aa r X i v : . [ m a t h . G R ] O c t ON THE CENTRALIZER OF A BALANCED NILPOTENTSECTION
WILLIAM HARDESTY
Abstract.
Let G be a split reductive algebraic group defined over a completediscrete valuation ring O , with residue field F and fraction field K , where thefiber G F is geometrically standard. A balanced nilpotent section x ∈ Lie( G )can roughly be thought of as an O -point in a K nilpotent orbit such that thecorresponding orbits over K and F have the same Bala–Carter label. In thispaper, we will establish a number of results on the structure of the centralizer G x ⊆ G of x . This includes a proof that G x is a smooth group scheme, andthat the component groups of the geometric fibers G x K and G x F are isomorphic. Introduction
Let p be a prime, and let O be a complete discrete-valuation-ring (DVR) withuniformizer ω , fraction field K , and perfect residue field F of characteristic p . For an O -scheme X , and an O -algebra A , we use the notation X A := X × Spec( O ) Spec( A ).Let G be a split reductive algebraic group scheme over Spec( O ), with a (lower)Borel subgroup B and a maximal torus T . We will assume that G K and G F are geometrically standard (cf. [Mc3, § Example . If G = SL n , then G F is geometrically standard provided p ∤ n .Let g = Lie( G ) be the Lie algebra of G , regarded as a scheme, and let g O := g ( O )denote the O -points of g , which gives a lattice g O ⊂ g K . We will refer to the elementsof g O as sections , since such an element corresponds to a map x : Spec( O ) → g .For each section x ∈ g O , let x K ∈ g K and x F ∈ g F denote the values of this map atthe generic point and closed point of Spec( O ) respectively.The adjoint action gives g O the structure of a rational G -module (equivalentlyan O [ G ]-comodule). For any x ∈ g O , let G x ⊆ G denote the scheme-theoreticcentralizer of x , and let G x K K ⊆ G K and G x F F ⊆ G F denote the scheme-theoreticcentralizers of x K and x F (see [J1, I.2.12(1)] for the definition). In fact, it can bededuced from the definition that for any O -algebra A , G x A A ∼ = ( G x ) A . To improve notation, we will often omit the parenthesis on the right hand side, andtake G xA ⊆ G A to mean the centralizer of x A ∈ g A . Definition 1.2.
We say that a section x is balanced if G x K K and G x F F are smoothgroup schemes and dim G x K K = dim G x F F (cf. [Mc3]). To simplify notation, we will often write G x K and G x F in place of G x K K and G x F F respectively. Remark . The definition of balanced in particular implies that the scheme-theoretic centralizers of x K and x F are actually reduced, since smooth implies re-duced. Thus, over the algebraic closures K and F , the base changes G x K and G x F coincide with the “classical” centralizers, which are defined only on the geometricpoints as in [LS]. Remark . It follows from [Mc3, Theorem 4.5.2 and Corollary 7.3.2] and the proofof [Mc3, Corollary 9.2.2], that the orbits of x K and x F have the same Bala–Carterlabel.It is worth mentioning that that balanced nilpotent sections exist for every F -orbit by [Mc3, Theorem 4.5.2]. Thus, since there exist only finitely many nilpotent F -orbits, it is possible to enlarge O by a finite extension so that for every F nilpo-tent orbit C F , there exists a balanced section x ∈ g O such that x F ∈ C F (thecorresponding extension of F is also finite, and thus will remain perfect).1.1. Smoothness of G x . By a smooth morphism f : X → Y of schemes, we willmean a morphism which satisfies the definition given in [St, Tag 01V8]. We haveincluded a proof of the following lemma due to the lack of a proper reference. Lemma 1.5.
A morphism f : X → Y between schemes of finite-type is smooth ifand only if (1) f is flat , (2) for every geometric point y → Y , the fiber product X × Y y is a smoothvariety.Proof. By first applying [St, Tag 01V4], we can reduce down to checking smoothnessat every fiber X × Y y for any point { y } → Y . By definition, { y } = Spec( k ) forsome field k , and so by [Ht, Theorem III.10.2], X × Y Spec( k ) is smooth if and onlyif X ⊗ Y Spec( k ) is smooth. Combining these two results gives the lemma. (cid:3) We will prove the following theorem.
Theorem 1.6. If x ∈ g O is a balanced nilpotent section, then G x → Spec( O ) is asmooth morphism.Remark . We already know that the morphism G x → Spec( O ) is finite-typewith geometric fibers G x K and G x F , which are smooth varieties by Remark 1.3, soaccording to Lemma 1.5, it suffices to show that O [ G x ] is a flat O -module. However,since O is a DVR, this is equivalent to proving that O [ G x ] is torsion-free over O .This will be proven in § Results on component groups.
Let A ( x K ) = G x K ( K ) / ( G x K ) ◦ ( K ) and A ( x F ) = G x F ( F ) / ( G x F ) ◦ ( F ) denote the (discrete) component groups of the geometric fibers G x K and G x F respectively. In the case where G is simple and of adjoint type, it followsfrom Remark 1.4 and [MS] that there is an isomorphism of groups A ( x K ) ∼ = A ( x F ).We will extend this result to arbitrary split reductive groups G with geometricallystandard fiber G F . Theorem 1.8. If x ∈ g O is a balanced nilpotent section, then there is a groupisomorphism A ( x K ) ∼ = A ( x F ) .Remark . This will be proven in § N THE CENTRALIZER OF A BALANCED NILPOTENT SECTION 3
We will also prove in Theorem 3.12, that it always possible to enlarge O so thatthe identity components for the fibers can be lifted to a normal subgroup scheme( G x ) ◦ E G x , where the quotient G x / ( G x ) ◦ is an affine group scheme whose geometricfibers are A ( x K ) and A ( x F ).1.3. Centralizers for the G × G m action. In §
4, we consider the centralizersfor a certain action of G × G m on g . For instance, we will show that if x ∈ g O isbalanced for the G action, then the centralizer ( G × G m ) x ⊆ G × G m is also smooth,and there exists an isomorphism of component groups for the geometric fibers (seeTheorem 4.4). An application of this will be given in Proposition 4.7, which can beused to relate the representation theory for the reductive quotients of the K and F centralizers by considering the representation theory of ( G × G m ) x .1.4. Additional comments.
The main source of motivation for this project orig-inated from the author’s work with P. Achar and S. Riche on the modular Lustig–Vogan bijection in [AHR1], where the structure and representation theory of G x plays a crucial role.It should also be mentioned that the smoothness of G x result has been verifiedin a number of cases, and by very different arguments, in [AHR1] and [B].1.5. Acknowledgements.
The author wishes to express his gratitude to P. Achar,S. Riche and G. McNinch for their helpful comments and suggestions.2.
The torsion-free subgroup scheme
Fix a balanced nilpotent section x ∈ g O and let I tor = ker( O [ G x ] → K ⊗ O O [ G x ] ∼ = K [ G x ]) . denote the ideal consisting of all the ω -torsion elements of O [ G x ], then V ( I tor ) is aclosed torsion-free subscheme of G x .2.1. We will begin by establishing some general properties of this subscheme. Lemma 2.1.
Let M an arbitrary O -module, and let N ≤ M be any free finite-rank O -submodule, then the natural morphism N ⊗ O N υ −→ M ⊗ O M is injective.Proof. Since the functor − ⊗ O K : O -mod → K -mod is exact, the induced map N K −→ M K is also injective, so we can regard N K as a submodule of M K . There is a commutativediagram( N ⊗ O N ) ⊗ K N ⊗ O N K N ⊗ O ( K ⊗ K N K ) N K ⊗ K N K υ ⊗ y y y y ( M ⊗ O M ) ⊗ K M ⊗ O M K M ⊗ O ( K ⊗ K M K ) M K ⊗ K M K where the “ = ” symbols denote natural isomorphisms. The rightmost arrow is thecanonical injection N K ⊗ K N K ֒ → M K ⊗ K M K , so in particular, υ ⊗ WILLIAM HARDESTY
Composing − ⊗ O K with the forgetful functor to O -mod, and letting S = ker( υ ),gives the commutative diagram0 −−−−→ S −−−−→ N ⊗ O N υ −−−−→ M ⊗ O M y y y −−−−→ S ⊗ O K −−−−→ ( N ⊗ O N ) ⊗ O K υ ⊗ −−−−→ ( M ⊗ O M ) ⊗ O K . The exactness of the top row implies exactness of the bottom row since − ⊗ O K isexact. However, the injectivity of υ ⊗ S ⊗ O K = 0. Therefore, S ⊆ N ⊗ O N isa torsion submodule of the free finite-rank O -module N ⊗ O N , and hence, S = 0. (cid:3) We will also require the following lemma.
