ x, T ( y )] for all y ∈ c implies that ev is injective, hence an isomorphism. In particular, the image of ϕ lies inthe smooth locus of Ab r ( g ) and dϕ is surjective. This proves that ϕ is smooth. (cid:3) We remark that the proof of the previous proposition did not rely on the symmetric spacebeing quasi-split. Taking this into account gives a commutative diagram g reg g reg G /N G/N , ϕ ϕ where the bottom arrow is given by c z g ( c ) . We note that the vertical arrows are smooth.We now define G /T ⊂ G /N × F l G to be the space of pairs( a , b ) , a ⊂ b , under the restriction that b is maximally split, which we recall means that b ( θ ) = b ∩ θ ( b ) be aregular θ -stable Borel subalgebra of z g ( a ss ). Here a = a ss ⊕ a nil is the Jordan decomposition ofthe algebra a . As before this comes equipped with a natural closed immersion G /T ⊂ G/T .This may be constructed as follows: we have the diagram e g reg G /N × G/N
G/T G/T g reg G /N G/N , φπ ϕ
ROTHENDIECK-SPRINGER FOR SYMMETRIC SPACES 17 where the arrow φ : e g reg → G /N × G/N
G/T is given by φ ( X, B ) = ( z g ( X ) , ( z g ( X ) , b )) . Then G /T is given by the image of the top row of arrows, and we have the following theorem. Theorem 4.3.
The diagram e g reg G /T g reg G /N . ϕ ϕ is Cartesian. In particular, the W a -covers a → a /W a and G /T → G /N are ´etale-locallyisomorphic.Proof. Note that we have a morphism e g reg → g reg × G /N G /T given by( X, B ) ( X, ( z g ( X ) , b )) . There is clearly a map the other direction, namely the map which sends a triple ( X, ( z g ( X ) , b ))to ( X, B ), where B is the unique Borel subgroup with Lie algebra b . This is obviously an inversemap on geometric points, which suffices to show it is an isomorphism since e g reg is smooth, hencereduced.Now, we show that the diagram of Cartesian squares a e g reg G /T a /W a g reg G /N , e χ ϕ χ ϕ implies that G /T → G /N is ´etale-locally (with respect to ´etale covers of G /N ) a pullback a → a /W a . A similar argument proves that a → a /W a is ´etale-locally a pull-back of G /T → G /N . The smoothness of the horizontal arrows implies that for any x ∈ g reg , we may finda suitable affine open neighborhood x ∈ U and an affine neighborhood V ⊂ G /N containing ϕ ( x ) such that there is a commutative diagram g reg U A kV G /N V, πp for some integer k . Here ϕ | U = p ◦ π and π is ´etale [Sta18, Lemma 28.34.20]. Using thezero section splitting V → A rV , for any x ∈ G /N , we obtain an ´etale neighborhood V ′ = U × A kV V → G /N of x equipped with a locally-closed immersion V ′ → U → g reg such thatthe diagram g reg V ′ G /N (9)commutes. Forming the fiber products V ′ × g reg e g reg and V ′ × G /N G /T , the commutativityof (9) implied that the natural map V ′ × g reg e g reg → V ′ × G /N G /T is an isomorphism. Labeling U ′ = V ′ × G /N G /T , the W a -cover U ′ → V ′ is thus a pullbackof a → a /W a by Theorem 3.1. (cid:3) Example . For the case ( g , g ) = ( sl (2) , so (2)), it is shown in [DG02, Example] that G/N ∼ = P , G/T ∼ = P × P with the map P × P → P ([ x : x ] , [ y : y ]) [ x y + x y : x y : x y ] . The involution induced on P is [ a : b : c ] [ − a : b : c ]. It is easy to see that G /T ∼ = G /N ∼ = P with P → P being the unique degree two map ramified over 0 and ∞ . Thesepoints correspond to the two nilpotent centralizers contained in g .4.2. Sheaves of abelian groups.
