Birational geometry for the covering of a nilpotent orbit closure
aa r X i v : . [ m a t h . AG ] J u l Birational geometry for the covering of a nilpotent orbit closureYoshinori NamikawaIntroduction
Let g be a complex semisimple Lie algebra and let O be a nilpotent orbit of g . Then O admits a symplectic 2-form ω KK called the Kirillov-Kostant 2-form. In general O is notsimply connected, but π ( O ) is finite. Let π : X → O be a finite etale covering. Thefunction field C ( X ) of X is a finite algebraic extension of C ( O ). Let ¯ O be the closureof O in g . The normalization X of ¯ O in C ( X ) determines a finite covering π : X → ¯ O .Then X is contained in X as a Zariski open subset so that π | X = π . If G is a simplyconnected complex semisimple Lie group with Lie ( G ) = g , then the adjoint G -actionon O (resp. ¯ O ) lifts to a G action on X (resp. X ). The 2-form ω := ( π ) ∗ ω KK is a G -invariant symplectic 2-form on X . The pair ( X, ω ) is then a symplectic variety in thesense of Beauville [Be]. Brylinski and Kostant [B-K] extensively studied such varieties inthe context of shared orbits. A nilpotent orbit closure ¯ O has a natural scaling C ∗ -actionfor which ω KK has weight 1. Let s : C ∗ → Aut( ¯ O ) be a homomorphism determined bythe scaling action. In general this C ∗ -action does not lift to a C ∗ -action on X . But, ifwe instead define a new C ∗ -action on ¯ O by the composite σ of C ∗ → C ∗ ( t → t ) , and s : C ∗ → Aut( ¯ O ) , then the new C ∗ -action σ always lifts to a C ∗ -action on X by [B-K, § wt ( ω ) = 2 with respect to this C ∗ -action. Therefore ( X, ω ) is a conical symplectic G -variety with wt ( ω ) = 2.A main purpose of this article is to study the birational geometry for the resolutionsof ( X, ω ). A crepant projective resolution f : Y → X of X is, by definition, a projectivebirational morphism f from a nonsingular variety Y to X such that K Y = f ∗ K X . Ingeneral X does not have a crepant projective resolution. But, instead, X always has anice crepant projective partial resolution f : Y → X called a Q -factorial terminalization by [BCHM]. A Q -factorial terminalization f is, by definition, a projective birationalmorphism from a normal variety Y to X such that Y has only Q -factorial terminalsingularities and K Y = f ∗ K X . It is natural to expect that such a Q -factorial terminal-ization can be constructed very explicitly in a group theoretic manner when ( X, ω ) isthe above.When X is the nilpotent orbit O itself, X is nothing but the normalization ˜ O of ¯ O .Namikawa [Na 1] and Fu [Fu 1] respectively constructed a Q -factorial terminalization of˜ O quite explicitly when g is a classical simple Lie allgebra and when g is an exceptionalsimple Lie algebra (See also [Lo] for a unified treatment). However, when deg( π ) > π ) is odd. 1n this article we construct a Q -factorial terminalization of X when O is a nilpotentorbit of a classical simple Lie algebra g and X is the universal covering of O . We shallexplain here a basic idea for the construction. Let Q ⊂ G be a parabolic subgroup of G and let Q = U · L be a Levi decomposition of Q by the unipotent radical U and aLevi subgroup L . Correspondingly the Lie algebra q decomposes q = n ⊕ l as a directsum of n := Lie ( U ) and l := Lie ( L ). Let O ′ be a nilpotent orbit of l . Then there isa unique nilpotent orbit O of g such that O meets n + O ′ in a Zariski open subset of n + O ′ . In such a case we say that O is induced from O ′ and write O = Ind gl ( O ′ ). Thereis a generically finite map µ : G × Q ( n + ¯ O ′ ) → ¯ O ([ g, z ] → Ad g ( z )) , which we call a generalized Springer map. Let ( X ′ ) → O ′ be an etale covering (whichis not necessarily the universal covering) and let X ′ → ¯ O ′ be the associated finite cover.Then we can consider the space n + X ′ which is, by definition, a product of an affinespace n and the affine variety ¯ O ′ . There is a finite cover n + X ′ → n + ¯ O ′ . If the Q -actionon n + ¯ O ′ lifts to a Q-action on n + X ′ , then we can make G × Q ( n + X ′ ) and get acommutative diagram G × Q ( n + X ′ ) µ ′ −−−→ Z π ′ y y G × Q ( n + ¯ O ′ ) µ −−−→ ¯ O, (1)where Z is the Stein factorization of µ ◦ π ′ . In this article we will find suitable Q , O ′ and X ′ for an arbitrary nilpotent orbit O of a classical Lie algebra g so that(1) O = Ind gl ( O ′ ),(2) X ′ has only Q -factorial terminal singularities, and(3) the Q-action on n + ¯ O ′ lifts to a Q-action on n + X ′ and the finite covering Z → ¯ O in the diagram coincides with the finite covering π : X → ¯ O associated with theuniversal covering X of O .Then µ ′ gives a Q -factorial terminalization of X . It is easy to see if G × Q ( n + X ′ )is nonsingular or not (cf. Lemma (1.6)). If it is nonsingular, µ ′ is a crepant projectiveresolution of X . If it is singular, then any other Q -factorial terminalizations of X arealso singular by [Na 3, Corollary 25]; hence X has no crepant projective resolution insuch a case.In the present article, we have not yet answered how many different Q -factorialterminalizations X has. When a Q -factorial terminalization has a form of T ∗ ( G/Q ) wealready know the answer by [Na 4, Corollary 3.4]. It would be interesting to generalizethe method in [Na 4] to have an answer in a general case.
Preliminaries (P.1)
Symplectic varieties
Let X be a normal variety over C . Let ω be a regular 2-form on X reg . Then ( X, ω )is a symplectic variety if(a) ω is non-degenerate and d -closed, and2b) for a resolution f : ˜ X → X of X , the 2-form f ∗ ω on f − ( X reg ) extends to aregular 2-form on ˜ X .A symplectic variety X has only canonical singularities; hence it has only rationalsingularities. The following properties of a symplectic variety will be frequently used inthis article. Proposition (0.1) . Let ( X, ω ) be a symplectic variety of dim 2 n and let f : ˜ X → X be a resolution of X . Write K Y = f ∗ K X + P a i E i with f -exceptional prime divisors E i (each coefficient a i being called the discrepancy of E i ). Let E i be an f -exceptionalprime divisor with a i = 0 . Then dim f ( E i ) = 2 n − .Proof . Put S := f ( E i ). We blow up ˜ X further to get a projective birationalmorphism ν : Z → ˜ X such that F := ( f ◦ ν ) − ( S ) is a simple normal crossing divisorof Z . F contains the proper transform F of E i by ν as an irreducible component. Bydefinition ω lifts to a regular 2-form ˜ ω on Z . Since the discrepancy of F is zero, we seethat ∧ n − ˜ ω | F = 0. Put ˆΩ pF := Ω pF /τ p , where τ p is the subsheaf of Ω pF consisting of thesections supported on Sing( F ). Then ˜ ω | F determines a nonzero element of H ( F, ˆΩ F ).Take a (non-empty) smooth open set U of S so that the fiber F x of ( f ◦ ν ) | F : F → S over x ∈ U is a simple normal crossing variety for any x ∈ U . Replace S by U and F by( f ◦ ν | F ) − ( U ). There is an exact sequence0 → F → ˆΩ F → ˆΩ F/S → → ( f ◦ ν ) | ∗ F Ω S → F → ( f ◦ ν ) | ∗ F Ω S ⊗ ˆΩ F/S → . We derive a contradiction by assuming that i := Codim X S ≥
3. ˜ ω | F ∈ H ( F, ˆΩ F )cannot be written as the pull-back of a 2-form ω S on S . In fact, since dim S ≤ n − ∧ n − ω S = 0. Then ∧ n − ˜ ω | F = 0, which means that ∧ n − ˜ ω | F = 0. This contradicts that ∧ n − ˜ ω | F = 0. Since ˜ ω | F ∈ H ( F, ˆΩ F ) is not the pull-back of a 2-form on S , we see that H ( F x , ˆΩ F x ) = 0 or H ( F x , ˆΩ F x ) = 0 for a general point x ∈ S by the exact sequencesabove. On the other hand, since ( X, x ) is a rational singularity, we have H ( F x , ˆΩ pF x ) = 0for all p > X ( S ) = 2. (cid:3) Corollary (0.2) . Let π : Y → X be a crepant partial resolution of a symplecticvariety X of dim 2 n and let E be a π -exceptional prime divisor. Then dim π ( E ) = 2 n − .Proof . Let ν : ˜ X → Y be a resolution and let ˜ E be the proper transform of E by ν .Put f = π ◦ ν . Then the discrepancy of ˜ E is 0. By Proposition (0.1) dim f ( ˜ E ) = 2 n − (cid:3) Corollary (0.3) . Let ( X, ω ) be a symplectic variety of dim 2 n with Codim X Sing( X ) ≥ . Then X has only terminal singularities.Proof . Let f : ˜ X → X be a resolution of X such that f | f − ( X reg ) : f − ( X reg ) ∼ = X reg .Assume that X does not have terminal singularities. Then there is an f -exceptionalprime divisor F such that its discrepancy is 0. By Proposition (0.1), dim f ( F ) = 2 n − X Sing( X ) ≥ (cid:3) Let (
X, ω ) be an affine symplectic variety with R = Γ( X, O X ). Assume that R is apositively graded ring R = ⊕ i ≥ R i with R = C . This means that X has a C ∗ -action3uch that the closed point 0 ∈ X corresponding to the maximal ideal m R := ⊕ i> R i is aunique fixed point of the C ∗ -action. If ω is homogeneous with respect to this C ∗ -action(i.e. t ∗ ω = t l ω for some integer l and for t ∈ C ∗ ), then we call ( X, ω ) a conical symplecticvariety . By the property (b) of a symplectic variety, the weight l is a positive integer.(P.2) Nilpotent orbits and their finite coverings