Lemma 2.2.
Let H be any affine algebraic group scheme over Spec O , then theclosed subscheme V ( I tor ) ⊆ H is actually a (torsion-free) subgroup scheme, whichwe will denote by H tf .Proof. By [J1, I.2.4(6)] it suffices to show(2.1) ∆( I tor ) ⊆ I tor ⊗ O [ H ] + O [ H ] ⊗ I tor ,ε ( I tor ) = 0 and σ ( I tor ) ⊆ I tor where ∆, ε and σ are the comultiplication, counitand antipode for O [ H ] respectively.Clearly I tor ⊆ I , since O [ H ] /I ∼ = O is torsion-free, where I = ker( ε ). Also,since σ : O [ H ] → O [ H ] is an O -linear map, it must preserve torsion. So the secondand third identities are verified and we are left to verify (2.1).Let f ∈ I tor be arbitrary, then ∆( f ) is also torsion since ∆ is O -linear. Nowsuppose, ∆( f ) = m X i =1 f i ⊗ h i , and let M ≤ O [ H ] be the O -submodule generated by f i , h i , 1 ≤ i ≤ m . Since M is finitely-generated and O is a DVR, then there exists a decomposition M = M tor ⊕ M free . Let { e , . . . , e r } be a basis for M free , then for i = 1 , . . . , m , f i = m X j =1 a i,j e j + ϕ i , h i = m X j =1 b i,j e j + ψ i , where a i,j , b i,j ∈ O and ϕ i , ψ i ∈ M tor . Let α i = P mj =1 a i,j e j and β i = P mj =1 b i,j e j ,so that f i ⊗ h i = α i ⊗ β i + ( α i ⊗ ψ i + ϕ i ⊗ β i + ϕ i ⊗ ψ i ) . Clearly, f i ⊗ h i − α i ⊗ β i ∈ I tor ⊗ O [ G x ] + O [ G x ] ⊗ I tor , and is, in particular, tor-sion.On the other hand, the α i ⊗ β i terms are in the image of the natural map M free ⊗ M free −→ O [ H ] ⊗ O [ H ] . By Lemma 2.1, this map is injective, and thus M free ⊗ M free can be regarded as afree finite-rank O -submodule of O [ H ] ⊗ O [ H ]. It follows that m X i =1 α i ⊗ β i = ∆( f ) − m X i =1 ( α i ⊗ ψ i + ϕ i ⊗ β i + ϕ i ⊗ ψ i ) N THE CENTRALIZER OF A BALANCED NILPOTENT SECTION 5 is both torsion and lies in the free submodule M free ⊗ M free , and therefore must bezero. (cid:3) Remark . The property that V ( I tor ) is a subgroup scheme, also appears to followfrom a more general property, stated at the beginning of § G x tf is smooth, and that G x tf , F containsthe identity component ( G x F ) ◦ (see [Mi, Definition 13.12] for the definition of theidentity component over a general field). By Remark 1.3, the identity componentof G xF is also reduced. Our strategy will be to compare the distribution algebras ofthe K and F centralizers.Let us first recall that the K and F centralizers admit the Levi decompositions(2.2) G x K = G x K , red ⋉ G x K , unip , G x F = G x F , red ⋉ G x F , unip (cf. [Mc1, Corollary 29]). By [Mc3, Corollary 9.2.2], the K and F reductive quotientsalso have the same root datum. Moreover, by [Mc1, Theorem 28], the unipotentradicals G x K , unip and G x F , unip are both split , and have the same rank (i.e. as schemes,they are both isomorphic to affine spaces of the same dimension). Thus, there isan isomorphism of graded vector spaces(2.3) Dist( G x K , unip ) = Dist( A d K , , Dist( G x F , unip ) = Dist( A d F , d = dim G x K , unip = dim G x F , unip . These distributions are explicitly calculatedin [J1, I.7.3]. Lemma 2.4.
For all n ≥ , dim Dist n ( G x K ) = dim Dist n ( G x F ) . Proof.
Apply (2.2), and observe that since G x red , K and G x red , F have the same root-datum, then the classification results from [J1, II.1] imply the existence of a Kostant O -form U O ⊆ Dist( G x red , K ) which satisfies F ⊗ O U O ∼ = Dist( G x red , F ). So in particular,dim Dist n ( G x red , K ) = dim Dist n ( G x red , F )for all n ≥ (cid:3) Definition 2.5.
By [J1, I.7.4], an affine scheme X over O is said to be infinitesi-mally flat at z ∈ X ( O ) provided O [ X ] /I n +1 z are flat O -modules for all n ≥
0, where I z is the ideal corresponding to z .The infinitesimal flatness property is necessary in order for distribution algebrasto work “nicely” for an O -group scheme. Lemma 2.6.
Both G x and G x tf are infinitesimally flat at ∈ G x tf ( O ) = G x ( O ) , and I tor ⊆ T n ≥ I n .Proof. Let I , K and I , F be the augmentation ideals for the corresponding fibers.From the map O [ G x ] → F ⊗ O O [ G x ] ∼ = O [ G x ] /ω O [ G x ] , we can see that I , F = I + ω O [ G x ] ω O [ G x ] , and more generally, I n , F = I n + ω O [ G x ] ω O [ G x ] . WILLIAM HARDESTY
This gives F [ G x ] /I n , F ∼ = O [ G x ] I n + ω O [ G x ] . On the other hand, F ⊗ O ( O [ G x ] /I n ) ∼ = O [ G x ] /I n ω ( O [ G x ] /I n ) ∼ = O [ G x ] /I n ( ω O [ G x ] + I n ) /I n ∼ = F [ G x ] /I n , F . So there exist morphisms K [ G x ] /I n , K ∼ = K ⊗ O ( O [ G x ] /I n ) ← O [ G x ] /I n → F ⊗ O ( O [ G x ] /I n ) ∼ = F [ G x ] /I n , F . By Lemma 2.4, the dimensions of the left and right side are equal, so that O [ G x ] /I n are torsion-free, and hence flat, for all n . Therefore, G x i infinitesimally flat at 1.Finally, since I tor ⊆ T n ≥ I n , it also follows that G x tf is infinitesimally flat at 1. (cid:3) The preceding lemma also implies that Dist n ( G x tf ) ∼ = Dist n ( G x ) for all n ≥ n ( G x tf , F ) ∼ = F ⊗ O Dist n ( G x tf ) ∼ = F ⊗ O Dist n ( G x ) ∼ = Dist n ( G x F ) , for all n ≥
0. In particular, Dist( G x tf , F ) ∼ = Dist( G x F ) as filtered algebras.We may now prove the main result of this section. Proposition 2.7.