The final goal of this section is to prove a relative analogueof Theorem 11.6 in [DG02]. This is an isomorphism between the tautological sheaf of regularstabilizers on G /N and a certain subsheaf of the restriction of scalars from G /T , and willbe useful in any attempt to generalize the results of Ngˆo [Ngo06] to the case of a relative traceformula associated to a symmetric variety.The first sheaf to consider is the sheaf of θ -fixed stabilizers C ⊂ G × G /N given by C = { ( g, a ) : Ad ( g ) x = x for all x ∈ a } . For the second group scheme, let T denote the universal Cartan of G .As noted in Corolllary1.13, the torus T may be equipped with a canonical involution θ can : T → T . Let T := T θ can be the fixed points of this involution. Note that the neutral component T ◦ is a torus, but wewish to consider the entire fixed-point subgroup. For example, if ( g , g ) is split, then this is afinite subgroup. This component group will play a role in the study of relative trace formulaeassociated to split involutions.We also consider the group scheme T over G /N defined as T = (cid:16) Res G /T /G /N ( T ) (cid:17) W a . That is, for any G /N -scheme S T ( S ) = Hom W a ( e S , T ) , where e S = S × G /N G /T . This functor is representable by a group scheme, giving our T . Lemma 4.5. [Kno96, Lemmas 2.1,2.2]
The group scheme T exists and is a smooth, commuta-tive affine group scheme over G /N . We have the following analogue of [DG02, Theorem 11.6].
Theorem 4.6.
There is an isomorphism of smooth commutative group schemes ι : C ∼ −→ T We are currently working under the assumption that G der is simply connected. In Section4.3, we explain how to extend the result to the general case. Proof.
Recall the isomorphism ι : C ∼ −→ T over G/N [DG02]. This morphism is defined asfollows: for any
G/N -scheme S , we take an S -point of C to the composition e S = S × G/N
G/T → C ×
G/N
G/T ι ′ −→ T, which is an arrow S → T . On geometric points, the isomorphism with T takes ( g, a ) ∈ C to the W -equivariant map ι ( g, a ) : F l a G → T b → g (mod [ B, B ]) , ROTHENDIECK-SPRINGER FOR SYMMETRIC SPACES 19 where F l a G is the fiber over a in G/T the reduced subscheme of which consists of the relevantBorel subalgebras, and Lie( B ) = b . We have a natural involution on C by restricting theinvolution θ ( g, a ) = ( θ ( g ) , θ ( a )) on G × G/N to C . We are interested in the fiber products C ′ := G /N × G/N
C C G /N G/N , and the corresponding diagram defining T ′ := T ×
G/N G /N . Then we have ι : C ′ ∼ −→ T ′ is anisomorphism of smooth groups schemes over G /N . Note that θ : C ′ → C ′ is given by θ ( g, a ) = ( θ ( g ) , a ) . In particular, the fixed-point subgroup scheme is precisely C . By [Edi92, Proposition 3.4], itfollows that C is smooth over G /N . The corresponding involution on T ′ sends ι ( g, a ) to ι ( θ ( g ) , a ). Lemma 4.7.
With respect to this involution, there is an isomorphism ( T ′ ) θ ∼ −→ T .Proof. We first construct the map. Let S be a G /N -scheme and let x : S → C be a θ -fixedpoint. The corresponding S -point of T is a W -equivariant map ϕ x : e S = S × G/N
G/T → T. Note that there is a natural inclusion e S = S × G /N G /T ֒ → S × G /N (cid:16) G /N × G/N
G/T (cid:17) = e S, so that by restriction we have a morphism ϕ x : e S → T which is W a -equivariant. It remains toshow that the image lies in T ⊂ T . For each geometric point s ∈ S let x ( s ) = ( g, a ) ∈ C bethe corresponding geometric point of C . The gives rise to a map ( F l a G ) split → T given by ϕ x ( b ) = t b = g (mod [ B, B ]) , for all maximally split Borel subgroups with a ⊂ b = Lie( B ) ∈ ( F l a G ) split . Since B is maximallysplit, the proof of Proposition 1.12 implies we may choose h ∈ Z G ( a ss ) such that B is θ h -split.Since g ∈ Z G ( a ), if we write g = tn for the Jordan decomposition, then