Let O ⊂ g be a nilpotent orbit of a complex semisimple Lie algebra g . Take a simplyconnected complex algebraic group G with Lie ( G ) = g . Let π : X → O be a finite etalecovering. Then the G -action on O extends to a G -action on X and π is a G -equivariantcovering. Let ¯ O be the closure of O ⊂ g . We can extend π to a finite covering map π : X → ¯ O . Since X = SpecΓ( X , O X ) and G acts on X , G acts on X in such a waythat π is a G -equivariant.For a point x ∈ O , let G x ⊂ G be the stabilizer group of x . We take the universalcovering of O as X and pick a point ˜ x ∈ X such that π (˜ x ) = x . Let G ˜ x be the stabilizergroup of ˜ x for the G -action on X . Then G ˜ x coincides with the identity component of G x . Therefore π ( O ) ∼ = G x / ( G x ) . By the Jacobson-Morozov theorem we take an sl (2)-triple x , y and h in g . We denote by φ the map sl (2) → g determined by the sl (2)-triple.Put g φ := { z ∈ g | [ z, x ] = [ z, y ] = [ z, h ] = 0 } . Obviously g φ ⊂ g x . Let u x be the nilradical of g x . Note that u x is the Lie algebra ofthe unipotent radical of G x . By Barbasch-Vogan and Kostant (cf. [C-M], Lemma 3.7.3),there is a direct sum decomposition g x = u x ⊕ g φ . Correspondingly we have a semi-directproduct G x = U x · G φ . Moreover, the inclusion G φ → G x induces an isomorphism G φ / ( G φ ) ∼ = G x / ( G x ) , which, in particular, means that π ( O ) ∼ = G φ / ( G φ ) .A nilpotent orbit closure ¯ O has a natural scaling C ∗ -action for which ω KK has weight1. Let s : C ∗ → Aut( ¯ O ) be a homomorphism determined by the scaling action. Ingeneral this C ∗ -action does not lift to a C ∗ -action on X . But, if we instead define a new C ∗ -action on ¯ O by the composite σ of C ∗ → C ∗ ( t → t ) , and s : C ∗ → Aut( ¯ O ) , then the new C ∗ -action σ always lifts to a C ∗ -action on X by [B-K, § wt ( ω ) = 2 with respect to this C ∗ -action. π is etale in codimension one becauseCodim ¯ O ¯ O − O ≥
2. The map π factors through the normalization ˜ O of ¯ O . Since ˜ O hascanonical singularities, X also has canonical singularities. Put ω := π ∗ ω KK . Then thismeans that ( X, ω ) is a symplectic variety. Since wt ( ω ) = 2, ( X, ω ) is a conical symplecticvariety. § g = sl ( d )A nilpotent orbit O of sl ( d ) is uniquely determined by its Jordan type. If the Jordannormal form of x ∈ O has j Jordan blocks of size d , j Jordan blocks of size d , ...,and j k Jordan blocks of size d k , then the Jordan type of O is a partition [ d j , ..., d j k k ] of4 . In the remainder we assume that d > d > ... > d k . We indicate by O [ d j ,...,d jkk ] thenilpotent orbit with Jordan type [ d j , ..., d j k k ].We first consider the case when O is the regular nilpotent orbit O [ d ] of sl ( d ). In thiscase ¯ O [ d ] is the nilpotent cone of sl ( d ). Proposition (1.1) .(1) π ( O [ d ] ) ∼ = Z /d Z .(2) X is Q -factorial for any etale covering π : X → O [ d ] .Proof . Take x ∈ O [ d ] and consider an sl (2)-triple φ .(1) As already remarked above, π ( O [ d ] ) = G φ / ( G φ ) . By Springer and Steinberg (cf.[C-M], Theorem 6.1.3), one has G φ ∼ = { ( ζ , ..., ζ ) ∈ GL (1) d | ζ d = 1 } ( ∼ = Z /d Z ) . (2) It is enough to prove that Pic( X ) is a finite group for X . X can be written as G/H with a subgroup H with ( G x ) ⊂ H ⊂ G x . Then we have an exact sequence (cf.[K-K-V, Proposition 3.2]) χ ( H ) → Pic(
G/H ) → Pic( G ) , where χ ( H ) = Hom( H, C ∗ ). Since Pic( G ) is finite, we need to show that χ ( H ) is finite.By the exact sequence 1 → U x → ( G x ) → ( G φ ) → χ (( G φ ) ) → χ (( G x ) ) → χ ( U x ) . Since ( G φ ) = 1, the 1-st term is zero. The 3-rd term is zero because U x is unipotent.Hence χ (( G x ) ) = 0. Now, by the exact sequence χ ( H/ ( G x ) ) → χ ( H ) → χ (( G x ) )we see that χ ( H ) is finite because χ ( H/ ( G x ) ) is finite. Proposition (1.2) . Assume that π : X → O [ d ] is the universal covering. Then Codim X Sing( X ) ≥ .In particular, X has only terminal singularities.Proof . Take a point z from the subregular nilpotent orbit O [ d − , and let S be a(complex analytic) transverse slice for O [ d − , ⊂ ¯ O [ d ] at z . Then S is a surface withan A d − -singularity at z . This means that the complex analytic germ ( S, z ) is a 2dimensional quotient singularity V d , where V d := ( C / ( Z /d Z ) , . Here ¯1 ∈ Z /d Z acts on C by ( x, y ) → ( ζ x, ζ − y )5ith ζ a primitive d -th root of unity. Note that, for any e with e | d , the quotient map( C , → V d factorizes as ( C , → V e → V d .The inclusion map S − { z } → O [ d ] induces a homomorphism π ( S − { z } ) → π ( O [ d ] ).We prove that it is an isomorphism. Suppose to the contrary. Then π − ( S ) splits intomore than one connected components, each of which is a copy of V e with some divisor e ( = d ) of d ; namely, π − ( S ) = V (1) e ⊔ ... ⊔ V ( e ) e ,π − ( S − { z } ) = ( V (1) e − { } ) ⊔ ... ⊔ ( V ( e ) e − { } ) . If we put f := d/e , then V ( i ) e → S is a cyclic cover of degree f .First we shall construct a cyclic covering ν : Y → O [ d ] of degree e in such a way that ν is etale not only over O [ d ] but also over O [ d − , . Take a point p ∈ O [ d ] . Then π − ( p )consists of exactly d points { p , ..., p d } and π ( O [ d ] ) acts on them. We may renumberthese points so that(a) each V ( i ) e contains p j with j = i (mod e ), and(b) π ( O [ d ] )(= Z /d Z ) acts on these d points so that the generator ¯1 acts as thepermutation (1 , , ..., d ); namely, p → p , p → p , ..., p d − → p d , p d → p .The natural surjection Z /d Z → Z /e Z determines a Z /e Z -covering ν : Y → ¯ O [ d ] . Bydefinition ν − ( S ) splits up into e copies of V d . This is the desired cyclic covering.On the other hand, ¯ O [ d ] can be resolved by the cotangent bundle W := T ∗ ( SL ( d ) /B )of the full flag variety SL ( d ) /B . We call this resolution the Springer resolution anddenote it by s : W → ¯ O [ d ] . The Springer resolution s is a symplectic resolution; hence, s is a semi-small map. Put Z := ¯ O [ d ] − O [ d ] − O [ d − , . Then, by the semi-smallness of s , wehave Codim W s − ( Z ) ≥
2. Since π ( W ) = { } , we have π ( W − s − ( Z )) = { } . By theconstruction ν : Y → ¯ O [ d ] is etale over ¯ O [ d ] − Z . Then one can construct a (connected)non-trivial etale cover of W − s − ( Z ) by pulling back ν by the map W − s − ( Z ) → ¯ O [ d ] − Z .This contradicts that π ( W − s − ( Z )) = { } .In the following we construct a Q -factorial terminalization of X for an arbitrary etalecover π : X → O [ d ] . Let us begin with the simplest cases. Examples (1.3) . (1) When g = sl (2), π ( O [2] ) = Z / Z . Then X = C for theuniversal covering X → O [2] , and π : C → ¯ O [2] is the quotient map of Z / Z by theaction ( x , x ) → ( − x , − x ).(2) When g = sl (4), π ( O [4] ) = Z / Z . When X is the universal cover of O [4] , wealready know that X has only Q -factorial terminal singularities by Proposition (1.1),(2) and Proposition (1.2).Next assume that X is a double cover of O [4] . The nilpotent cone ¯ O [4] has a Springerresolution T ∗ ( SL (4) /Q , , , ). Here Q , , , is a parabolic subgroup of SL (4) stabilizinga flag 0 ⊂ F ⊂ F ⊂ F ⊂ C with dim Gr iF = 1 for all i . Let n , , , be the nilradicalof Q , , , . By using the Killing form of sl (4), we have an identification T ∗ ( SL (4) /Q , , , ) ∼ = SL (4) × Q , , , n , , , . Let Q , be the parabolic subgroup of SL (4) stabilizing the flag 0 ⊂ F ⊂ C . Then Q , has a Levi decomposition Q , = U · L, U is the unipotent radical of Q , and L is a Levi part of Q , . In our case U = { (cid:18) I ∗ I (cid:19) } .L = { (cid:18) A B (cid:19) | det( A )det( B ) = 1 } . Corresponding to the decomposition we have a direct sum decomposition of Liealgebras q , = n ⊕ l . In our case l = sl (2) ⊕ ⊕ z , where z is a 1-dimensional center of l . Take a nilpotent orbit closure ¯ O [2] × ¯ O [2] in sl (2) ⊕ ⊕ z . The parabolic subgroup Q , acts on q , . Since n is an ideal of q , , n isstable under the Q , -action. On the other hand, l is not stable. Let z ∈ l and q ∈ Q , .Then Ad q ( z ) decomposes into the sum of the nilradical part ( Ad q ( z )) n and the Levi part( Ad q ( z )) l . Let Q , → Q , /U = L be the quotient map and let ¯ q ∈ L be the image of q ∈ Q , by this map. The Levi subgroup L acts on l by the adjoint action. Then( Ad q ( z )) l = Ad ¯ q ( z ) . In particular, if z ∈ ¯ O [2] × ¯ O [2] , then ( Ad q ( z )) l ∈ ¯ O [2] × ¯ O [2] . Therefore Q , acts on n + ( ¯ O [2] × ¯ O [2] ). Then SL (4) × Q , ( n + ¯ O [2] × ¯ O [2] ) gives a crepant partial resolution of¯ O [4] . Moreover, the Springer resolution of O [4] factors through this partial resolution: SL (4) × Q , , , n , , , → SL (4) × Q , ( n + ¯ O [2] × ¯ O [2] ) → ¯ O [4] . We want to make a double cover X ′ of SL (4) × Q , ( n + ¯ O [2] × ¯ O [2] ) so that the diagram X ′ µ −−−→ X π ′ y π y SL (4) × Q , ( n + ¯ O [2] × ¯ O [2] ) −−−→ ¯ O [4] (2)commutes and µ gives a Q -factorial terminalization of X . By (1) we have a finitecovering C × C → ¯ O [2] × ¯ O [2] of degree 4. Then Z / Z acts on C (= C × C ) by ( x , x , y , y ) → ( − x , − x , − y , − y ).We denote by C / h +1 , − i the quotient space. The covering map