The group scheme G x tf is smooth and ( G x F ) ◦ ⊆ G x tf , F .Proof. By construction, G x tf , F ⊆ G x F , and thus by [J1, I.7.17(7)], Dist( G x tf , F ) ⊆ Dist( G x F ) as filtered algebras. So Dist n ( G x tf , F ) ⊆ Dist n ( G x F ) for all n ≥
0. How-ever, by (2.4), it follows that dim F Dist n ( G x tf , F ) = dim F Dist n ( G x F ), and hence,Dist n ( G x tf , F ) = Dist n ( G x F ) for all n ≥
0. Therefore,(2.5) Dist( G x tf , F ) = Dist( G x F ) . Let us now base change to F , where we note that (2.5) also holds over F by[J1, I.7.4(1)]. The definition of the identity component implies 1 G x F ∈ ( G x F ) ◦ ( F ),and so it follows that Dist(( G x F ) ◦ ) = Dist( G x tf , F ) . Now since ( G x F ) ◦ is an irreduciblesubgroup scheme of G x F , then by (2.5) and [J1, I.7.17(7)], ( G x F ) ◦ ⊆ G x tf , F , provingthe second statement of the proposition.To prove the first statement, we first recall that by [J1, I.7.1(2)],dim F Dist + n ( G x tf , F ) = dim F Dist + n ( G x F )for all n ≥
1. Thus, by [J1, I.7.7], the Lie algebras of G x tf , F and G x F have the samedimension. On the other hand, ( G x F ) ◦ ⊆ G x tf , F ⊆ G x F implies dim G x tf , F = dim G x F , and hence, G x tf , F is smooth by [J1, I.7.17(1)]. Likewise, G x tf , K is automatically smooth by [J1, I.7.17(2)]. Thus, by Lemma 1.5, G x tf willbe smooth provided O [ G x tf ] is torsion-free. However, this property holds from thedefinition of G x tf , and so we are done. (cid:3) Smoothness and component groups of centralizers
To simplify our arguments, we will assume throughout this section that O is suchthat F = F is algebraically closed, unless specified otherwise. N THE CENTRALIZER OF A BALANCED NILPOTENT SECTION 7
Diagonalizable group schemes.
For any commutative group Λ and a com-mutative ring k , the diagonalizable group scheme over k associated to Λ, denotedDiag(Λ), is defined by setting k [Diag(Λ)] := k [Λ], where k [Λ] is the group algebrafor Λ with the usual Hopf algebra structure (cf. [J1, I.2.5]). Lemma 3.1.
Let k be a commutative integral ring, and suppose X = Diag(Λ X ) , Y = Diag(Λ Y ) and Z = Diag(Λ Z ) are diagonalizable group schemes over k withmorphisms X ϕ −→ Z and Y ψ −→ Z , then X × Z Y ∼ = Diag(Λ) , where Λ is a pushout induced by two uniquely determined group homomorphisms Λ Z ϕ ′ −→ Λ X and Λ Z ψ ′ −→ Λ Y .Proof. Since k is integral, then the isomorphism [J1, I.2.5(2)] ensures that themorphisms ϕ and ψ are induced by unique group homomorphisms Λ Z ϕ ′ −→ Λ X and Λ Z ψ ′ −→ Λ Y . More precisely, if we identify k [ X ] = k [Λ X ], k [ Y ] = k [Λ Y ] and k [ Z ] = k [Λ Z ], where the Hopf algebras on the right are the group algebras for Λ X ,Λ Y and Λ Z respectively, then the comorphisms ϕ ∗ and ψ ∗ are the group algebrahomomorphisms induced by ϕ ′ and ψ ′ respectively.Now let Λ be the pushout of ϕ ′ and ψ ′ , then(3.1) Λ = Λ X × Λ Y h ( ϕ ′ ( λ Z ) , , ψ ′ ( λ Z ) − ) | λ Z ∈ Λ Z i , where we use multiplicative notation to denote the group structure. Let us employthe natural identification Λ ′ ⊆ k [Λ ′ ] for an abelian group Λ ′ (i.e. identifying Λ ′ with the “group-like” elements of k [Λ ′ ]), so that by definition, Λ ′ gives a k -basis of k [Λ ′ ]. We will also make use of the isomorphism of algebras(3.2) k [Λ X ] ⊗ k k [Λ Y ] ∼ −→ k [Λ X × Λ Y ] , induced by sending λ X ⊗ k λ Y ( λ X , λ Y ) for any λ X ∈ Λ X and λ Y ∈ Λ Y .Now let us note that the tensor product k [Λ X ] ⊗ k [Λ Z ] k [Λ Y ] coincides with thepushout along ϕ ∗ and ψ ∗ in the category of commutative k -algebras. By definition k [Λ X ] ⊗ k [Λ Z ] k [Λ Y ] ∼ = k [Λ X ] ⊗ k k [Λ Y ] /I , where I is the k -submodule generated by f φ ∗ ( h ) ⊗ g − f ⊗ ψ ∗ ( h ) g for all f ∈ k [ X ], g ∈ k [ Y ] and h ∈ k [ Z ]. In particular, I is spanned by λ X ϕ ′ ( λ Z ) ⊗ λ Y − λ X ⊗ ψ ′ ( λ Z ) λ Y , for all λ X , λ Y and λ Z .Observe now that λ X ϕ ′ ( λ Z ) ⊗ λ Y = λ X ⊗ ψ ′ ( λ Z ) λ Y ⇐⇒ ϕ ′ ( λ Z ) ⊗ ψ ′ ( λ Z ) − = 1 ⊗ ⇐⇒ ( ϕ ′ ( λ Z ) , ψ ′ ( λ Z ) − ) = 1for λ X , λ Y and λ Z , where the second “ ⇐⇒ ” arises from (3.2). From this, wecan see that k [Λ X ] ⊗ k [Λ Z ] k [Λ Y ] can be equivalently obtained from k [Λ X × Λ Y ] byimposing the relation ( ϕ ′ ( λ Z ) , ψ ′ ( λ Z ) − ) = 1 for all λ Z ∈ Λ Z . Comparing with thedescription in (3.1) gives the group algebra k [Λ]. Therefore, we are done. (cid:3) Let k be any commutative ring, then we say that an affine k -scheme X is constant ,provided the associated k -functor X : { k -algebras } → { sets } WILLIAM HARDESTY is a constant functor. If the cardinality | X ( k ) | = r < ∞ , then there exists analgebra isomorphism k [ X ] ∼ = k × r , where the right hand side is an r -fold directproduct with pointwise multiplication. An affine group scheme is called constant ifit is constant as a scheme. Lemma 3.2.
Let Λ be a finite abelian group with r = | Λ | . If p ∤ r , then Diag(Λ) isa constant O -group scheme (i.e. O [Λ] ∼ = O × r as an O -algebra).Proof. Since Λ is a finite abelian group, thenΛ ∼ = Z /n Z × Z /n Z × · · · × Z /n t Z , where the n , . . . , n t ∈ Z may have repeated multiplicities. In particular, O [Λ] ∼ = O [ Z /n Z ] ⊗ O [ Z /n Z ] ⊗ · · · ⊗ O [ Z /n t Z ] . Thus, Diag(Λ) is constant if Diag( Z /n i Z ) is constant for all i . The condition p ∤ r clearly implies p ∤ n i for all i . Therefore, without loss of generality, we may assumeΛ is cyclic.Supposing now that Λ = Z /r Z , gives O [Λ] = O [ z ] / h z r − i . Our assumption on O at the beginning of this section, implies that O (and hence K ) contain all roots of unity which are co-prime to p . Thus, z r − z − ζ )( z − ζ ) · · · ( z − ζ r ) , where ζ , . . . , ζ r ∈ O ∗ ⊂ K ∗ are the primitive r -th roots of unity. We now define aring homomorphism ϕ : O [Λ] → O × r f ( f ( ζ ) , f ( ζ ) , . . . , f ( ζ r )) . The result will follow if we can prove that ϕ is an isomorphism.It suffices to prove that ϕ is surjective as an O -module homomorphism. For i = 1 , . . . , r , set f i = Q j = i ( z − ζ j ). Then f ( ζ j ) = 0 for all j = i and f i ( ζ i ) = Y j = i ( ζ i − ζ j ) . Our assumption p ∤ r implies that the reductions ζ , ζ , . . . ζ r ∈ F = O / h ω i are alldistinct (where ω ∈ O is the uniformizer). In particular, for i = j , ζ i − ζ j = 0, andhence, ω ∤ ( ζ i − ζ j ) so that ( ζ i − ζ j ) is invertible in O . Thus, f i ( ζ i ) ∈ O is alsoinvertible for all i .Finally, if ǫ i ∈ O × r denotes the i -th coordinate function for i = 1 , . . . , r , then ϕ ( f i ) = f i ( ζ i ) ǫ i for all i . The invertibility of the f i ( ζ i ) imply that the f i ( ζ i ) ǫ i forma basis for the free O -module O × r , and therefore, ϕ is surjective. (cid:3) Following the convention from [Mc3, p. 5], we say that a subgroup scheme S ⊆ H of an O -group scheme H is a maximal torus if the subgroup schemes S K ⊆ H K and S F ⊆ H F are both maximal tori. Lemma 3.3.