t ∈ Z ( Z G ( a ss )). Thisfollows from the corresponding fact about centralizers of regular nilpotent elements and [KR71,Theorem 7]. We may now compute θ can ( t b ) = θ h ( g ) (mod [ B, B ])= θ h ( t ) (mod [ B, B ])= θ ( t ) (mod [ B, B ])= g (mod [ B, B ]) = t b , where we used the fact that x ( s ) = ( g, a ) ∈ C is a fixed point. Therefore, the morphism ϕ x : e S → T factors through the inclusion of T ⊂ T , and we have a morphism T θ −→ T .We now show that this morphism is an isomorphism over the regular semi-simple locus.Since T is smooth (hence reduced), this suffices. Note that W -equivariance implies that forany S → G /N , a morphism e S → T determines a unique morphism e S → T . This is because W × W a G /T ∼ −→ G /N × G/N
G/T , where the map is given on geometric points by [( w, gT )] ( gN , gw − T ). This gives a naturalmap T → T ′ . Since the map Z G ( a ) → B/ [ B, B ] is injective over the regular semi-simple locus,the previous argument implies that θ ( g ) = g . This implies that the above morphism factorsthrough T → ( T ′ ) θ , and it gives an inverse morphism on this locus. This shows that ( T ′ ) θ → T is an isomorphism. (cid:3) This completes the proof of Theorem 4.6. Indeed we already have seen that T is smooth andthat there is an isomorphism C ∼ −→ ( T ′ ) θ . (cid:3) Given the inclusion of subgroups T = T − θ can ⊂ T , we may form the following subgroupscheme of C over G /N : C = { ( g, a ) ∈ C : θ ( g ) = g − } , We may similarly define T ⊂ T and form the corresponding W a -invariant restriction of scalarsgroup schemes T . Corollary 4.8.
We also have isomorphisms C ∼ −→ T .Proof. The argument above goes through verbatim in this case. We leave the details to thereader. (cid:3)
When G der is not simply connected. In [DG02], the authors do not assume that G der issimply connected. That they work in full generality is of the utmost importance for applicationsto the Langlands program. In this subsection, we describe the analogous result in the symmetricspace setting when we relax the simple-connectedness assumption.Donagi and Gaitsgory first define T = (cid:16) Res
G/T /G/N ( T ) (cid:17) W , as in the preceding section, then define a subgroup group scheme T ⊂ T by imposing certaineigenvalues occur on the branching locus of the map
G/T → G/N to obtain an isomorphism C ∼ −→ T . More precisely, let Φ = Φ( g , t ) denote the set of roots of ( G, T ). For any root α of T ,let D α ⊂ G/T denote the fixed-point locus of the involution s α . For any S → G/N and S -point t : e S = S × G/N
G/T → T of T , the composition S × G/N D α ֒ → e S t −→ T α −→ G m ( C α )has image ±
1. The group subscheme T is defined to be the subgroup of maps avoiding − C α ). They then show that C ∼ −→ T .Under the assumption that G der is simply connected, this subscheme is actually the entiregroup T . Nevertheless, the argument in the proof of Theorem 4.6 did not depend on thisrestriction, so to generalize we need only explicate the appropriate restrictions on the points ofthe group scheme T for Lemma 4.7 to hold.To make this precise, we drop the assumption that G der is simply connected and now set T = (cid:16) Res G /T /G /N ( T ) (cid:17) W a , and describe a subgroup scheme T ⊂ T such that we have an isomorphism C ∼ −→ T . For each α ∈ Φ, we form the fiber product D θα = G /T × G/T D α . This is never empty since it containsthe pairs ( a , b ) where a is nilpotent, for example. Then for any scheme S → G /N , the proofof Lemma 4.7 makes clear that for an element t ∈ T θ ( S ), we have a commutative diagram S × G/N D α e S TS × G /N D θα e S T G m . t αt α Since t satisfies the condition ( C α ), we conclude that the composition λ ◦ t : S × G /N D θα → G m avoids −
1. In particular, we have the following characterization.
ROTHENDIECK-SPRINGER FOR SYMMETRIC SPACES 21
Corollary 4.9.