above then factorsthrough C / h +1 , − i : C → C / h +1 , − i → ¯ O [2] × ¯ O [2] . We want to make n + C / h +1 , − i into a Q , -space and to define X ′ to be SL (4) × Q , ( n + C / h +1 , − i ). Claim (1.3.1) . The adjoint action of L on ¯ O [2] × ¯ O [2] lifts to an action on C / h +1 , − i . roof . As already remarked above, L = { (cid:18) A B (cid:19) | det( A )det( B ) = 1 } . Define a subgroup T of L by T = { (cid:18) λI λ − I (cid:19) | λ ∈ C ∗ } . We identify SL (2) × SL (2) with a subgroup of L by { (cid:18) A B (cid:19) | A, B ∈ SL (2) } . Then the inclusion map SL (2) × SL (2) ⊂ L induces an isomorphism SL (2) × SL (2) / h +1 , − i ∼ = L/T.SL (2) × SL (2) naturally acts on C ; hence SL (2) × SL (2) / h +1 , − i acts on C / h +1 , − i .As a consequence, L/T acts on C / h +1 , − i . In particular, L acts on C / h +1 , − i , whichis a lift of the adjoint action of L on ¯ O [2] × ¯ O [2] . Q.E.D.Now Q , acts on the space n + C / h +1 , − i as follows. Take a point z + v from thespace. Here z ∈ n and v ∈ C / h +1 , − i . We denote by ¯ v the image of v by the map C / h +1 , − i → ¯ O [2] × ¯ O [2] . For q ∈ Q , we denote by ¯ q ∈ L the image of q by the map Q , → Q , /U = L . We define q · ( z + v ) := ( Ad q ( z + ¯ v )) n + ¯ q · v ∈ n + C / h +1 , − i . Here ¯ q ∈ L acts on v ∈ C / h +1 , − i as described in Claim (1.3.1). Let us consider thecomposed map SL (4) × Q , ( n + C / h +1 , − i ) → SL (4) × Q , ( n + ¯ O [2] × ¯ O [2] ) → ¯ O [4] . The Stein factorization of this map is nothing but X . As a consequence we have acommutative diagram SL (4) × Q , ( n + C / h +1 , − i ) µ −−−→ X π ′ y π y SL (4) × Q , ( n + ¯ O [2] × ¯ O [2] ) −−−→ ¯ O [4] (3)We can generalize this construction to more general situations. Let us considerthe regular nilpotent orbit O [ d ] of sl ( d ). Assume that e is a divisor of d . We put f := d/e . By Proposition (1.1), (1) π ( O [ d ] ) = Z /d Z . The surjective homomorphism Z /d Z → Z /e Z determines an etale cover X → O [ d ] of degree e . We will construct a Q -factorial terminalization of X by the same idea of Examples (1.3). Let Q e,e,...,e bea parabolic subgroup of SL ( d ) with flag type ( e, ..., e ). Let Q e,...,e = U · L be a Levidecomposition where 8 = { I e ∗ ∗ ∗ ∗ I e ∗ ∗ ∗ ... ... ... ... ... I e ∗ I e } ,L = { A A ... ... ... ... ... A f −
00 0 0 0 A f | A i ∈ GL ( e ) , det( A ) · · · det( A f ) = 1 } . We have q e,...,e = n ⊕ l , l = sl ( e ) ⊕ f ⊕ z , where z is the f − l . We take a nilpotent orbit closure ¯ O × f [ e ] inside l . Then SL ( d ) × Q e,...,e ( n + ( ¯ O × f [ e ] )is a crepant partial resolution of ¯ O [ d ] . Let X [ e ] → ¯ O [ e ] be the finite covering of degree e corresponding to the universal covering of O [ e ] . The adjoint action of SL ( e ) on ¯ O [ e ] extends to an action on X [ e ] . The center Z of SL ( e ) is written as { ζ I e ∈ SL ( e ) | ζ e = 1 } . Then Z acts effectively on X [ e ] as covering transformations of X [ e ] → ¯ O [ e ] . In fact, let x ∈ O [ e ] and put G := SL ( e ). Then the universal cover X e ] of O [ e ] is written as G/ ( G x ) .Now G x / ( G x ) ∼ = G φ / ( G φ ) . By the description of G φ in the proof of Proposition (1.1),(1) we see that G φ = Z and ( G φ ) = { } . Since G x / ( G x ) acts effectively on G/ ( G x ) ,we see that Z acts effectively on G/ ( G x ) ; hence, acts effectively on X [ e ] . Then Z × f actson X × f [ e ] . Define a subgroup S ( Z × f ) of Z × f by S ( Z × f ) := { ( ζ I e , ..., ζ f I e ) ∈ Z × f | ζ · · ζ f = 1 } . Notice that S ( Z × f ) ∼ = ( Z /e Z ) ⊕ f − . Then the map X × f [ e ] → ¯ O × f [ e ] factors through X × f [ e ] /S ( Z × f ): X × f [ e ] → X × f [ e ] /S ( Z × f ) → ¯ O × f [ e ] . By definition the 2-nd map is a Z /e Z -Galois covering. Claim (1.3.2) . The adjoint action of L on ¯ O × f [ e ] lifts to an action on X × f [ e ] /S ( Z × f ) .Proof . Define a subgroup T of L by T := { ζ I e ζ I e ... ... ... ... ... ζ f − I e
00 0 0 0 ζ f I e | ζ i ∈ C ∗ , ζ · · · ζ f = 1 } .
9e identify SL ( e ) × f with a subgroup of L defined by { A A ... ... ... ... ... A f −
00 0 0 0 A f | A i ∈ SL ( e ) ∀ i } . Then the inclusion SL ( e ) × f → L induces an isomorphism SL ( e ) × f /S ( Z × f ) ∼ = L/T . As SL ( e ) × f acts on X × f [ e ] , L/T acts on X × f [ e ] /S ( Z × f ). Hence L acts on X × f [ e ] /S ( Z × f ). Q.E.D.Now Q e,...,e acts on n + X × f [ e ] /S ( Z × f ) as follows. Take a point z + v ∈ n + X × f [ e ] /S ( Z × f ).Here z ∈ n and v ∈ X × f [ e ] /S ( Z × f ). We denote by ¯ v the image of v by the map X × f [ e ] /S ( Z × f ) → ¯ O × f [2] . For q ∈ Q e,...,e we denote by ¯ q ∈ L the image of q by themap Q e,...,e → Q e,...,e /U = L . We define q · ( z + v ) := ( Ad q ( z + ¯ v )) n + ¯ q · v ∈ n + X × f [ e ] /S ( Z × f ) . Here ¯ q ∈ L acts on v ∈ X × f [ e ] /S ( Z × f ) as described in Claim (1.3.2).Recall that π : X → ¯ O [ d ] is a finite Z /e Z -cover. We have a commutative diagram SL ( d ) × Q e,...,e ( n + X × f [ e ] /S ( Z × f )) µ −−−→ X π ′ y π y SL ( d ) × Q e,...,e ( n + ¯ O × f [2] ) s −−−→ ¯ O [ d ] (4)Here µ is the Stein factorization of s ◦ π ′ . X × f [ e ] is Q -factorial by Proposition (1.1), (2).Then X × f [ e ] /S ( Z × f ) is also Q -factorial by the following lemma. Lemma (1.4) . Let f : V → W be a finite Galois covering of normal varieties. If V is Q -factorial, then W is also Q -factorial.Proof . Let D be a prime Weil divisor of W . We need to show that mD is a Cartierdivisor for a suitable m . Put E := f − ( D ) and regard it as a reduced Weil divisor. Bythe assumption rE is a Cartier divisor on V for some r >
0. Take a point x ∈ W . By[Mum, Lecture 10, Lemma B], one can choose an open neighborhood U of x ∈ W suchthat rE is a principal divisor of f − ( U ). We take a defining equation ϕ of rE | f − ( U ) . Let G be the Galois group of f . Then Φ := Y g ∈ G ϕ g is a local equation of the Cartier divisor | G | rE . Since Φ is G -invariant, it can be regardedas an element of Γ( U, O W ). Then Φ is a local equation of some multiple of D on U .Q.E.D. 10he variety SL ( d ) × Q e,...,e ( n + X × f [ e ] /S ( Z × f )) is a fiber bundle over SL ( d ) /Q e,...,e witha typical fiber n + X × f [ e ] /S ( Z × f ). By [Ha II, Proposition 6.6], we haveCl( X × f [ e ] /S ( Z × f ) ∼ = Cl( n + X × f [ e ] /S ( Z × f )) , where Cl denotes the divisor class group. By using this we see that n + X × f [ e ] /S ( Z × f ) is Q -factorial since X × f [ e ] /S ( Z × f ) is Q -factorial,. Then SL ( d ) × Q e,...,e ( n + X × f [ e ] /S ( Z × f )) isalso Q -factorial by the following lemma. Lemma (1.5) . Let f : V → T be an etale fiber bundle over a nonsingular variety T with a typical fiber Y . Assume that(1) Y is a Q -factorial normal variety.(2) Y has only rational singularities with Codim Y Sing( Y ) ≥ .Then V is also Q -factorial.Proof . Take a closed point v ∈ V and put t = f ( v ). Replace T by a suitable openneighborhood of t . Put n = dim T . Then one has a sequence of nonsingular subvarieties { t } ⊂ T ⊂ T ⊂ ... ⊂ T n − ⊂ T n = T with dim T i = i . Put V i := V × T T i . Then we geta sequence V (= Y ) ⊂ V ⊂ V ⊂ ... ⊂ V n = V . Here each V i is a Cartier divisor of V i +1 .We apply [Ko-Mo, Corollary (12.1.9)] for Y ⊂ V to see that V is Q -factorial around V . Now V satisfies the conditions (1) and (2). Therefore we can apply [ibid, Corollary(12.1.9)] repeatedly for V i ⊂ V i +1 and finally see that V is Q -factorial around V n − . Inparticular, V is Q -factorial around f − ( t ) ⊂ V . Since v is an arbitrary closed point, V is Q -factorial. (cid:3) Lemma (1.6) . For e > , both X [ e ] and X × f [ e ] /S ( Z × f ) are singular.Proof . Assume that X [ e ] is smooth. Since X [ e ] → ¯ O [ e ] is a finite quotient map, ¯ O [ e ] would be a symplectic quotient singularity. Then the closure of any symplectic leafof ¯ O [ e ] would be again a quotient singularity On the other hand, the closure ¯ O [2 , e − ] of the minimal nilpotent orbit O [2 , e − ] is not a quotient singularity if e >
1. In fact,the Springer resolution of ¯ O [2 , e − ] is given by the cotangent bundle T ∗ P e of P e . Theexceptional locus of the Springer resolution is the zero locus of the cotangent bundle,which has codimension e ( > O [2 , e − ] is not Q -factorial. In particular,¯ O [2 , e − ] is not a quotient singularity. We can prove similarly that X × f [ e ] /S ( Z × f ) is notsmooth by using the quotient map X × f [ e ] /S ( Z × f ) → ¯ O × f [ e ] . Q.E.D.Since X [ e ] has terminal singularities, the product X × f [ e ] has terminal singularities. Thefixed locus of any nonzero element of S ( Z × f ) has codimension ≥
4; hence X × f [ e ] /S ( Z × f )has only terminal singularities. Corollary (1.7) . Let O [ d ] ⊂ sl ( d ) be the regular nilpotent orbit. Assume that X → O [ d ] is an etale covering of degree e > and let X → ¯ O [ d ] be the associated finite coverof ¯ O [ d ] . Then the map SL ( d ) × Q e,...,e ( n + X × f [ e ] /S ( Z × f )) → X is a Q -factorial terminalization of X . In particular, X has no crepant resolutions.