Let C F be a nilpotent orbit, then there exists a balanced nilpotentsection x ∈ g O such that x F ∈ C F and S F = ( T F ∩ G x F ) ◦ is a maximal torus for G x F . N THE CENTRALIZER OF A BALANCED NILPOTENT SECTION 9
Proof.
By [Mc3, Theorem 1.2.1(a)], any x F ∈ C F lifts to some balanced nilpotentsection x ∈ g O . Suppose now that y F ∈ C F is arbitrary, and let y ∈ g O , be acorresponding balanced nilpotent section which lifts y F . If S ′ F ⊆ G y F F is a maximaltorus for the centralizer, then there must exist a maximal torus T ′ F of G F suchthat S ′ F ⊆ T ′ F . Now since the maximal tori for G F are conjugate, then there exists g ∈ G F ( F ) such that T F = gT ′ F g − . Let x F = g − y F g and fix a balanced lift x ∈ g O ,then x F ∈ C F and S F = g − S ′ F g ⊆ T F , must also be a maximal torus in G x F F .It will follow that S F = ( T F ∩ G x F ) ◦ provided T F ∩ G x F is reduced. To see why thisis true, let us first fix the root space decomposition g F = t F ⊕ M α ∈ Φ ⊂ X ( g F ) α , so that the structure of g F as a T F -module is given by the comodule map∆ g F : g F → F [ T ] ⊗ g F v X µ ∈ Φ ∪{ } µ ( t ) ⊗ v µ , where v µ denotes the projection onto the µ weight space of g F . The centralizer of x F in T F is then determined by the abelian group X x F = X / h µ ∈ X | ( x F ) µ = 0 i , (i.e. T ∩ G x F ∼ = Diag( X x F )). From the properties of diagonalizable group schemes,this is reduced if and only if X x F contains no p -torsion. However, the latter propertyfollows immediately from [He, Definition 2.11 and Theorem 5.2]. (cid:3) The following proposition will enable us to relate the torus characters between G x K and G x F representations. Proposition 3.4.
Let x ∈ g O and S F ⊆ G x F be as in Lemma 3.3, then S F lifts to asplit torus S ⊆ G x such that S K ⊆ G x K is a maximal torus which is split.Proof. First observe that since S F ⊆ G x F is irreducible, and Dist( G x F ) = Dist( G x tf , F )by (2.5), then from [J1, I.7.17(7)], it follows that S F ⊆ G x tf , F . The fact that G x tf is smooth by Proposition 2.7 now allows us to apply [Mc3, Theorem 2.1.1(c)] tolift the embedding ϕ : G r m ֒ → G x tf , F (corresponding to the inclusion S F ⊆ G x tf , F ), toa map ψ : G r m → G x tf ⊆ G x . Moreover, by the same reasoning as in the proof of[AHR2, Lemma 4.1], it can be verified that ψ is actually a closed embedding.Thus, if we set S = ψ ( G r m ) ⊆ G x tf , where S ∼ = G r m , then S K = ψ K ( G r m ) ⊆ G x K is asplit rank r -torus contained in G x K . Finally, since the maximal tori for G x K and G x F have the same rank by [Mc3, Corollary 9.2.2], then S K must also be maximal. (cid:3) Remark . In particular, S K ⊆ ( G x K ) ◦ and S F ⊆ ( G x F ) ◦ are maximal tori for theconnected components of the geometric fibers.In order to reduce our component group calculations to the case of diagonalizablegroup schemes, we will need the following proposition. Proposition 3.6.
Let Z = Z ( G ) denote the center of G , let Z ′ ⊆ Z be anydiagonalizable subgroup scheme, and let S ⊆ G be any split O -torus, then S ∩ Z ′ ⊆ G is diagonalizable. Proof.
First recall that the center Z is diagonalizable (see [J1, II.1.6, II.1.8]). Thestrategy of the proof will be to first construct a split O -torus T ′ ⊆ G with S ⊆ T ′ and Z ′ ⊆ T ′ , this will allow the intersection of S ∩ Z ′ to be taken inside of T ′ . Theresult will then follow from Lemma 3.1.To construct T ′ , first set H = C G ( S ), where C G ( S ) denotes the centralizer of S in G . By [Mc3, Proposition 2.2.1], H is a smooth reductive group scheme over O with connected fibers. Now let T ′ F ⊆ H F be a maximal split torus (of rank- r ) for H (recall that F = F by our assumption, so such a torus exists). As in the proofof Proposition 3.4, T ′ F can be lifted to a rank- r split torus of H . Let T ′ denote thislift. Thus, T ′ K ⊆ H K is a rank- r split torus of H K . By [Mc3, Corollary 2.1.2], themaximal tori for H K and H F must have the same dimension, and hence the samerank if they are split. Therefore, T ′ K is maximal in H K .Let Z ( H ) ⊆ H denote the center of H . By definition, S ⊆ Z ( H ) and Z ⊆ Z ( H )(and hence Z ′ ⊆ Z ( H )). For k ∈ { K , F } ,(3.3) Z ( H ) k = Z ( H k ) ⊆ T ′ k , where the containment on the right holds because T ′ k ⊆ H k is a maximal torus overa field and Z ( H ) k is the center. So in particular, S k ⊆ T ′ k .To prove S ⊆ T ′ , first consider the inclusion ϕ : S ∩ T ′ ֒ → S, and observe that S ⊆ T ′ if and only if ϕ is an isomorphism. Since base-changecommutes with taking fiber products, and thus commutes with taking intersections,the base-changes of ϕ to k , ϕ k : ( S ∩ T ′ ) k → S k , coincide with the inclusion S k ∩ T ′ k ֒ → S k . Thus, by (3.3), ϕ k is an isomorphism(i.e. is surjective). On the level of algebras, O [ S ∩ T ′ ] = O [ S ] ⊗ O [ H ] O [ T ′ ] wherethe comorphism ϕ ∗ is surjective, so it suffices to show that ϕ ∗ is injective (as an O -module morphism). Base changing to K induces a commutative diagram O [ S ] ϕ ∗ −−−−→ O [ S ] ⊗ O [ H ] O [ T ′ ] ψ y y ψ ′ K [ S ] ϕ ∗ K −−−−→ ∼ K [ S ] ⊗ K [ H ] K [ T ′ ] . The diagonalizability of S implies that O [ S ] is torsion-free, and hence ψ = 1 ⊗ idmust be injective. Thus ϕ ∗ K ◦ ψ is injective, and by commutativity, ψ ′ ◦ ϕ ∗ is injective.It then follows that ϕ ∗ must be injective, and is therefore an isomorphism.By a similar argument, we can show that the map Z ′ ∩ T ′ ֒ → Z ′ is an isomorphism(using the fact that Z ′ is diagonalizable), and thus Z ′ ⊆ T ′ .We have established that S ⊆ T ′ and Z ′ ⊆ T ′ . This allows us to identify S ∩ Z ′ = S × T ′ Z ′ . Thus, it follows from Lemma 3.1 that S ∩ Z ′ is diagonalizable. (cid:3) Component groups of the geometric fibers.