Define subgroup T ⊂ T so that for any G /N -scheme S , the set of S -points T ( S ) consists of W a -equivariant arrows t : e S → T such that for every α ∈ Φ the composition S × G /N D θα ֒ → e S t −→ T α | T −−−→ G m avoids − ∈ G m . Then we have an isomorphism C ∼ −→ T .Example . In the case of ( sl (2) , so (2)), we need only consider one root α : T → G m . In thiscase, D θα = Spec k ⊔ Spec k = 0 ⊔ ∞ is the disjoint union of points corresponding to the two nilpotent regular centralizers containedin g and associated θ -stable Borel subalgebras.Working with G = SL(2) gives T = Z ( G ) = {± Id } . For either nilpotent closed point n , n × G /N D θα = Spec k is the corresponding pair and there are two morphisms t : Spec k → T .Since α ( ± I ) = 1, both are admissible and we find ( C ) n ∼ −→ {± } . On the other hand, if we work with G = PGL(2), then T = { ω ( ± } , where ω : G m → T isthe fundamental coweight. While there are two maps t : Spec k → T , only the one with image Id = ω (1) is admissible since α ( ω ( − −
1. Thus ( C ) n ∼ −→ { } in this case.5. Smoothness and resolution of singularities
In this final section, we consider the question of whether e g reg has a partial compactification e g res ⊂ e g that plays the a role analogous to the Grothendieck-Springer resolution over the entirespace g . That is, we ask if there is a smooth family of resolutions of the singularities of theadjoint quotient map. For simplicity, we assume now that G is semi-simple and continue toassume that it is simply connected (see [Ste68, 9.16] and [Lev07, Lemma 1.3]).Toward this question, we consider a subspace which we show recovers the classical Grothendieck-Springer resolution in the case of the case of the diagonal symmetric space ( g ⊕ g , ∆ g ). Wealso show that our proposal does indeed form a family of resolutions of the singularities of thequotient map g → g //G , and give a sufficient criterion for this space to be smooth.However, there are very basic cases when the morphism χ : g → g //G does not admit asimultaneous resolution after base change to any finite ramified cover of g //G . In such cases,our space e g res will not give rise to an irreducible scheme. For example, assume that k = C so that we may work topologically. If we consider the split involution of type A associated tothe symmetric pair ( sl ( n ) , so ( n )) ( n >
2) we may see that no simultaneous resolution exists asfollows: consider the subregular Slodowy slice S ⊂ g studied in [Tho13]. Then f : S → g //G is a family of plane curves with an isolated singularity at 0 of type A n . The monodromyrepresentation on R f ∗ Z has image the principle congruence subgroup Γ(2) ⊂ Sp g ( Z ), where g is the genus of the curves [AVGL88], so no finite base change can remove this obstruction. Sincea simultaneous resolution of g → g //G would pull back to one of S → g //G , it followsthat no such resolution can exist. The author wishes to thank Jack Thorne for explaining thisexample to him.Our proposal for e g res is quite natural: we simply extend the construction of e g reg from Propo-sition 3.6 to all of g . That is, we consider the following subspace of g × g e g : e g res := { ( X, B ) ∈ g × F l G : B ( θ ) = Z B ( X ss ) is a regular θ -stable Borel of Z G ( X ss ) } , where the superscript res stands for resolution. The next proposition shows that this construc-tion recovers the Grothendieck-Springer resolution for the diagonal symmetric space. Proposition 5.1.
Consider the diagonal symmetric space ( g × g , ∆ g ) . Then φ : e g res −→ e g (( X, − X ) , ( B , B )) ( X, B ) is an isomorphism, where this latter variety is the Grothendieck-Springer resolution of g . Proof.