11e next consider the nilpotent orbit O [ d i ] of sl ( di ). Put G = SL ( di ). As before ,take an element x from O [ d i ] and fix an sl (2)-triple φ containing x . Then G φ = { ( A, ..., A ) ∈ GL ( i ) × d | det( A ) d = 1 } .G φ has exactly d connected components. Let ζ be a primitive d -th root of unity. Theneach connected component is given by { ( A, ..., A ) ∈ GL ( i ) × d | det( A ) = ζ i } , i = 0 , , ..., d − . Note that G φ / ( G φ ) ∼ = Z /d Z . Proposition (1.8) .(1) π ( O [ d i ] ) ∼ = Z /d Z .(2) X is Q -factorial for any etale covering π : X → O [ d i ] .Proof . (1) We have already seen that (1) holds.(2) We can prove (2) in the same manner as in Proposition (1.1), (2). Note that χ (( G φ ) ) = { } because ( G φ ) ∼ = SL ( i ) × d . Q.E.D.The closure ¯ O [ d i ] contains the largest orbit O [ d i − ,d − , in ¯ O [ d i ] − O [ d i ] . Note that¯ O [ d i ] has A d − -surface singularities along O [ d i − ,d − , . Moreover, ¯ O [ d i ] has a Springerresolution. Since the universal covering of O [ d i ] is a cyclic covering of degree d , we canprove the following in the same way as Proposition (1.2). Proposition (1.9) . Assume that π : X → O [ d i ] is the universal covering. Then Codim X Sing( X ) ≥ . In particular, X has only terminal singularities. For a partition [ d j , ..., d j k k ] of j d + ... + j k d k , consider the nilpotent orbit O [ d j ,...,d jkk ] ⊂ sl ( j d + ... + j k d k ). Let d := gcd ( d , ..., d k ). By [C-M], Corollary 6.1.6, we have π ( O [ d j ,...,d jkk ] ) ∼ = Z /d Z . Let π : X → O [ d j ,...,d jkk ] be the universal covering and let π : X → ¯ O [ d j ,...,d jkk ] be the associated finite cover. We will construct a Q -factorialterminalization of X by using Proposition (1.9). Example (1.10) . Let Q i d,...,i r d ⊂ SL (( i + ... + i r ) d ) be parabolic subgroup of flagtype ( i d, ..., i r d ). We assume that gcd ( i , ..., i r ) = 1. Let Q i d,...,i r d = U · L be a Levidecomposition with U = { I i d ∗ . ∗ ... ∗ I i d ∗ ... ∗ ... ... ... ... ... ... I i r − d ∗ ... I i r d } and L = { A . ... A ... ... ... ... ... ... ... A r −
00 0 ... A r | A ∈ GL ( i d ) , ..., A r ∈ GL ( i r d ) , det( A ) ··· det( A r ) = 1 } . L consisting of the matrices with A ∈ SL ( i d ), ..., A r ∈ SL ( i r d ) isisomorphic to SL ( i d ) × ... × SL ( i r d ). In the remainder we identify SL ( i d ) × ... × SL ( i r d )with this subgroup of L . Let us consider the product of the nilpotent orbits ¯ O [ d i ] × ... × ¯ O [ d ir ] in sl ( i d ) × ... × sl ( i r d ). Let X [ d iα ] → ¯ O [ d iα ] be the finite cover associated with theuniversal covering of O [ d iα ] for each 1 ≤ α ≤ r . Then we have a finite covering f : X [ d i ] × ... × X [ d ir ] → ¯ O [ d i ] × ... × ¯ O [ d ir ] . We consider the finite subgroup µ i d × ... × µ i r d of SL ( i d ) × ... × SL ( i r d ) consisting ofthe elements ( t I i d , ..., t r I i r d ) with t i d = ... = t i r dr = 1. Moreover let H be the subgroupof µ i d × ... × µ i r d determined by t i ...t i r r = 1. Define a surjection α : µ i d × ... × µ i r d → µ d × ... × µ d by ( t , ..., t r ) → ( t i , ..., t i r r ). Similarly, define a surjection β : µ d × ... × µ d → µ d by ( t , ..., t r ) → t ...t r . Then H is nothing but the kernel of the composed map µ i d × ... × µ i r d → µ d . y y −−−→ µ i × ... × µ i r ι −−−→ H −−−→ Coker( ι ) y y −−−→ µ i d × ... × µ i r d ∼ = −−−→ µ i d × ... × µ i r d −−−→ α y β ◦ α y Ker( β ) −−−→ µ d × ... × µ d β −−−→ µ d −−−→ y y α ) = µ i × ... × µ i r and ι is the natural inclusion. By the snake lemma,Coker( ι ) ∼ = Ker( β ). The subgroup µ i d × ... × µ i r d of SL ( i d ) × ... × SL ( i r d ) acts on X [ d i ] × ... × X [ d ir ] as covering transformations of f . But Ker( α ) acts trivially on X [ d i ] × ... × X [ d ir ] . This means that µ d × ...µ d acts on X [ d i ] × ... × X [ d ir ] , which is nothing butthe Galois group of the map f . Since Coker( ι ) ∼ = Ker( β ), the finite covering( X [ d i ] × ... × X [ d ir ] ) /H → ¯ O [ d i ] × ... × ¯ O [ d ir ] is a cyclic covering of degree d . Claim (1.10.1) . The adjoint action of L on ¯ O [ d i ] × ... × ¯ O [ d ir ] lifts to an action on ( X [ d i ] × ... × X [ d ir ] ) /H . roof . Define a subgroup T of L by T = { t I i d . ... t I [ i d ] ... ... ... ... ... ... ... t r − I i r − d
00 0 ... t r I i r d | t i ...t i r r = 1 } . Then the inclusion SL ( i d ) × ... × SL ( i r d ) → L induces an inclusion ( SL ( i d ) × ... × SL ( i r d )) /H → L/T . Since ( SL ( i d ) × ... × SL ( i r d )) /H acts on ( X [ d i ] × ... × X [ d ir ] ) /H , L acts on ( X [ d i ] × ... × X [ d ir ] ) /H , which gives a lift of the adjoint action. Q.E.D.By Claim (1.10.1), Q i d,...,i r d acts on n + ( X [ d i ] × ... × X [ d ir ] ) /H . Then we have acyclic covering SL (( i + ... + i r ) d ) × Q i d,...,ird ( n + ( X [ d i ] × ... × X [ d ir ] ) /H ) → SL (( i + ... + i r ) d ) × Q i d,...,ird ( n + ¯ O [ d i ] × ... × ¯ O [ d ir ] )of degree d . (cid:3) Now look at a nilpotent orbit O [ d j ,...,d jkk ] ⊂ sl ( j d + ... + j k d k ). Put d := gcd( d , ..., d k ).Then the dual partition [ d j , ..., d j k k ] can be written as the form [ i d , ...., i dr ] by using suitableset of positive integers { i , ...i r } . Here the same number may possibly appears more thanonce in { i , ..., i r } . For example, the partition [9 ,
6] has the dual partition [2 , , ].Note that 3 = gcd(9 , i dj ( j = 1 , ..., r ) equals [ d i j ] ( j = 1 , ..., r ).Now let us consider the product of the nilpotent orbit closures ¯ O [ d i ] × ... × ¯ O [ d ik ] in sl ( i i d ) × ... × sl ( i k d ). Then SL (( i + ... + i r ) d ) × Q i d,...,ird ( n + ¯ O [ d i ] × ... × ¯ O [ d ir ] ) gives acrepant partial resolution of ¯ O [ d j ,...,d jkk ] .Let X → ¯ O [ d j ,...,d jkk ] be the finite covering associated with the universal covering of O [ d j ,...,d jkk ] . By using Example (1.10), we have a commutative diagram SL (( i + ... + i r ) d ) × Q i d,...,ird ( n + ( X [ d i ] × ... × X [ d ir ] ) /H ) µ −−−→ X π ′ y π y SL (( i + ... + i r ) d ) × Q i d,...,ird ( n + ¯ O [ d i ] × ... × ¯ O [ d ir ] ) −−−→ ¯ O [ d j ,...,d jkk ] (6)Since X [ d i ] × ... × X [ d ir ] has terminal singularities and the fixed locus of each nonzeroelement of H has codimension ≥
4, ( X [ d i ] × ... × X [ d ir ] ) /H has only terminal singularities.Since X [ d i ] × ... × X [ d ir ] is Q -factorial, ( X [ d i ] × ... × X [ d ir ] ) /H is Q -factorial by Lemma(1.4). Therefore µ gives a Q -factorial terminalization of X .Next assume that X → ¯ O [ d j ,...,d jkk ] is the finite covering associated with an etalecovering of O [ d j ,...,d jkk ] of degree e . We put f := d/e . In this case we take instead theproduct of nilpotent orbit closures ¯ O × f [ e i ] × ... × ¯ O × f [ e ir ] and consider the finite covering X × f [ e i ] × ... × X × f [ e ik ] → ¯ O × f [ e i ] × ... × ¯ O × f [ e ir ] . H of µ × fi e × ... × µ × i r e . Then we have a commutative diagram SL (( i + ... + i r ) d ) × Q ( i e ) f ,..., ( ire ) f ( n + ( X × f [ e i ] × ... × X × f [ e ir ] ) /H ) µ −−−→ X π ′ y π y SL (( i + ... + i r ) d ) × Q ( i e ) f ,..., ( ire ) f ( n + ¯ O × f [ e i ] × ... × ¯ O × f [ e ir ] ) −−−→ ¯ O [ d j ,...,d jkk ] (7)and µ gives a Q -factorial terminalization. Corollary (1.11) . Let π : X → ¯ O be the finite covering associated with a nontrivialetale covering of a nilpotent orbit O of sl ( d ) . Then X has no crepant resolutions exceptwhen π is the double covering C → ¯ O [2] ⊂ sl (2) . § g = sp (2 n )We write a partition p of 2 n as [ d r d , ( d − r d − , ..., r , r ] with r d = 0. Other r i maypossibly be zero; in such a case i does not appear in the partition. If r i >
0, then we call i a member of the partition. Each nilpotent orbit of sp (2 n ) is uniquely determined byits Jordan type. Such a Jordan type is a partition p of 2 n with all odd members havingeven multiplicities. Conversely, for each such partition p of 2 n , there is a nilpotent orbitwith Jordan type p . Let b be the number of distinct even members of p . Then we have Proposition (2.1) . (1) π ( O p ) ∼ = ( Z / Z ) ⊕ b .(2) Assume that r i = 2 for all even members i of p . Then X is Q -factorial for anyetale covering X → O p .Proof . Put G = Sp (2 n ). Let x ∈ O p and take an sl (2)-triple φ in sp (2 n ) containing x . Put Sp ( r i ) × i ∆ := { ( A, ..., A ) ∈ Sp ( r i ) × i | A ∈ Sp ( r i ) } and O ( r i ) × i ∆ := { ( A, ..., A ) ∈ O ( r i ) × i | A ∈ O ( r i ) } . By [C-M, Theorem (6.1.3)] we have G φ ∼ = Y i :odd Sp ( r i ) × i ∆ × Y i :even O ( r i ) × i ∆ . Hence ( G φ ) ∼ = Y i :odd Sp ( r i ) × i ∆ × Y i :even SO ( r i ) × i ∆ . An important remark is that each factor of the right hand side is a simple Lie groupexcept that SO (2) ∼ = C ∗ and SO (4) is a semisimple Lie group of type A + A . Since π ( O p ) ∼ = G φ / ( G φ ) , (1) is clear from the above. If the condition of (2) holds, then( G φ ) does not have SO (2) as a factor; hence χ (( G φ ) ) = 0. The etale covering X of O p can be written as G/H for a suitable subgroup H with ( G φ ) ⊂ H ⊂ G φ . By the15ame argument as in Proposition (1.1), (2) we see that Pic( G/H ) is a finite group, whichmeans that X is Q -factorial. Q.E.D. Example (2.2) . Let O [2 n ] be the regular nilpotent orbit. By Proposition (2.1), (1)we have π ( O [2 n ] ) ∼ = Z / Z . Let X → ¯ O [2 n ] be the double covering associated with theuniversal covering X → O [2 n ] . Put the n × n matrix J n = ...