We will now prove thatthe component groups A ( x K ) and A ( x F ) of the geometric fibers, have the samecardinality. N THE CENTRALIZER OF A BALANCED NILPOTENT SECTION 11
Lemma 3.7. If G is a simple group (with G F geometrically standard), then | A ( x K ) | = | A ( x F ) | .Proof. Let k ∈ { K , F } and denote H = G x for a balanced nilpotent section x ∈ g O and S ⊆ H as in Proposition 3.4, so that S k is a maximal torus for H ◦ k byRemark 3.5.Let G ′ = G ad and H ′ = G x ad , then the arguments in the proof of [LS, Lemma 2.33]can be used to show that there exists an isogeny1 → Z k → H k → H ′ k → , obtained by restricting the “covering space” isogeny1 → Z k → G k → G ′ k → , down to H k . (In particular, Z k ⊆ G k is the center of G k .)Now since H ◦ k gets mapped onto H ′◦ k , there exists a sequence(3.4) 1 → Z k H ◦ k H ◦ k → A ( x k ) → A ′ ( x k ) → , where A ′ ( x k ) = H ′ k H ′◦ k . We can also identify Z k H ◦ k H ◦ k ∼ = Z k Z k ∩ H ◦ k . By Remark 1.4and [MS, Theorem 36], | A ′ ( x K ) | = | A ′ ( x F ) | . Thus, by (3.4), it suffices to show | Z K ∩ H ◦ K | = | Z F ∩ H ◦ F | .However, since S k is a maximal torus, and Z k ∩ H ◦ k is central, then it followsthat Z k ∩ H ◦ k ⊆ S k . This gives, Z k ∩ H ◦ k = Z k ∩ S k . Observe that the intersection on the right handside actually arises from an intersection of O -group schemes since Z k and S k arethe (geometric) fibers of the subgroup schemes Z ⊆ G and S ⊆ G . In other words,( Z ∩ S ) k = Z k ∩ S k . By Proposition 3.6, Z ∩ S is a diagonalizable subgroup scheme of Z . Now observe Z ∼ = Diag(Λ) with Λ = X / Z Φ. The geometrically standard assumption implies that p ∤ | X / Z Φ | . In particular, identifying Z ∩ S = Diag(Λ ′ ) gives p ∤ | Λ ′ | since Λ ′ isa quotient of Λ. Thus, Lemma 3.2 now implies that Z ∩ S constant, and hence | ( Z ∩ S ) K | = | ( Z ∩ S ) F | . Therefore, we are done. (cid:3) Now we consider the case of semisimple groups.
Lemma 3.8.
Let G be a semisimple group (with G F geometrically standard), then | A ( x K ) | = | A ( x F ) | .Proof. In this proof k will denote either K or F .First suppose that G = G × · · · × G r , where the G i are simple (and hence geometrically standard over F since G is ge-ometrically standard of F by [Mc3, § G act independently on the corresponding factors of g O , then for any nilpotentsection x = ( x , . . . , x r ) ∈ ( g ) O × · · · × ( g r ) O , the centralizer G x satisfies G x = G x × · · · × G x r r , where G x r i ⊆ G i is the centralizer of x i . To simplify notation, set H i = G xi for all i , and set H = G x . Thus, H k = ( H ) k × · · · × ( H r ) k , and hence, H k is smooth if and only if ( H i ) k is smooth for all i . This implies that x ∈ g O is a balanced for G if and only if for all i , x i is balanced for G i . Thus, ifwe assume that x is balanced, then from the identity H ◦ k = ( H ) ◦ k × · · · × ( H r ) ◦ k , itfollows that A ( x k ) = A (( x ) k ) × · · · × A (( x r ) k ) . Therefore, | A ( x K ) | = | A ( x F ) | by Lemma 3.7.More generally, if G is an arbitrary semisimple group, then there exists an isogeny(3.5) 1 → Z ′ → r Y i =1 G i π −→ G → , for some central, diagonalizable subgroup scheme Z ′ (see [J1, II.1.6]). Since G ′ k is geometrically standard, then Z ′ k is both smooth by [He, Definition 2.11 andTheorem 5.2], and finite since G ′ k is a finite product of quasi-simple group schemes.In particular, Z ′ is constant.Let G ′ = Q ri =1 G i , and let x ′ = ( x ′ , . . . , x ′ r ) ∈ g ′ O be a balanced nilpotent section,and let x = dπ ( x ′ ) ∈ g O . Let H ′ denote the centralizer of x ′ , and let H denotethe centralizer of x . By the same argument as in [LS, Lemma 2.33], there exists anisogeny(3.6) 1 → Z ′ k → H ′ k → H k → , induced by base changing (3.5) to k , and restricting down to the centralizer H ′ k .Now since Z ′ k is smooth, and H ′ k is smooth since x ′ is balanced, then H k must alsobe smooth. It follows that the section x ∈ g O is also balanced.Observe now that the isogeny sends H ′◦ k onto H ◦ k , and thus induces a sequence1 → Z ′ k H ′◦ k H ′◦ k → A ′ ( x k ) → A ( x k ) → , where we identify Z ′ k H ′◦ k H ′◦ k ∼ = Z ′ k Z ′ k ∩ H ′◦ k .Without loss of generality, we can assume that x ′ ∈ g ′ O also satisfies the condi-tions of Lemma 3.3. Applying Proposition 3.4 to x ′ and G ′ now provides a split,maximal torus S ′ ⊆ H ′ . As in the proof of Lemma 3.7, the proof of this lemmawill follow by showing that Z ′ ∩ S ′ is both diagonalizable and constant. The formerproperty holds by Proposition 3.6, and the latter propery can be deduced fromLemma 3.2 since Z ′ is constant and diagonalizable, and hence any diagonalizablesubgroup scheme of Z ′ must also be constant. (cid:3) The preceding argument can also be extended to arbitrary reductive groups G with G F geometrically standard. Proposition 3.9.
Let G be a reductive group (with G F geometrically standard),then | A ( x K ) | = | A ( x F ) | . N THE CENTRALIZER OF A BALANCED NILPOTENT SECTION 13
Proof.
In this proof k will denote either K or F .Let G ′ = [ G, G ], then by [J1, II.1.18] there exists an isogeny(3.7) 1 → T ∩ T → G ′ × T π −→ G → , where T and T are tori and T ∩ T ⊆ Z ( G ′ × T ) is central, diagonalizable andfinite. Let Z ′ = T ∩ T . The arguments appearing immediately below (3.5) in theproof of Lemma 3.8, also imply that Z ′ is diagonalizable and constant.Let h O denote the Lie algebra of T , and note that there must exist a balancednilpotent section x ∈ g O which satisfies the conditions of Lemma 3.3, and is ofthe form x = dπ ( x ′ ) for some balanced nilpotent section x ′ ∈ g ′ O × h O , whichalso satisfies the conditions of Lemma 3.3. (This again follows from the argumentsappearing immediately below (3.6).)Let H ′ ⊆ G ′ × T be the centralizer of x ′ , and let H ⊆ G be the centralizer of x .Again, as in [LS, Lemma 2.33], there is an isogeny1 → Z ′ k → H ′ k → H k → , obtained from (3.7) by base-changing to k , and restricting down to H ′ k . Now justas in the proof of Lemma 3.8, it suffices to show that the finite groups Z ′ K H ′◦ K H ′◦ K ∼ = Z ′ K Z ′ K ∩ H ′◦ K , Z ′ F H ′◦ F H ′◦ F ∼ = Z ′ F Z ′ F ∩ H ′◦ F have the same order. However, this follows from the argument given in the lastparagraph of the proof of Lemma 3.8. (cid:3) Remark . In Theorem 3.12, it will be proven that the component groups are ac-tually isomorphic (not just of the same order). The isomorphism of the componentgroups for general reductive G with geometrically standard fiber G F was originallyclaimed in [Mc2, Theorem B], however an error was later found in the proof.3.3. Proof of Theorem 1.6.
In this subsection, we will establish the smoothnessof G x . The key step will be to show that G x tf , F contains all the connected componentsof G x F . (We are still maintaining our assumption that F = F .)Label the connected components of G x F by ( G x F ) i for i = 0 , . . . , m −
1, where m = | A ( x F ) | and ( G x F ) := ( G x F ) ◦ is the identity component subgroup scheme. Thenthe decomposition of G x F into its connected components induces the decomposition F [ G x F ] = F [( G x F ) ◦ ] × F [( G x F ) ] × · · · × F [( G x F ) m − ] , where F [( G x F ) ◦ ] ∼ = F [( G x F ) i ] for all i ≥ G x tf , F is smoothimplies F ⊗ O [ G x tf ] = F [( G x F ) ◦ ] × F [( G x F ) ] × · · · × F [( G x F ) k − ] , where 1 ≤ k ≤ m . In particular, F ⊗ O [ G x tf ] ⊆ F [ G x F ] and G x tf , F has precisely k connected components. Our goal is to show k = m .If necessary, let us enlarge O , and consequently K , by a finite extension (theassumption F = F implies that F remains unchanged), so that by Proposition 3.9(3.8) K [ G x K ] = K [( G x K ) ◦ ] × K [( G x K ) ] × · · · × K [( G x K ) m − ] . The integral closure of a (complete) DVR in a finite algebraic extension is a finitely-generated(complete) DVR (cf. [Se, Chap. 1, Proposition 8, and Chap. 2, Proposition 3]).