First, note that the property of (( X, − X ) , ( B , B )) lying in e g res is that B ∩ B = Z B ( X ss ) = Z B ( X ss ) , since ( X, − X ) ss = ( X ss , − X ss ) so that Z G × G (( X, − X ) ss ) = Z G ( X ss ) × Z G ( X ss ) . We construct an inverse to φ : Let ( X, B ) ∈ e g and suppose that X = X ss + X nil . Consider theparabolic subgroup P ( X ) = Z G ( X ss ) B ⊃ B with Levi subgroup Z G ( X ss ). Note that if P ( X ) = Z G ( X ss ) U P is the Levi decomposition of P ( X ), then B = Z B ( X ss ) U P . It is standard theorythat there exists a unique parabolic subgroup P ( X ) op such that P ( X ) ∩ P ( X ) op = Z G ( X ss ); let U opP be its unipotent radical. Then, the group B opX = Z B ( X ss ) U opP is also a Borel subgroup of G . By construction, B ∩ B opX = Z B ( X ss ). Thus, we define the morphism ψ : e g −→ e g res ( X, B ) (( X, − X ) , ( B, B opX )) . Clearly, φ ◦ ψ = Id . We claim also that ψ ◦ φ = Id . Suppose that ψ ◦ φ (( X, − X ) , ( B, B )) = (( X, − X ) , ( B, B )) . This implies that B ∩ B = B ∩ B = Z B ( X ss ) . The Borel subgroup Z B ( X ss ) contains a maximal torus S centralizing X ss , so B = wB w − forsome w ∈ W S . The claim now follows since, for fixed Borel subgroup B containing a maximaltorus S , the set of subgroups B ∩ B ′ as B ′ ranges over the W S -torsor of Borel subgroupscontaining S are all distinct. This final statement is true as the sets Φ + w = { α ∈ Φ + : wα < } for w ∈ W S are distinct subsets of Φ + . (cid:3) We now consider the fibers of the map e χ : e g res → a . Let a ∈ a , and recall the Kostant-Weierstrass section κ : a /W a → g , which depends on a choice of regular nilpotent element.Setting X ( a ) = κ ( a ) ss , we have the identification χ − ( a ) red ∼ = G × Z G ( a ) θ ( X ( a ) + N ( a ) ) , where N ( a ) = N ( z g ( a )) is the nilpotent cone in the ( − z g ( a ). This schemedecomposes into finitely many irreducible components N ( a ) = ∪ i N ( a ) i . Since G and Z G ( a )are connected, we have a decomposition into irreducible components χ − ( a ) red = [ i ∈ π ( N ( a ) ) χ − ( a ) i where χ − ( a ) i ∼ = G × Z G ( a ) θ N ( a ) i . Theorem 5.2.
There is a decomposition into connected components e χ − ( a ) red = G i ∈ π ( N ( a ) ) e χ − ( a ) w ( i ) such that each component is smooth and the map e χ − ( a ) w ( i ) → χ − ( a ) i is a resolution ofsingularities. In particular, e χ − ( a ) red is smooth.Proof. Recall that θ | Z G ( a ) is a quasi-split involution, which we also denote by θ . Let Z G ( a ) θ denote the fixed point subgroup of θ in Z G ( a ). Note also that Z G ( a ) θ = Z G ( a ) = Z G ( a ) ∩ G is connected since the derived subgroup Z G ( a ) (1) is simply connected [Ste68].By [Ree95, Proposition 2.3.4], the fixed point set of F l Z G ( a ) is a disjoint union of varietiesisomorphic to F l Z G ( a ) . The above morphism only maps to the regular θ -stable Borel subgroupsof Z G ( a ), denoted by ( F l θZ G ( a ) ) reg . Using the notation from Proposition 2.1, we have( F l θZ G ( a ) ) reg = G i ∈ π ( N ( a ) ) C w ( i ) , ROTHENDIECK-SPRINGER FOR SYMMETRIC SPACES 23 where C w ( i ) = { B ∈ F l Z G ( a ) : Lie( B ) ∩ ( N ( a ) i ) reg = ∅} is the closed Z G ( a ) θ -orbit of regular θ -stable Borel subgroups whose Lie algebras meet the regular locus of the component N ( a ) i ⊂N ( a ) .For simplicity, we adopt the notation g B = g − Bg . Let ( X, B ) ∈ e χ − ( a ). Then there exists g ∈ G and n ∈ N ( a ) such that X = Ad( g )( X ( a ) + n ) so that ( X ( a ) + n, g B ) ∈ e χ − ( a ). If g ′ ∈ G is another element such that X = Ad( g ′ )( X ( a ) + n ′ ), then ( X ( a ) + n ′ , g ′ B ) ∈ e χ − ( a )and g − g ′ ∈ Z G ( a ) ∩ G = Z G ( a ) θ , and n = Ad( g − g ′ )( n ′ ) . Then ( g B )( θ ) = ( g − g ′ ) g ′ B ( θ )( g − g ′ ) − , so that the regular θ -stable Borel subgroups g B ( θ ) , g ′ B ( θ ) ⊂ Z G ( a ) are in the same Z G ( a ) θ -orbit.Since Z G ( a ) θ is connected, this implies a decomposition e χ − ( a ) = G i ∈ π ( N ( a ) ) e χ − ( a ) w ( i ) into connected components. It is clear that the restriction of π to any component gives amorphism e χ − ( a ) w ( i ) → χ − ( a ) w ( i ) .We need the following lemma. Lemma 5.3.