10 0 ... ... ... ... ... ... ... ... . Then Sp (2 n ) = { A ∈ GL (2 n ) | A t (cid:18) J n − J n (cid:19) A = (cid:18) J n − J n (cid:19) } . Now let us consider the isotropic flag0 ⊂ h e i ⊂ h e , e i ... ⊂ h e , e , ..., e n − i ⊂ C n and let Q n − , , n − be the parabolic subgroup of Sp (2 n ) stabilizing the flag. Let U bethe unipotent radical of Q n − , , n − . One has a Levi decomposition Q n − , , n − = U · L with L = { t ... ... ... ... ... t ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... t n − ... ... ... ... ... A ... ... ... ... ... t − n − ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... t − ... ... ... ... ... ... t − | t , ..., t n − ∈ C ∗ , A ∈ Sp (2) } . The Lie algebra l decomposes into the direct sum l = gl (1) ⊕ n − ⊕ sp (2) . Take a nilpotent orbit O [2] of sp (2). Then we have a crepant partial resolution Sp (2 n ) × Q n − , , n − ( n × ¯ O [2] ) → ¯ O [2 n ] . Let C → ¯ O [2] be the double covering associated with the universal covering of O [2] .The adjoint action of Sp (2) on ¯ O [2] lifts to an action on C . Since there is a naturalprojection L → Sp (2), L acts in the adjoint way on ¯ O [2] , which lifts to an L -action on C . This means that Q n − , , n − acts on n × C . Then we have a commutative diagram Sp (2 n ) × Q n − , , n − ( n × C ) µ −−−→ X π ′ y π y Sp (2 n ) × Q n − , , n − ( n × ¯ O [2] ) −−−→ ¯ O [2 n ] (8)16he map µ is a crepant resolution of X . (cid:3) Let us consider a partition p = [ d r d , ( d − r d − , ..., r , r ] of 2 n such that(i) r i is even for each odd i , and(ii) r i = 0 for each even i .For example, the partitions [5 , ,
2] and [6 , , , ] satisfy these conditions, but[8 , , , ] does not satisfy (ii) because 6 does not appear. If d is even, then all d , d − d −
4, ..., 2 must appear in the partition. If d is odd, then all d − d −
3, ..., 2must appear in the partition. The following is a key proposition.
Proposition (2.3) . Let p be a partition satisfying (i) and (ii). Let π : X → ¯ O p bethe finite covering associated with the universal covering of O p . Then Codim X Sing( X ) ≥ . Proof . Let i > p such that r i − = 0. This means that p has theform [ d r d , ..., i r i , ( i − r i − , ( i − r i − , ..., r ]Such an i is called a gap member . By the condition (ii) any gap member must be even.For a gap member i , one can find a nilpotent orbit O ¯ p ⊂ ¯ O p with¯ p = [ d r d , ..., i r i − , ( i − , ( i − r i − − , ( i − r i − , ..., r ] . By Kraft and Procesi [K-P, 3.4] we see that Codim ¯ O p O ¯ p = 2 and the transversal slice S for O ¯ p ⊂ ¯ O p is an A -surface singularity. Note that S is a quotient singularity( C / ( Z / Z ) , C , → S .Let { i , i , ..., i k } be the set of all gap members of p . As defined above, for these gapmembers, we have nilpotent orbits O ¯ p , ..., O ¯ p k . Notice that ¯ O ¯ p , ..., ¯ O ¯ p k are nothing but the irreducible components of Sing( ¯ O p ) whichhave codimension 2 in ¯ O p . To prove that Codim X Sing( X ) ≥
4, we only have to showthat X is smooth along π − ( O ¯ p j ) for each 1 ≤ j ≤ k . Let S j be the transversal slicesfor O ¯ p j ⊂ ¯ O p . Then it is equivalent to showing that π − ( S j ) are disjoint union of finitecopies of ( C , induced orbits . There are twotypes of inductions:(Type I): Let i be a gap member of p (which may possibly be odd or even). Put r := r d + ... + r i and let Q ⊂ Sp (2 n ) be a parabolic subgroup of flag type ( r, n − r, r )with Levi decomposition q = n ⊕ l . Notice that l = gl ( r ) ⊕ sp (2 n − r ). There is anilpotent orbit O p ′ of sp (2 n − r ) with Jordan type p ′ = [( d − r d , ..., ( i − r i + r i − , ( i − r i − , ..., r ]such that O p = Ind gl ( O p ′ ) . Notice that p ′ satisfies (i). Claim (2.3.1) (cf. [He, Theorem 7.1, (d)]).
The generalized Springer map µ : Sp (2 n ) × Q ( n + ¯ O p ′ ) → ¯ O p s a birational map.Proof . This is true for any partition p with the condition (i). We write p as[ d , d , ..., d s ] with d ≥ d ≥ ... ≥ d s >
0. The partition p determines a Young di-agram Y ( p ). By definition Y ( p ) is a subset of Z > such that ( i, j ) ∈ Y ( p ) if and only if( i, j ) satisfies 1 ≤ i ≤ d j .We can take a basis { e(i,j) } ( i,j ) ∈ Y ( p ) of C n so that(a) { e ( i, j ) } is a Jordan basis for x , i.e. x · e ( i, j ) = e ( i − , j ) for i > x · e (1 , j ) = 0.(b) h e ( i, j ) , e ( p, q ) i 6 = 0 if and only if p = d j − i + 1 and q = β ( j ). Here β is apermutation of { , , ..., s } such that β = id , d β ( j ) = λ j , and β ( j ) = j if d j is odd (cf.[S-S], p.259, see also [C-M], 5.1).Put F := P ≤ j ≤ r C e (1 , j ). Then F ⊂ F ⊥ is an isotropic flag such that x · F = 0 and x · C n ⊂ F ⊥ and x is an endomorphism of F ⊥ /F with Jordan type p ′ . This is actuallya unique isotropic flag of type ( r, n − r, r ) satisfying these properties. Hence µ − ( x )consists of one element. (cid:3) (Type II): Let i be an even member of p with r i = 2. Put r := r d + ... + r i − +1 and let Q ⊂ Sp (2 n ) be a parabolic subgroup of flag type ( r, n − r, r ) with Levi decomposition q = n ⊕ l . Notice that l = gl ( r ) ⊕ sp (2 n − r ). There is a nilpotent orbit O p ′ of sp (2 n − r )with Jordan type p ′ = [( d − r d , ..., i r i +2 , ( i − r i +1 +2+ r i − , ( i − r i − , ..., r ]such that O p = Ind gl ( O p ′ ) . Notice that r i +1 + 2 + r i − is even because i is even; hence p ′ satisfies the condition (i). Claim (2.3.2) (cf. [He, Theorem 7.1, (d)]).
The generalized Springer map µ : Sp (2 n ) × Q ( n + ¯ O p ′ ) → ¯ O p is generically finite of degree .Proof . Fix an element x ∈ O p and take the same basis { e ( i, j ) } of C m as the previousclaim. Then the permutation β satisfies β ( r ) = r + 1 and β ( r + 1) = r .We put F := P ≤ j ≤ r C e (1 , j ). Then F ⊂ F ⊥ is an isotropic flag such that x · F = 0and x · C n ⊂ F ⊥ and x is an endomorphism of F ⊥ /F with Jordan type p ′ .On the other hand, put F ′ := P ≤ j ≤ r +1 , j = r C e (1 , j ). Then F ′ ⊂ ( F ′ ) ⊥ is an isotropicflag such that x · F ′ = 0 and x · C n ⊂ ( F ′ ) ⊥ and x is an endomorphism of ( F ′ ) ⊥ /F ′ with Jordan type p ′ .Assume that Q is the parabolic subgroup of Sp (2 n ) stabilizing the flag F ⊂ F ⊥ . Wehave an Sp (2 n )-equivariant (locally closed) immersion ι : Sp (2 n ) × Q ( n + O p ′ ) ⊂ Sp (2 n ) /Q × sp (2 n ) , [ g, y ] → ( gQ, Ad g ( y )) . Consider ( F ⊂ F ⊥ , x ) and ( F ′ ⊂ ( F ′ ) ⊥ , x ) as elements of Sp (2 n ) /Q × sp (2 n ). Note that ι ([1 , x ]) = ( Q, x ) = ( F ⊂ F ⊥ , x ).We want to prove that ( F ′ ⊂ ( F ′ ) ⊥ , x ) is also contained in Im( ι ). Let Q ′ be theparabolic subgroup stabilizing the flag F ′ ⊂ ( F ′ ) ⊥ . Then one can write Q ′ = gQg − for18ome g ∈ Sp (2 n ). Let q ′ = n ′ ⊕ l ′ be a Levi decomposition. By definition x ∈ n ′ + O ′ p ′ ,where O ′ p ′ is a nilpotent orbit in l ′ with Jordan type p ′ . Then g − xg ∈ n + g − O ′ p ′ g ,where g − O ′ p ′ g ∈ g − l ′ g . The Lie algebra g − O ′ p ′ g is a Levi subalgebra of q . Hence l and g − O ′ p ′ g are conjugate by an element of Q . By changing g by gq ′ for a suitable q ′ ∈ Q ,we may assume from the first that g − O ′ p ′ g ⊂ l . Since g − O ′ p ′ g and O p ′ have the sameJordan type and the nilpotent orbits of sp (2 n − r ) are completely determined by theirJordan types, we see that g − O ′ p ′ g = O p ′ . This means that x, g − xg ∈ n + O p ′ . The Q -orbit of x and the Q -orbit of g − xg are both dense in n + O p ′ . Hence theyintersects. In other words, there is an element q ∈ Q such that g − xg = qxp − . Then gq ∈ Z Sp (2 n ) and Q ′ = ( gq ) Q ( gq ) − . Now let us consider an element [ gq, x ] ∈ Sp (2 n ) × Q ( n + ¯ O p ′ ). Then ι ([ gq, x ]) = ( gQ, x ) = ( F ′ ⊂ ( F ′ ) ⊥ , x ). (cid:3) Let { i , i , ..., i k } be the set of all gap members of p and let i j be one of them. Put r := r d + ... + r i j and let Q j ⊂ Sp (2 n ) be a parabolic subgroup of flag type ( r, n − r, r )with Levi decomposition q j = n j ⊕ l j . There is a nilpotent orbit O p ′ j of l j with Jordantype p ′ j = [( d − r d , ..., ( i j − r ij +1 , ( i j − r ij + r ij − , ( i j − r ij − , ..., r ]such that O p = Ind gl ( O p ′ j ) We have a generically finite morphism called the generalizedSpringer map µ j : Sp (2 n ) × Q j ( n j + ¯ O p ′ j ) → ¯ O p . By the previous claim, µ j is a birational morphism. Let b ′ j be the number of distincteven members of p ′ j . Then b ′ j = b −
1. This, in particular, means that π ( O p ′ j ) ∼ = ( Z / Z ) ⊕ b − . Let X p ′ j → ¯ O p ′ j be the finite covering associated with the universal covering of O p ′ j .Then n j + X p ′ j is a Q j -space; hence we get a ( Z / Z ) ⊕ b − -covering map π ′ j : Sp (2 n ) × Q j ( n j + X p ′ j ) → Sp (2 n ) × Q j ( n j + ¯ O p ′ j ) → ¯ O p Let X j be the Stein factorization of µ j ◦ π ′ j . Then we have a commutative diagram Sp (2 n ) × Q j ( n j + X p ′ j ) µ ′ j −−−→ X jπ ′ j y π j y Sp (2 n ) × Q j ( n j + ¯ O p ′ j ) µ j −−−→ ¯ O p (9)By definition π j is a ( Z / Z ) ⊕ b − -covering and X j factorizes π as π : X ρ j → X j π j → ¯ O p , where ρ j is a Z / Z -covering. 19 laim (2.3.3) . (1) µ j is a crepant resolution around O ¯ p j ⊂ ¯ O p . In other words, µ − j ( S j ) is the minimal resolution of the A -surface singularity S j . On the other hand, µ j is an isomorphism over open neighborhoods of O ¯ p , ..., O ¯ p j − , O ¯ p j +1 , ..., O ¯ p k . Inother words, the maps µ − j ( S ) → S , ..., µ − j ( S j − ) → S j − , µ − j ( S j +1 ) → S j +1 , ..., µ − j ( S k ) → S k are all isomorphisms. (2) π − j ( S j ) is a disjoint union of b − copies of S j . π − ( S j ) is a disjoint union of b − copies of ( C , . In other words, π j is an etale cover over an open neighborhood of O ¯ p j ⊂ ¯ O p , and ρ j is a ramified double covering over an open neighborhood of π − j ( O ¯ p j ) ⊂ X j . Notice that Claim (2.3.3), (2) implies Proposition (2.3).