This decomposition is given by a set of orthogonal idempotents ǫ , ǫ , . . . , ǫ m − ∈ K [ G x K ] with ǫ i = ǫ i for i ≥ ǫ i ǫ j = 0 for i = j and ǫ + · · · + ǫ m − = 1. Now ǫ i = h i ω n i where h i ∈ O [ G x tf ] and n i ≥ i . Assume that each n i is chosen minimally sothat h i ω O [ G x tf ]. Thus,(3.9) ǫ i = ǫ i = ⇒ h i = ω n i h i . On the other hand, h i ω O [ G x tf ] implies that h i = 0 ∈ O [ G x tf ] /ω O [ G x tf ] ∼ = F [ G x tf , F ].But by (3.9), h i = 0 if n i > F [ G x tf , F ] has nononzero nilpotent elements since it is reduced), so n i = 0 for all i , and therefore, ǫ i ∈ O [ G x tf ] for all i .This gives an internal decomposition O [ G x tf ] = A × A × · · · A m − , where A i = ǫ i O [ G x tf ]. Tensoring with F gives an internal algebra decomposition F ⊗ O [ G x tf ] = F ⊗ A × F ⊗ A × · · · × F ⊗ A m − , since for all i , ǫ i = 0 and ǫ i = ǫ i , also ǫ i ǫ j = 0 for i = j and ǫ + · · · + ǫ m − = 1.Thus, G x tf , F has at least m connected components, and hence k = m . In particular, G x tf , F = G x F , and hence, G x tf = G x so G x is smooth in this case. Remark . We have just shown that Theorem 1.6 will hold if O is enlarged by aDVR O ⊂ O ′ where the residue field of O ′ is algebraically closed and the fractionfield of O ′ is large enough so that (3.8) holds. Proof of Theorem 1.6.
For the proof, we return to the setup from §
1, where theonly condition on O is that it is complete and F = O /ω .Let O denote the completion of the maximal unramified extension of O . Bydefinition, ω is also the uniformizer for O and O /ω O = F . It also possible toenlarge O by a finite purely ramified integral extension O ′ ⊇ O (cf. [Se, § K ′ of O ′ , where we note O ′ must have thesame residue field as O . By Remark 3.11, the base change G x O ′ is smooth, and inparticular, O ′ [ G x O ′ ] = O ′ ⊗ O [ G x ]is torsion-free. Observe that O ′ is torsion-free as an O -module, since O and O ′ areintegral domains and O ⊆ O ′ . The fact that O is a DVR now implies O ′ is alsoa flat O -module (since these properties are equivalent for discrete valuation rings).Thus, the functor O ′ ⊗ O − is exact, and the natural map(3.10) O [ G x ] ⊗ id −−−→ O ′ ⊗ O [ G x ]is injective. To see why (3.10) is injective, first let 0 = f ∈ O [ G x ], be arbitrary.There is a commutative diagram O f −−−−→ O [ G x ] y y ⊗ id O ′ ⊗ O f −−−−→ O ′ ⊗ O [ G x ] , where the bottom map is injective by exactness of O ′ ⊗ O − . N THE CENTRALIZER OF A BALANCED NILPOTENT SECTION 15
Now suppose that f is torsion, then since O f is cyclic and non-zero, we musthave O f ∼ = O /ω k for some k ≥
1. If ω ′ is the uniformizer of O ′ , then ( ω ′ ) r = ω forsome r ≥
1. Thus O ′ ⊗ O f ∼ = O ′ /ω r ∼ = O ′ / ( ω ′ ) rk , where rk ≥
1, and so must be non-zero. The injectivity of the bottom map inthe preceding diagram implies O ′ (1 ⊗ f ) ∼ = O ′ / ( ω ′ ) rk , therefore 1 ⊗ f must also benon-zero, and torsion. But this is not possible since O ′ ⊗ O [ G x ] is torsion-free.This allows us to identify identify O [ G x ] ⊆ O ′ ⊗ O [ G x ], which implies that O [ G x ]must be torsion-free, since O ′ ⊗ O [ G x ] is torsion-free and any non-zero O -submoduleof a torsion-free module is torsion-free. (cid:3) The component group scheme.
We will continue to maintain our assump-tion that F = F .By [Mi, Definition 13.12], for any field k over O , there exists a component groupscheme , denoted A k ( x ). This is defined to be the spectrum of the largest ´etalesubalgebra of k [ G x k ] (cf. [Mi, 13b]). Moreover, the coordinate algebra k [ A k ( x )] isnaturally a Hopf subalgebra of k [ G x k ]. The inclusion k [ A k ( x )] ֒ → k [ G x k ], induces asurjective group scheme homomorphism G x → A k ( x ). The connected component of G x k , denoted ( G x k ) ◦ , can now be defined as the normal subgroup scheme given bythe kernel of this morphism.It follows from [Mi, Proposition 13.18], that for any field extension k ′ ⊇ k , A k ′ ( x ) ∼ = A k ( x ) k ′ , and ( G x k ′ ) ◦ ∼ = ( G x k ) ◦ k ′ . In particular, ( A K ( x ) K )( K ) ∼ = A ( x K ) and ( A F ( x ))( F ) ∼ = A ( x F ), where the discretegroups on the right-hand-side were considered in § § x ∈ g O is a balanced nilpotent section, and K satisfies(3.8), then there exist orthogonal idempotents ǫ , . . . , ǫ m − ∈ O [ G x ] so that(3.11) O [ G x ] = ǫ O [( G x )] × ǫ O [( G x )] × · · · × ǫ m − O [( G x )] , where m = | A ( x ) K | = | A ( x ) F | .Therefore, the O -scheme G x has a decomposition(3.12) G x = ( G x ) ◦ ⊔ ( G x ) ⊔ · · · ⊔ ( G x ) m − , where ( G x ) i = Spec( ǫ i O [ G x ]) and ( G x ) i K and ( G x ) i F give the complete set of con-nected components for G x K and G x F respectively. Theorem 3.12.
Let x ∈ g O be a balanced nilpotent section, and suppose that O issuch that F = F and K satisfies (3.8) . If we let A ( x ) be the constant scheme definedby the subalgebra (3.13) O [ A ( x )] := m − X i =0 O ǫ i ⊆ O [ G x ] , where ǫ , . . . , ǫ m − ∈ O [ G x ] are as in (3.11) , then (1) for any field k over O , A ( x ) k ∼ = A k ( x ) , (2) O [ A ( x )] is a Hopf subalgebra of O [ G x ] , (3) ( G x ) ◦ is the kernel of the induced homomorphism G x ։ A ( x ) , so that A ( x ) ∼ = G x / ( G x ) ◦ and ( G x ) ◦ E G x is a normal subgroup scheme. Proof.