The image of e χ − ( a ) w ( i ) under the projection π : e g → F l G lies in a single G -orbit.Proof. If (
X, B ) , ( Y, B ) ∈ e χ − ( a ) w ( i ) , then there exists g , g ∈ G such that X = Ad( g )( X ( a ) + n ) , and Y = Ad( g )( X ( a ) + n ) , and g B ( θ ) , g B ( θ ) ∈ C w ( i ) . We may assume X = X ( a ) + n so that g = 1. Then since C w ( i ) is a single Z G ( a ) θ -orbit, we find that there is g ∈ G such that, replacing g B by g B , B ( θ ) = g B ( θ ). Since ( X ( a ) , B ) , ( X ( a ) , g B ) ∈ e g , Corollary 3.7 thus implies that B lies inthe same G -orbit as g B in F l G , so that B and B do as well. (cid:3) Now fix a Borel B such that ( X ( a ) , B ) ∈ e χ − ( a ) with B ( θ ) ∈ C w ( i ) . Then for every ( X, P ) ∈ e χ − ( a ) w ( i ) , the previous lemma says that we may write P = g B = g − Bg for some g ∈ G .This implies that (Ad( g )( X ) , B ) ∈ e χ − ( a ) so thatAd( g )( X ) ≡ X ( a ) (mod [ b ( θ ) , b ( θ )]) . Thus, the difference Ad( g )( X ) − X ( a ) ∈ [ b ( θ ) , b ( θ )] is nilpotent, implying Ad( g )( X ) ∈ X ( a ) + n ( θ ) . We have proven the following lemma. Lemma 5.4.
There is an isomorphism e χ − ( a ) w ( i ) ∼ = { ( X, gB ) ∈ g × G · B : X ∈ Ad( g ) ( X ( a ) + n ( θ ) ) } , which we may identify with G × B ( θ ) ( X ( a ) + n ( θ ) ) . (cid:3) Let us now consider the resolution of singularities of N ( a ) . Using Proposition 2.1, we seethat e N ( a ) = { ( X, B ) ∈ N ( a ) × F l Z G ( a ) : X ∈ Lie( B ) , B regular θ -stable Borel } has a similar decomposition into components e N ( a ) = G i ∈ π ( N ( a ) ) E w ( i ) −→ G i ∈ π ( N ( a ) ) C w ( i ) . Fix a component N ( a ) i , and restrict the previous map to the fiber over this component. By[Ree95, Proposition 3.2], π i : E w ( i ) −→ N ( a ) i is a resolution of singularities. More explicitly,let e ∈ N ( a ) i,reg . In Section 2, we constructed a Borel subgroup P ⊂ Z G ( a ) with Lie algebra p = Lie( P ) such that if q i = N ( a ) ∩ p , then e ∈ q i , and E w ( i ) ∼ = Z G ( a ) θ × P θ ( X ( a ) + q i ) . It follows that G × Z G ( a ) θ π i : G × Z G ( a ) θ (cid:16) Z G ( a ) θ × P θ ( X ( a ) + q i ) (cid:17) −→ G × Z G ( a ) θ ( X ( a ) + N ( a ) i )is a resolution of singularities of an irreducible component of χ − ( a ) red . The natural map f i : G × Z G ( a ) θ (cid:16) Z G ( a ) θ × P θ ( X ( a ) + q i ) (cid:17) → G × P θ ( X ( a ) + q i ) , is an isomorphism. For any Borel subgroup B ⊂ G such that X ( a ) + e ∈ Lie( B ) and B ( θ ) = B ∩ θ ( B ) = P , we may identify n ( θ ) = q i and B ( θ ) = P θ so that Lemma 5.4 implies that f i induces an isomorphism f i : G × Z G ( a ) θ (cid:16) Z G ( a ) θ × P θ ( X ( a ) + q i ) (cid:17) ∼ −→ e χ − ( a ) w ( i ) , and thus a commutative diagram e χ − ( a ) w ( i ) G × Z G ( a ) θ (cid:16) Z G ( a ) θ × P θ ( X ( a ) + q i ) (cid:17) χ − ( a ) w ( i ) G × Z G ( a ) θ N ( a ) i , ∼ π G × ZG ( a ) θ π i ∼ showing that π : e χ − ( a ) w ( i ) → χ − ( a ) w ( i ) is a resolution of singularities. (cid:3) Consider the morphism e χ : e g res → a . We wish to know if e g res may be endowed with anatural scheme structure such that this morphism is smooth. Our analysis of the fibers of thismorphism shows that their reduced subschemes are all smooth of dimension r = dim( a ). Touse our analysis of the fibers to conclude smoothness, we require the following technical lemma. Lemma 5.5.