Proof . (1) The gap members of p ′ j are i − , ..., i j − − , i j +1 , ..., i k . For the later convenience we put i ′ := i − , ..., i ′ j − := i j − − , i ′ j +1 := i j +1 , ..., i ′ k := i k . Corresponding to these gap members, we get nilpotent orbits O ( p ′ j ) , ..., O ( p ′ j ) j − , O ( p ′ j ) j +1 ,..., O ( p ′ j ) k in ¯ O p ′ j . These are irreducible components of Sing( ¯ O p ′ j ) which have codimension2 in ¯ O p ′ j . For each 1 ≤ l ≤ k ( k = j ), we have a natural embedding Sp (2 n ) × Q j ( n j + ¯ O ( p ′ j ) l ) ⊂ Sp (2 n ) × Q j ( n j + ¯ O p ′ j ) . One can check that µ j ( Sp (2 n ) × Q j ( n j + ¯ O ( p ′ j ) l ) ) = ¯ O ¯ p l . From this fact we see that there is an irreducible components of Sing( Sp (2 n ) × Q j ( n j +¯ O p ′ j ) ) which dominates ¯ O ¯ p l for each l ( = j ), but there is no irreducible components ofSing( Sp (2 n ) × Q j ( n j + ¯ O p ′ j ) ) which dominates ¯ O ¯ p j .Since µ j is a crepant partial resolution of ¯ O p and ¯ O p has A -surface singularity along O ¯ p l , this means that µ j is an isomorphism over an open neighborhood of O ¯ p l ⊂ ¯ O p for l = j , but µ j is a crepant resolution around O ¯ p j ⊂ ¯ O p .(2) Since µ j is a crepant partial resolution and π ′ j is etale in codimension 1, the partialresolution µ ′ j is a crepant partial resolution. By (1) there is an µ j -exceptional divisor E of Sp (2 n ) × Q j ( n j + ¯ O p ′ j ) which dominates ¯ O ¯ p j . Then ( π ′ j ) − ( E ) is a µ ′ j -exceptionaldivisor of Sp (2 n ) × Q j ( n j + X p ′ j ) which dominates π − j ( ¯ O ¯ p j ). If π j is ramified over O ¯ p j , then X j is smooth along π − j ( O ¯ p j ). This contradicts that µ ′ j is a crepant partialresolution. Hence π is unramified over ¯ O ¯ p j , which is nothing but the first statementof (2). Next suppose that the second statement of (2) does not hold. Then ρ j is etaleover an open neighborhood of π − j ( O ¯ p j ) ⊂ X j . Let ( ¯ O p ) be an open set obtained from¯ O p by excluding all irreducible components of Sing( ¯ O p ) different from ¯ O ¯ p j . We put X j := π − j (( ¯ O p ) ). Then ρ − j ( X j ) → X j is an etale cover. On the other hand, let X p ′ j
20e the universal covering of O p ′ j . Then Sp (2 n ) × Q j ( n j + X p ′ j ) is simply connected. Infact, we have an exact sequence π ( n j + X p ′ j ) → π ( Sp (2 n ) × Q j ( n j + X p ′ j )) → π ( Sp (2 n ) /Q j ) → π ( n j + X p ′ j ) = { } and π ( Sp (2 n ) /Q j ) = { } , we have the result.For l = j , µ j is an isomorphism over an open neighborhood of O ¯ p l ⊂ ¯ O p by (1).By Zariski’s Main Theorem µ ′ j is also an isomorphism over an open neighborhood of π − j ( O ¯ p l ) ⊂ X j . This means that( µ ′ j ) − ( π − j ( O p ∪ O ¯ p j )) ⊂ Sp (2 n ) × Q j ( n j + X p ′ j )and the complement of the inclusion is contained in ( µ ′ j ) − ( π ′ j − ( F )), where F is theunion of all nilpotent orbits contained O ′ in ¯ O p with Codim ¯ O p O ′ ≥
4. By Corollary(0.2), ( µ ′ j ) − ( π j − ( F )) has codimension ≥ Sp (2 n ) × Q j ( n j + X p ′ j ). In particular,( µ ′ j ) − ( X j ) is obtained from a smooth variety Sp (2 n ) × Q j ( n j + X p ′ j ) by removing aclosed subset of codimension ≥
2. Hence π (( µ ′ j ) − ( X j )) ∼ = π ( Sp (2 n ) × Q j ( n j + X p ′ j ) = { } . Since µ ′ j is birational, π ( X j ) = { } . This contradicts that ρ − j ( X j ) is a (connected)etale cover of X j of degree 2. (cid:3) We here consider an additional condition for p :(iii) r i = 2 for each even i . Corollary (2.4) . Let p be a partition satisfying (i), (ii) and (iii). Let π : X → ¯ O p be the finite covering associated with the universal covering of O p . Then X has only Q -factorial terminal singularities.Proof . By the condition (iii), X is Q -factorial by Proposition (2.1), (2). Then theresult follows from Proposition (2.3). (cid:3) Let ¯ O p be an arbitrary nilpotent orbit closure of sp (2 n ), and let X → ¯ O p be the finitecovering associated with the universal covering of O p . We shall construct explicitly a Q -factorial terminalization of X . Since p is a Jordan type of a nilpotent orbit, p satisfiesthe condition (i). Let b be the number of distinct even members of p .By using the inductions of type (I) repeatedly for p , we can finally find a parabolicsubgroup Q of Sp (2 n ) and a nilpotent orbit O p ′ of a Levi part l of q such that(a) O p = Ind sp (2 n ) l ( O p ′ ) . (b) p ′ satisfies the condition (ii), and b ′ = b (where b ′ is the number of distinct evenmembers of p ′ ).Notice that l is a direct sum of a simple Lie algebra sp (2 n ′ ), some simple Lie algebrasof type A and the center. O p ′ is a nilpotent orbit of sp (2 n ′ ).For the partition p ′ thus obtained, we let e be the number of even members i of p ′ such that r i = 2. By using the inductions of type (II) repeatedly for p ′ , we can finallyfind a parabolic subgroup Q ′ of Sp (2 n ′ ) and a nilpotent orbit O p ′′ of a Levi part l ′ of q ′ such that 21a’) O p ′ = Ind sp (2 n ′ ) l ′ ( O p ′′ ) , (b’) p ′′ satisfies the conditions (ii), (iii), and b ′′ = b ′ − e (where b ′′ is the number ofdistinct even members of p ′′ ).All together, we get a generalized Springer map µ ′′ : Sp (2 n ) × Q ′′ ( n ′′ + ¯ O p ′′ ) → ¯ O p , where µ ′′ is generically finite of degree 2 b − b ′′ . Let π ′′ : X ′′ → ¯ O p ′′ be the finite coveringassociated with the universal covering of O p ′′ . By Corollary (2.4) X ′′ has only Q -factorialterminal singularities. Note that deg( π ′′ ) = 2 b ′′ . There is a finite cover π ′′ : Sp (2 n ) × Q ′′ ( n ′′ + X ′′ ) → Sp (2 n ) × Q ′′ ( n ′′ + ¯ O p ′′ )of degree 2 b ′′ . Then the Stein factorization of µ ′′ ◦ π ′′ coincides with X . We have acommutative diagram Sp (2 n ) × Q ′′ ( n ′′ + X ′′ ) −−−→ X π ′′ y π y Sp (2 n ) × Q ′′ ( n ′′ + ¯ O p ′′ ) µ ′′ −−−→ ¯ O p (10)The map Sp (2 n ) × Q ′′ ( n ′′ + X ′′ ) → X here obtained is a Q -factorial terminalizationof X . Examples (2.5) . (1) Let us consider the nilpotent orbit O [6 , ] ⊂ sp (20) andlet π : X → ¯ O [6 , ] be the finite covering associated with the universal covering of O [6 , ] . We have deg( π ) = 4 by Proposition (2.1), (1). We shall construct a Q -factorialterminalization of X and we shall show that it is actually a crepant resolution.Let Q , , be a parabolic subgroup of Sp (20) with flag type (4 , , q , , = n , , ⊕ l , , be a Levi decomposition. Then l , , = sp (12) ⊕ gl (1) ⊕ and O [6 , ] isinduced from the nilpotent orbit O [4 , ] ⊂ sp (12). This is a type I induction, and thegeneralized Springer map Sp (20) × Q , , ( n , , + ¯ O [4 , ] ) → ¯ O [6 , ] is a birational map. Let Q , , be a parabolic subgroup of Sp (12). with a Levi decom-position q , , = n , , ⊕ l , , . Then l , , = sp (10) ⊕ gl (1) and O [4 , ] is induced from O [3 , ] ⊂ sp (10). This is a type II induction. Hence we get a generically finite map ofdegree 2 Sp (20) × Q , , , , ( n , , , , + ¯ O [3 , ] ) → Sp (20) × Q , , ( n , , + ¯ O [4 , ] ) . Let Q , , be a parabolic subgroup of Sp (10) with flag type (3 , ,
3) with a Levi decom-position q , , = n , , ⊕ l , , . Then l , , = sp (4) ⊕ gl (1) ⊕ and O [3 , ] is induced from O [1 ] ⊂ sp (4). This is a type II induction, and we get a generically finite map of degree2 Sp (20) × Q , , , , , , n , , , , , , → Sp (20) × Q , , , , ( n , , , , + ¯ O [3 , ] ) .