We begin by proving (1). Let us first consider the case where k = K , thenit suffices to show that K [ A ( x ) K ] = K ⊗ O [ A ( x )] ⊆ K [ G x k ] is the largest ´etalesubalgebra of K [ G x K ]. To see why this is the case, let R ⊆ k [ G x K ] be the maximal´etale subalgebra, so that by definition, R ⊇ K [ A ( x ) K ] (such a subalgebra alwaysexists). By [Mi, Proposition 13.8], K ⊗ R ⊆ K [ G x K ] also gives the maximal ´etalesubalgebra. Now, [Mi, Corollary 13.9] and [Mi, Lemma 13.4] implydim K R = dim K ( K ⊗ R ) = |{ connected components of G x K }| = m, where the rightmost equality follows from assumption (3.8). Thus, dim K R =dim K K [ A K ( x )], and therefore, R = K [ A K ( x )]. The same argument also show that F [ A ( x ) F ] is the maximal ´etale subalgebra of F [ G x F ]. Finally, suppose k is any field over O . Then either k ⊇ K or k ⊇ F , and in both cases [Mi, Proposition 13.8]implies that k [ A ( x ) k ] is the maximal ´etale subalgebra of k [ G x k ]. So we have verified(1).Now we will verify (2). Let ∆, σ and ε denote the coproduct, antipode andcounit for O [ G x ] respectively. It suffices to show∆( O [ A ( x )]) ⊆ O [ A ( x )] ⊗ O [ A ( x )] ,σ ( O [ A ( x )]) ⊆ O [ A ( x )] and ε ( O [ A ( x )]) = O . The third identity can be verifiedimmediately by observing that ε ( O [ A ( x )]) ⊆ O is an O -subalgebra of O , whichimplies equality since O cannot have any proper O -subalgebras.To verify the first two identities, we first note that for an O -algebra k , the Hopfalgebra structure of k [ G x k ] = k ⊗ O [ G x ] is given by ∆ k := id k ⊗ ∆, ε k := id k ⊗ ε , σ k := id k ⊗ σ . So, in particular, for k = K , the first two identities must hold for K ⊗ O [ A ( x )] by (1). To verify the second identity, it suffices to show S ( ǫ i ) ∈ O [ A ( x )]for all i since S is O -linear. However, the fact that O [ A ( x )] is torsion-free, allows usto identify O [ A ( x )] ⊂ K ⊗ O [ A ( x )], so that σ = ( σ K ) | O [ A ( x )] . Thus, if i is arbitrary,then σ ( ǫ i ) = σ K ( ǫ i ) = a ǫ + a ǫ + · · · + a m − ǫ m − ∈ ( K ⊗ O [ A ( x )]) ∩ σ ( O [ A ( x )])for some a , . . . , a m − ∈ K . But since σ is a morphism of O -algebras, and the ǫ , . . . , ǫ m − are pairwise orthogonal idempotents, then σ ( ǫ i ) = σ ( ǫ i ) = σ ( ǫ i )implies a ǫ + a ǫ + · · · + a m − ǫ m − = a ǫ + a ǫ + · · · + a m − ǫ m − , so a i = a i , and hence, a i ∈ { , } ⊂ O for all i . So the second identity is verified.Similarly, if i is arbitrary, then∆( ǫ i ) = ∆ K ( ǫ i ) = X ( j,k ) a jk ǫ j ⊗ ǫ k ∈ ( K ⊗ O [ A ( x )] ⊗ K ⊗ O [ A ( x )]) ∩ ∆( O [ A ( x )])for some a jk ∈ K . Now observe set of ǫ j ⊗ ǫ k gives a linearly independent set ofpairwise orthogonal idempotents for O [ G x ] ⊗ O [ G x ], and that ∆ is a morphismalgebras. Thus, ∆( ǫ i ) = ∆( ǫ i ) = ∆( ǫ i )implies X ( j,k ) a jk ǫ j ⊗ ǫ k = X ( j,k ) a jk ǫ j ⊗ ǫ k , N THE CENTRALIZER OF A BALANCED NILPOTENT SECTION 17 so that a jk = a jk , which forces a jk ∈ { , } ⊂ O . Therefore, O [ A ( x )] is a Hopfsubalgebra of O [ G x ].So the inclusion O [ A ( x )] ֒ → O [ G x ] of Hopf algebras, induces a surjective map ofgroup schemes G x ։ A ( x ) . Finally, let H be the kernel of this homomorphism. From the definition of thekernel of a group scheme homomorphism, we have O [ H ] = O [ G x ] / ( ǫ + · · · + ǫ m − ) O [ G x ] = O [( G x ) ◦ ] . Thus, H = ( G x ) ◦ , and in particular, ( G x ) ◦ is a normal subgroup scheme of G x . (cid:3) Now we can prove Theorem 1.8.
Prooof of Theorem 1.8.
Let us now return to our hypothesis on O , K and F from § O with the completionof its maximal unramified extension, which we denote by O . Now the residue fieldof O is the algebraic closure F of F , and let K ′′ ⊇ K be the fraction field of O . Alsolet O ′ be a finite integral extension of O (with fraction field K ′ ⊇ K ′′ ⊇ K ) suchthat the hypothesis of Theorem 3.12 is satisfied.Applying Theorem 3.12 to this setup, immediately implies A ( x K ′ ) ∼ = A ( x F ),where K ′ denotes the algebraic closure of K ′ . Finally, A K denote the componentgroup scheme, and noting that K ⊆ K ′ , then it follows from [Mi, Proposition 13.8]that A ( x K ) ∼ = A ( x K ′ ) as groups. Therefore, A ( x K ) ∼ = A ( x F ). (cid:3) Centralizers for the G × G m action Smoothness in the graded case.
For simplicity, we will introduce the no-tation G = G × G m . The scheme g is also equipped with a G action, where G m acts by the cohomological action , t · x = t − x for t ∈ G m ( k ) and x ∈ g ( k ) and any O -algebra k . This gives O [ g ] a non-negative, even grading which is generated indegree 2. Now, for any balanced nilpotent x ∈ g O , consider the centralizer G x , aswell as the centralizers for the base-changes G x K and G x F .Suppose now that x ∈ g O is a balanced nilpotent section, then by [Mc3, Theorem1.2.1], there exists an integral associated cocharacter φ x : G m −→ G x , such that φ x, K and φ x, F are the associated cocharacters arising from the Jacobson-Morozov triples for x K and x F respectively.Let Int : ( G x ) op −→ Aut( G x ), with Int h ( g ) = hgh − for h, g ∈ G x ( k ) be a mor-phism of group schemes. An action of G m on G x can then be given by(4.1) t · g = φ x ( t − ) gφ x ( t ) = Int φ x ( t − ) ( g ) , for g ∈ G x ( k ) and any O -algebra k . Let G m ⋉ φ x G x , be the semi-direct product formed from this action. Recall that g O := g ( O ). Proposition 4.1.
There is an isomorphism of group schemes G m ⋉ φ x G x ∼ −→ G x ⊆ G × G m , given by t ⋉ g ( gφ x ( t − ) , t − ) for any g ∈ G ( k ) , t ∈ G m ( k ) and any O -algebra k .Proof. Let us first note that this map is well-defined and natural for any O -algebra k . Now, letting k be arbitrary and identifying x ∈ g ( k ) with its image under themorphism g ( O ) → g ( k ), we observe that ( g, t ) ∈ G x ( k ) ⊆ ( G × G m )( k ) if and onlyif x = ( g, t ) · x = Ad (1 ,t ) ◦ Ad ( g, x = t − Ad g x, or equivalently, Ad g x = t x. Now since φ x is an associated cocharacter, then x ∈ g with respect to its induced grading on g (equivalently Ad φ x ( t ) x = t x for any t ∈ G m ( k )). Thus, Ad g x = Ad φ x ( t ) x ⇐⇒ Ad gφ x ( t − ) x = x, and so gφ x ( t − ) ∈ G x ( k ). Therefore,( g, t ) ∈ G x ( k ) ⇐⇒ ( g, t ) = ( hφ x ( t ) , t ) for h = gφ x ( t − ) ∈ G x ( k ) . Therefore, this map is canonically a group isomorphism for every O -algebra k , andhence is an isomorphism of group schemes. (cid:3) Corollary 4.2.
The scheme-theoretic centralizers G x K and G x F are smooth. Now recall that if k is a field over O , then there exists an additional Levi-decomposition of G x k . Let G x k , red × G m act on G x k , unip by( g, t ) · u = gφ x ( t − ) uφ x ( t ) g − , for ( g, t ) ∈ ( G xred, k × G m )( k ′ ), u ∈ G x k , unip ( k ′ ) and any k -algebra k ′ . Let( G x k , red × G m ) ⋉ G x k , unip be the semi-direct product formed from this action. Corollary 4.3.