Suppose that X is a variety (that is, a reduced, irreducible, separated scheme offinite type over an algebraically closed field k ) and suppose Y is a smooth affine k -scheme ofdimension m . Suppose that f : X → Y is a morphism such that(1) ( X y ) red is smooth of fixed dimension n > for all y ∈ Y ( k ) ,(2) the maximal open V y ⊂ X y which is a reduced scheme is dense in X y for all y ∈ Y ( k ) .Then f is smooth. In particular, X is smooth over k . We remark that the statement trivially holds for n = 0 once one assumes that f is surjective. Proof.
Denote by V ⊂ X the open subscheme on which the restriction f | V is smooth. Then V y is the fiber ( f | V ) − ( y ): this follows from [dJ96, 2.8]. Let n : X ′ → X denote the normalizationof X ; note that V ⊂ X ′ is an open subscheme of X ′ as well. We have the commutative diagram X ′ X Y. f ′ n f (10)First, we show that the assumptions imply that for each y ∈ Y ( k ), the induced map ( X ′ y ) red ∼ −→ ( X y ) red is an isomorphism. Indeed, this is a finite morphism that is an isomorphism over V y = ( V y ) red . Moreover, ( X ′ y ) red is equidimensional by Krull’s height theorem, so that the mapis birational. It is thus an isomorphism as the base is smooth, hence normal. In particular, f ′ : X ′ → Y also satisfied the assumptions of the lemma. This also implies a bijection betweenclosed points of X ′ and X .For any smooth effective Cartier divisor Z ⊂ Y , consider the morphism ( f ′ ) − ( Z ) → Z . Since X ′ normal, it follows that ( f ′ ) − ( Z ) is reduced [Sta18, Lemma 27.12.4]. If dim( Y ) = 1, thisshows that the fibers of f ′ are reduced, so that they are smooth by the preceding paragraph.But then f ′ : X ′ → Y is a morphism with smooth equidimensional fibers over a smooth base. Itis flat by [Sch10, Theorem 3.3.27], and thus smooth by [Har77, Theorem 10.2]. For dim( Y ) > y ∈ Y ( k ) ROTHENDIECK-SPRINGER FOR SYMMETRIC SPACES 25 we may choose Z such that y ∈ Z ( k ) and ( f ′ ) − ( Z ) is irreducible. Note that we have used thefact that f ′ is surjective. Then the map ( f ′ ) − ( Z ) → Z also satisfies (1) and (2). By inductionon the dimension of the base, ( f ′ ) − ( Z ) → Z is a smooth morphism. In particular, all the fibersof f ′ are smooth. By the argument above, f : X ′ → Y is a smooth morphism.To conclude, we show that n : X ′ → X is an isomorphism. Since we have seen that isis bijective on closed points, we need only check that it is injective on tangent vectors. Thediagram (10) implies that any vector in the kernel of dn must be vertical with respect to f ′ : X ′ → Y ; that is must lie in T ( X ′ y ) ⊂ T X for some y ∈ Y ( k ). But this is impossible since X ′ y = ( X ′ y ) red ∼ −→ ( X y ) red is an isomorphism of smooth varieties. (cid:3) Corollary 5.6. If e g res is a variety, the morphism e χ : e g res → a is smooth.Proof. This follows from Lemma 5.5. To see this, take f = e χ , X = e g , and Y = a . Then underthe assumption on G = e g , the spaces X and Y satisfy the criteria, (1) follows from Theorem5.2 above, and (2) follows from the Cartesian diagram in Proposition 3.6 which implies that( e χ reg ) − ( t ) ⊂ e χ − ( t ) , which is Zariski open and dense, is smooth. (cid:3) References [AVGL88] Vladimir Igorevich Arnol’d, Victor Anatolievich Vassiliev, Viktor Vladimirovich Goryunov, andOV Lyashko. Singularities. i. local and global theory.
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