22e can illustrate the induction step above by([1 ] , sp (4)) TypeII → ( O [3 , ] , sp (10)) TypeII → ( O [4 , ] , sp (12)) TypeI → ( O [6 , ] , sp (20)) . Composing these 3 maps together, we have a generically finite map of degree 4 Sp (20) × Q , , , , , , n , , , , , , → ¯ O [6 , ] . This map factors through X and Sp (20) × Q , , , , , , n , , , , , , gives a crepant resolutionof X .(2) Let π : X → ¯ O [8 , , , ] be the finite covering associated with the universal coveringof O [8 , , , ] . By Proposition (2.1), (1) we have deg( π ) = 4. We can take the followinginductions ( O [4 , , , ] , sp (14)) TypeI → ( O [6 , , , , sp (26)) TypeI → ( O [8 , , , ] , sp (28)) . Note that in each step the number of distinct even members does not change, and thepartition [4 , , , ] satisfies the conditions (i), (ii) and (iii). Let X [4 , , , ] → ¯ O [4 , , , ] be the finite covering associated with the universal covering of O [4 , , , ] . By Corol-lary (2.4), X [4 , , , ] has only Q -factorial terminal singularities. Then Sp (28) × Q , , , , ( n , , , , + X [4 , , , ] ) gives a Q -factorial terminalization of X . (cid:3) § g = so ( m )We write a partition p of m as [ d r d , ( d − r d − , ..., r , r ] with r d = 0. Other r i maybe possibly zero; in such a case i does not appear in the partition. If r i >
0, then we call i a member of the partition. Let O be a nilpotent orbit of g . Then its Jordan type p isa partition of m such that all even members have even multiplicities. When p consistsof only even members, we call p very even . If p is not very even, then the orbit O isuniquely determined by the Jordan type p , and we denote by O p the nilpotent orbit.On the other hand, if p is very even, there are two nilpotent orbits with Jordan type p .The two orbits are conjugate to each other by an element of O ( m ) \ SO ( m ). When wewant to distinguish them, we denote them by O + p , O − p , but we usually denote by O p oneof them. A partition p is called rather odd if all odd members have multiplicity 1. Notethat a very even partition is rather odd. Let a be the number of distinct odd members of p . When g = so (2 n + 1), we always have a >
0, but when g = so (2 n ), we may possiblyhave a = 0. Proposition (3.1) . (1) If p is not rather odd, then π ( O p ) ∼ = ( Z / Z ) ⊕ max( a − , . If p is rather odd, then there is a short exact sequence → Z / Z → π ( O p ) → ( Z / Z ) ⊕ max( a − , → so that Z / Z is contained in the center of π ( O p ) . Assume that r i = 2 for all odd members i of p . Then X is Q -factorial for anyetale covering X → O p .Proof . (1) Put G = Spin ( m ). There is a double covering ρ m : G → SO ( m ) and g = so ( m ). Take an element x from O p and take an sl (2)-triple φ in g containing x . By[C-M, Theorem (6.1.3)] SO ( m ) φ is isomorphic to S ( Y i :even Sp ( r i ) × i ∆ × Y i :odd O ( r i ) × i ∆ ) := { ( Y i :even A × ir i , Y i :odd B × ir i ) ∈ Y i :even Sp ( r i ) × i ∆ × Y i :odd O ( r i ) × i ∆ | Y i :odd det( B r i ) i = 1 } . Then G φ = ρ − m ( SO ( m ) φ ); hence G φ is a double cover of S ( Q i :even Sp ( r i ) × i ∆ × Q i :odd O ( r i ) × i ∆ ).When p has only even members, S ( Q i :even Sp ( r i ) × i ∆ × Q i :odd O ( r i ) × i ∆ ) = Q i :even Sp ( r i ) × i ∆ ,which is connected. When p has some odd members, S ( Q i :even Sp ( r i ) × i ∆ × Q i :odd O ( r i ) × i ∆ )has 2 a − connected components. Therefore, in any case, S ( Q i :even Sp ( r i ) × i ∆ × Q i :odd O ( r i ) × i ∆ )has 2 max( a − , connected components.If p is not rather odd, then the identity component of S ( Q i :even Sp ( r i ) × i ∆ × Q i :odd O ( r i ) × i ∆ )is Q i :even Sp ( r i ) × i ∆ × Q i :odd SO ( r i ) × i ∆ . Note that π ( SO ( r i )) = Z / Z for r i > π ( SO (2)) = Z . For an odd i with r i ≥
2, there is a unique surjective homomorphism φ i : π ( SO ( r i ) × i ∆ ) → Z / Z . We then have a surjection X φ i : Y i : odd, r i ≥ π ( SO ( r i ) × i ∆ ) → Z / Z . The left hand side is identified with π ( Q i :even Sp ( r i ) × i ∆ × Q i :odd SO ( r i ) × i ∆ ). Thereforeit determines a connected etale double covering of Q i :even Sp ( r i ) × i ∆ × Q i :odd SO ( r i ) × i ∆ .One can check that ρ − m ( Q i :even Sp ( r i ) × i ∆ × Q i :odd SO ( r i ) × i ∆ ) is such an etale covering.Hence ρ − m ( S ( Q i :even Sp ( r i ) × i ∆ × Q i :odd O ( r i ) × i ∆ )) has 2 max( a − , connected components. If p is rather odd, then each connected component of S ( Q i :even Sp ( r i ) × i ∆ × Q i :odd O ( r i ) × i ∆ ) isisomorphic to Q i :even Sp ( r i ) × i ∆ , which is simply connected. Hence ρ − m ( S ( Q i :even Sp ( r i ) × i ∆ × Q i :odd O ( r i ) × i ∆ )) has 2 · max( a − , connected components.(2) If p is rather odd, ( G φ ) is isomorphic to Q i :even Sp ( r i ) × i ∆ . Then χ (( G φ ) ) = 0. If p is not rather odd, then ( G φ ) is a double covering of Q i :even Sp ( r i ) × i ∆ × Q i :odd SO ( r i ) × i ∆ .Note that SO ( r ) is a simple Lie group except that SO (2) ∼ = C ∗ and SO (4) is a semisimpleLie group of type A + A . This means that, if r i = 2 for all odd i , then χ ( Y i :even Sp ( r i ) × i ∆ × Y i :odd SO ( r i ) × i ∆ ) = 0 . Then the short exact sequence1 → Z / Z → ( G φ ) → Y i :even Sp ( r i ) × i ∆ × Y i :odd SO ( r i ) × i ∆ → χ ( Y i :even Sp ( r i ) × i ∆ × Y i :odd SO ( r i ) × i ∆ ) ) → χ (( G φ ) ) → χ ( Z / Z ) . χ (( G φ ) ) is also finite. As a result, χ (( G φ ) ) is finite in any case under the assumption(2). The remainder is the same as in the proof of Proposition (1.1), (2). (cid:3) Let π : X → ¯ O p be a finite covering associated with the universal covering of O p and τ : Y → ¯ O p a finite covering determined by the surjection π ( O p ) → ( Z / Z ) ⊕ max( a ( p ) − , in Proposition (3.1), (1). Proposition (3.2) . The adjoint action of SO ( m ) on ¯ O p lifts to an SO ( m ) -actionon Y . If p is not rather odd, then X = Y . If p is rather odd, then X is a double coverof Y and the SO ( m ) action on Y does not lift to an SO ( m ) -action on X .Proof . Put X := π − ( O p ) and Y := τ − ( O p ). Choose x ∈ O p . By Proposition(3.1), the double cover ρ m : Spin ( m ) → SO ( m ) induces a double cover ( Spin ( m ) x ) → ( SO ( m ) x ) when p is not rather odd, and induces an isomorphism ( Spin ( m ) x ) ∼ =( SO ( m ) x ) when p is rather odd. In any case, Y = SO ( m ) / ( SO ( m ) x ) . Hence SO ( m )naturally acts on Y . Since Γ( Y, O Y ) = Γ( Y , O Y ), SO ( m ) acts on Y . The natural SO ( m ) action on SO ( m ) /SO ( m ) x is nothing but the adjoint action of SO ( m ) on O p .This means that the SO ( m )-action on Y is a lift of the adjoint SO ( m )-action on O p .Therefore the SO ( m )-action on Y is a lift of the adjoint action of SO ( m ) on ¯ O p .Assume that p is not rather odd. Then X = Spin ( m ) / ( Spin ( m ) x ) = SO ( m ) / ( SO ( m ) x ) = Y . Hence X = Y .Assume that p is rather odd. Then π factorizes as X ρ → Y τ → O p . Let us considerthe composite SO ( m ) × X → Y of the map SO ( m ) × X id × ρ → SO ( m ) × Y and the map SO ( m ) × Y → Y determined by the SO ( m )-action on Y . The map SO ( m ) × X → Y lifts to a map to X if and only if π ( SO ( m ) × X ) → π ( Y ) is the zero map. Take apoint ˜ x ∈ X such that π (˜ x ) = x . Then the maps SO ( m ) × { ˜ x } → SO ( m ) × X → Y induces homomorphisms of fundamental groups π ( SO ( m ) × { ˜ x } ) → π ( SO ( m ) × X ) → π ( Y ) = Z / Z . Since π ( X ) = 1, the first map is an isomorphism. Since SO ( m ) × { ˜ x } is a fibrebundle over Y with a typical fiber ( SO ( m ) x ) , the map π ( SO ( m ) × { ˜ x } ) → π ( Y )is a surjection. As a consequence, π ( SO ( m ) × X ) → π ( Y ) = Z / Z is a surjection.This means that the SO ( m )-action on Y does not lift to an SO ( m )-action on X . (cid:3) Lemma (3.3) . Assume that p is not very even. Then the adjoint action of O ( m ) on ¯ O p lifts to an O ( m ) -action on Y .Remark . A lifting of an O ( m )-action is not unique. Proof . We fix an element x of O p and an sl (2)-triple φ containing x . Then O ( m ) φ isisomorphic to Y i :even Sp ( r i ) × i ∆ × Y i :odd O ( r i ) × i ∆ :=25 ( Y i :even A × ir i , Y i :odd B × ir i ) ∈ Y i :even Sp ( r i ) × i × Y i :odd O ( r i ) × i . Note that ( O ( m ) φ ) = Y i :even Sp ( r i ) × i ∆ × Y i :odd SO ( r i ) × i ∆ . Since p is not very even, there is an odd member i . We put H φ = Y i :even Sp ( r i ) × i ∆ × Y i :odd = i SO ( r i ) × i ∆ × O ( r i ) × i ∆ , and H x = U x · H φ . Then we have an isomorphism Y := SO ( m ) / ( SO ( m ) x ) ∼ = O ( m ) /H x . The right hand side has a natural O ( m )-action. This O ( m )-action determines an O ( m )-action on Y . (cid:3) Let us consider the following conditions for a partition p of m .(i) r i is even for each even i .(ii) r i = 0 for every odd i . Proposition (3.4) . Let p be a partition of m which is not rather odd. Assume that p satisfies the conditions (i) and (ii). Let X → ¯ O p be the finite covering associated withthe universal covering of O p . Then Codim X Sing( X ) ≥ .Proof . Notice that the conditions (i) and (ii) are replacements of the conditions (i)and (ii) in the previous section where the roles of odd and even members are reversed.In this sense, this proposition is an SO ( m )-analogue of Proposition (2.3). By virtue ofProposition (3.2), this proposition is proved completely in the same way as Proposition(2.3). (cid:3) When p is rather odd, we encounter a different situation as the following exampleillustrates. Example (3.5) . In this example, we show that a usual induction step (cf. §
2) doesnot work well for a rather odd partition p . Put the m × m matrix J m = ...