For k ∈ { K , F } , there is an isomorphism ( G x k , red × G m ) ⋉ G x k , unip ∼ −→ G x k given by ( g, t ) ⋉ u ( gφ x ( t − ) u, t − ) for ( g, t ) ⋉ u ∈ (cid:0) ( G x k , red × G m ) ⋉ G x k , unip (cid:1) ( k ′ ) and any k -algebra k ′ . In particular, G x k , red ∼ = G x k , red × G m and G x k , unip ∼ = G x k , unip .Proof. This follows by observing that for any k -algebra k ′ , this map is canonicallyequivalent to the one in Proposition 4.1, and is therefore an isomorphism. (cid:3) The following theorem is now immediate.
Theorem 4.4.
Let x ∈ g O be a balanced nilpotent section, and let k ∈ { K , F } , thenthe morphism G x → Spec( O ) is smooth and G x k / ( G x k ) ◦ ( k ) ∼ = A ( x k ) . Furthermore, if the conditions of Theorem 3.12 are satisfied, then the morphism G x ։ A ( x ) lifts to a group scheme homomorphism G x ։ A ( x ) , where the basechanges A ( x ) k also give the component group schemes for G x k (i.e. there are groupscheme isomorphisms G x k / ( G x k ) ◦ ∼ = G x k / ( G x k ) ◦ . ) N THE CENTRALIZER OF A BALANCED NILPOTENT SECTION 19
Integral lattices.
For a balanced nilpotent section x ∈ g O , let H ∈ { G x , G x } ,then the flatness of H now implies that for any finite-dimensional H K -module V ,there exists an H -stable O -lattice M ⊂ V (cf. [J1, I.10.4]). Lemma 4.5.
Let k ∈ { K , F } . If N is a G x k = G m ⋉ φ x G x k -module such that ( t ⋉ g ) · n = (1 ⋉ g ) · n for any t ⋉ g ∈ ( G m ⋉ φ x G x k )( k ′ ) , n ∈ N ( k ′ ) and any k -algebra k ′ , then G x k , unip acts trivially on N .Proof. If we assume the hypothesis, then since G x k , G m and G x k are all reduced byCorollary 4.2, it suffices to show that G x k , unip ( k ) acts trivially on N ( k ) = N ⊗ k k .So without loss of generality, assume that k = k , so that it suffices to work withthe geometric points.The restriction of N to G x k corresponds to a group variety homomorphism ψ : G x k −→ GL ( N ) , and our goal is show ψ ( G x k , unip ) = { } . However, sinceLie ψ ( G x k , unip ) = dψ (Lie G x k , unip ) , then it suffices to show dψ (Lie G x k , unip ) = 0.The fact that N is a ( G m ⋉ φ x G x k )-module is equivalent to saying that ψ is G m -equivariant, where G m y G x k via (4.1) (e.g. by Int φ x ( t − ) ), and that G m y GL ( N )trivially by the hypothesis. Moreover, there are induced actions of G m on therespective Lie algebras (given by the adjoint action), so that dψ : g x k −→ gl ( N )is also G m -equivariant. Equivalent the respective Lie algebras are given gradings g x k = M k ∈ Z ( g x k ) k , gl ( N ) = M k ∈ Z gl ( N ) k , where the G m -equivariance of dψ induces a decomposition dψ = L k ∈ Z dψ k suchthat dψ k : ( g x k ) k −→ gl ( N ) k . Observe now that gl ( N ) = gl ( N ) and dψ k = 0 for all k = 0, since the action of G m on GL ( N ), and hence on gl ( N ), is trivial. By (4.1), the action of any t ∈ G m is given by the automorphism d (Int φ x ( t − )) = Ad φ x ( t − ) : g x k −→ g x k , and thus by [J2, Proposition 5.10],Lie( G x k , red ) = ( g x k ) , Lie( G x k , unip ) = M k< ( g x k ) k , with ( g x k ) k = 0 for all k >
0. Finally, we see that since dψ k = 0 for all k = 0, then dψ (Lie G x k , unip ) = 0. (cid:3) Remark . The reason for the sign difference between the Z -grading on g x k in thepreceding proof, and the Z -grading in [J2, Proposition 5.10] is due to the fact thatthe grading here is induced from the action (4.1), which gives Ad φ x ( t − ) , while thecited proposition is with respect to the inverse action given by Ad φ x ( t ) for t ∈ G m . Proposition 4.7. If V is any G x K -module which factors through G x K , red , then thereexists a G x -stable O -lattice M ⊂ V such that M F factors through G x F , red (i.e. G x F , unip acts trivially on M F ). Proof.
By Corollary 4.3, V can be lifted to a module for G x K which has trivial G m and G x K , unip actions. Comparing with Proposition 4.1, we observe that the restrictionof V to ( G m ⋉ φ x K ⊆ ( G m ⋉ φ x G x K ) K is also trivial. Now Theorem 4.4 ensures that G x is smooth (and hence flat over O ),therefore there must exist a G x -stable O -lattice M ⊂ V , and by Proposition 4.1, itfollows that M has the structure of a G m -module by restricting to G m ⋉ φ x
1. Thisstructure is the equivalent to giving M a grading M = L k ∈ Z M k . And moreover, V = M ⊗ K = M k ∈ Z ( M k ⊗ K ) , where V k = M k ⊗ K is the grading arising from the ( G m ⋉ φ x K module structure.Since this module structure is trivial, then it must be the case that V = V which,by the fact that M is free, implies M k = 0 for all k = 0. Finally, by base-changingto F , it can be observed that the ( G m ⋉ φ x F module structure on M F is given by M F = M ⊗ K = M k ∈ Z ( M k ⊗ F ) = M ⊗ F . This implies that M F satisfies the conditions of Lemma 4.5, and therefore, it factorsthrough G x F , red . (cid:3) References [AHR1] P. Achar, W. Hardesty, and S. Riche,
Integral exotic sheaves and the modular Lusztig–Vogan bijection , in preparation.[AHR2] P. Achar, W. Hardesty, and S. Riche,
Representation theory of disconnected reductivegroups , in preparation.[B] J. Booher,
Geometric Deformations of Orthogonal and Symplectic Galois Representa-tions , Ph.D. thesis, Stanford University, 2016.[GM] P. Gille, L. Moret-Bailly,
Actions alg´ebriques de groupes arithm´etiques , Torsors, ´etalehomotopy and applications to rational points, 231-249, London Math. Soc. Lecture NoteSer., 405, Cambridge Univ. Press, Cambridge, 2013.[Ht] R. Hartshorne,
Algebraic geometry , Graduate Texts in Mathematics, no. 52, Springer-Verlag, New York, 1977.[He] S. Herpel,
On the smoothness of centralizers in reductive groups , Trans. Amer. Math.Soc. (2013), 3753–3774.[J1] J. C. Jantzen,
Representations of algebraic groups, second edition , Mathematical surveysand monographs 107, Amer. Math. Soc., 2003.[J2] J. C. Jantzen,
Nilpotent orbits in representation theory , in
Lie theory , 1–211, Progr.Math. 228, Birkh¨auser Boston, 2004.[LS] M. Liebeck, G. Seitz,
Unipotent and nilpotent classes in simple algebraic groups and Liealgebras , Mathematical Surveys and Monographs 180, American Mathematical Society,2012.[Mc1] G. McNinch,
Nilpotent orbits over ground fields of good characteristic , Math. Ann. 329(2004), no. 1, 49-85.[Mc2] G. McNinch,
The centralizer of a nilpotent section , Nagoya J. Math 190 (2008), pp.129-181.[Mc3] G. McNinch,
On the nilpotent orbits of a reductive group over a local field , preprint,2016.[MS] G. McNinch and E. Sommers,
Component groups of unipotent centralizers in good char-acteristic , J. Algebra 260 (2003), pp. 323-337.[Mi] J. S. Milne,
Algebraic Groups, Lie Groups, and their Arithmetic Subgroups
Local fields , Graduate Texts in Mathematics, vol. 67, Springer-Verlag, NewYork-Berlin, 1979, Translated from the French by Marvin Jay Greenberg.
N THE CENTRALIZER OF A BALANCED NILPOTENT SECTION 21 [St] The Stacks Project Authors,
Stacks project , http://stacks.math.columbia.edu, 2018.
Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803,U.S.A.
E-mail address ::