10 0 ... ... ... ... ... ... ... ... . Then SO ( m ) = { A ∈ SL ( m ) | A t J A = J } . Fix positive integers s , ..., s k , q so that m = 2 P s i + q . Assume that q ≥
2. Let Q ′ be aparabolic subgroup of SO ( m ) fixing the isotropic flag of flag type ( s , ..., s k , q, s k , ..., s )0 ⊂ h e , ..., e s i ⊂ h e , ...e s + s i ... ⊂ ... h e , ..., e P k s i i ⊂ h e , ..., e P k s i + q i h e , ..., e P k s i + q + s k i ..., ⊂ h e , ..., e P k s i + q + P k s i i = C m One has a Levi decomposition Q ′ = U ′ · L ′ with L ′ = { A ... ... ... ... ... A ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... A k ... ... ... ... ... B ... ... ... ... ... A ′ k ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... A ′ ... ... ... ... ... ... A ′ | A i ∈ GL ( s i ) , A ′ i = J s i ( A ti ) − J s i , B ∈ SO ( q ) } . In particular, L ′ ∼ = Q GL ( s i ) × SO ( q ). Let Q := ρ − m ( Q ′ ) and L := ρ − m ( L ′ ) for ρ m : Spin ( m ) → SO ( m ). The Lie algebra l of L is isomorphic to ⊕ gl ( s i ) ⊕ so ( q ). Let p ′ be a rather odd partition of q such that a ( p ) = a ( p ′ ) and let O p ′ be a nilpotentorbit of so ( q ) with Jordan type p ′ . Assume that p ′ is obtained from p by a successionof type I inductions and O p = Ind so ( m ) l ( O p ′ ). For an odd s i , we consider the map SL ( s i ) × C ∗ → GL ( s i ) determined by ( X, λ ) → λ X . This is a cyclic covering oforder 2 s i . The cyclic group µ s i acts on SL ( s i ) × C ∗ so that ( X, λ ) → ( ζ − s i X, ζ s i λ )for a primitive 2 s i -th root ζ s i of unity. Let us consider a unique subgroup µ s i of µ s i of order s i . Then SL ( s i ) × C ∗ /µ s i → GL ( s i ) is a double cover of GL ( s i ). For aneven s i , we consider the map SL ( s i ) × C ∗ → GL ( s i ) determined by ( X, λ ) → λX .This is a cyclic covering of order s i . The cyclic group µ s i acts on SL ( s i ) × C ∗ so that( X, λ ) → ( ζ − s i X, ζ s i λ ) for a primitive s i -th root ζ s i of unity. Let us consider a uniquesubgroup µ s i / of µ s i of order s i /
2. Then SL ( s i ) × C ∗ /µ s i / → GL ( s i ) is a double coverof GL ( s i ). Here we put t i = s i when s i is odd, and put t i := s i / s i is even. Wethen have a covering map( SL ( s ) × C ∗ ) /µ t × ... × ( SL ( s k ) × C ∗ ) /µ t k × Spin ( q ) → GL ( s ) × ... × GL ( s k ) × SO ( q ) . The Galois group of this covering is ( Z / Z ) ⊕ k +1 . Put H := Ker[( Z / Z ) ⊕ k +1 P → Z / Z ],where P is defined by ( x , ..., x k +1 ) → P x i . Claim (3.5.1) . L = { ( SL ( s ) × C ∗ ) /µ t × ... × ( SL ( s k ) × C ∗ ) /µ t k × Spin ( q ) } / H. Proof . We first prove that ρ − m ( GL ( s i )) = ( SL ( s i ) × C ∗ ) /µ t i and ρ − m ( SO ( q )) = Spin ( q ). For each 1 ≤ j ≤ s + ... + s k , we consider the non-degenerate quadratic sub-space V j := h e j , e m +1 − j i of C m . Then SO ( V j ) is a subgroup of SO ( m ) and ρ − m ( SO ( V j )) = Spin ( V j ) (cf. [F-H, (20.31)]). Note that GL ( s i ) contains some SO ( V j ). Since ρ − m ( SO ( V j ))is connected, we see that ρ − m ( GL ( s i )) is a connected double cover of GL ( s i ). Since π ( GL ( s i )) = Z , we have a unique surjective homomorphism π ( GL ( s i )) → Z / Z .This means that ρ − m ( GL ( s i )) = ( SL ( s i ) × C ∗ ) /µ t i . Since the q dimensional sub-space h e P k s i +1 , ..., e P k s i + q i is a non-degenerate quadratic space, we have ρ − m ( SO ( q )) = Spin ( q ) by the same reason as above. 27e then have Cartesian diagrams( SL ( s i ) × C ∗ ) /µ t i ˜ ι i −−−→ L y y GL ( s i ) ι i −−−→ L ′ (11)and Spin ( q ) ˜ ι q −−−→ L y y SO ( q ) ι q −−−→ L ′ (12)Here ι i and ι q are natural injections and all vertical maps are induced by ρ m .By using the group structure of L , we have a commutative diagram Q ( SL ( s i ) × C ∗ ) /µ t i × Spin ( q ) ˜ ι · ... · ˜ ι k · ˜ ι q −−−−−→ L y yQ GL ( s i ) × SO ( q ) ∼ = −−−→ L ′ (13)The covering group of the left vertical map is ( Z / Z ) ⊕ k +1 . The etale covering L → L ′ is determined by a surjection P : ( Z / Z ) ⊕ k +1 → Z / Z . Let τ i : Z / Z → ( Z / Z ) ⊕ k +1 be the inclusion map into the i -th factor. By the previous Cartesian diagrams, thecomposite h ◦ τ i must be isomorphisms for all 1 ≤ i ≤ k + 1. This means that P isdefined by ( x , ..., x k +1 ) → P x i . (cid:3) Let π ′ : X ′ → ¯ O p ′ be a finite covering associated with the universal covering of O p ′ . Let τ ′ : Y ′ → ¯ O p ′ be a finite covering determined by the surjection π ( O p ′ → ( Z / Z ) ⊕ max( a ( p ) − , in Proposition (3.1), (1). Put ( Y ′ ) := τ ′− ( O p ) and ( X ′ ) := π ′− ( O p ). The adjoint action of SO ( q ) on O p ′ lifts to an SO ( q )-action on ( Y ′ ) . Since L ′ = Q GL ( s i ) × SO ( q ), L ′ acts on ( Y ′ ) (and on Y ′ ) by the projection map L ′ → SO ( q ).By the surjection L → L ′ , L also acts on ( Y ′ ) (and on Y ′ ).We prove that this L -action never lifts to an L -action on ( X ′ ) . Let p : L → SO ( q )be the composite of the map L → L ′ and the projection map L ′ → SO ( q ). Take a point y ∈ ( Y ′ ) and define a map L → ( Y ′ ) by g → g · y . By definition this map factorizesas L p → SO ( q ) → ( Y ′ ) . It induces homomorphisms π ( L ) p ∗ → π ( SO ( q )) → π (( Y ′ ) ) . If the L -action on ( Y ′ ) lifts to ( X ′ ) , then the composite of these homomorphisms is thezero map (cf. the proof of Proposition (3.2)). By Claim (3.5.1), the map p has connectedfibers. This means that p ∗ is a surjection. Hence the map π ( SO ( q )) → π (( Y ′ ) ) mustbe zero. But this map is not the zero map because the SO ( q )-action on ( Y ′ ) does notlift to ( X ′ ) by Proposition (3.2). 28ince L acts on Y ′ , Q acts on n + Y ′ . We have a commutative diagram Spin ( m ) × Q ( n + Y ′ ) −−−→ Z ˆ τ y y Spin ( m ) × Q ( n + ¯ O p ′ ) s −−−→ ¯ O p (14)Here Z is the Stein factorization of s ◦ ˆ τ . The horizontal maps are both crepant partialresolutions. Let π : X → ¯ O p be a finite covering of ¯ O p associated with the universalcovering of O p . Then π factorizes as Y ρ → Z → ¯ O p . By the construction deg( ρ ) = 2.Then we have the following claim. Claim (3.5.2) . The Q -action on n + ( Y ′ ) does not lift to a Q -action on n + ( X ′ ) .Moreover, we have π ( Spin ( m ) × Q ( n + ( Y ′ ) )) = { } . Proof . Applying the homotopy exact sequence to the Q -bundle: Spin ( m ) × ( n +( Y ′ ) ) → Spin ( m ) × Q ( n + ( Y ′ ) ), we get an exact sequence π ( Q ) ob → π ( Spin ( m ) × ( n + ( Y ′ ) )) → π ( Spin ( m ) × Q ( n + ( Y ′ ) )) → . The map ob is induced from a map β y : Q → Spin ( m ) × ( n + ( Y ′ ) ) q → ( q − , q (0 + y ))with some y ∈ ( Y ′ ) . We shall prove that ob is the obstruction map to lifting the Q action on n + ( Y ′ ) to a Q -action on n + ( X ′ ) . Let us consider the diagram Q × (n + ( X ′ ) ) n + ( X ′ ) y y Q × ( n + ( Y ′ ) ) −−−→ n + ( Y ′ ) (15)Here the vertical map on the bottom row is given by Q × ( n + ( Y ′ ) ) → n + ( Y ′ ) ( q, n + y ) → q ( n + y ) . The map α : Q × (n + ( X ′ ) ) → n + ( Y ′ ) lifts to a map ˜ α : Q × (n + ( X ′ ) ) → n + ( X ′ ) if and only if α ∗ : π ( Q × (n + ( X ′ ) ) → π ( n + ( Y ′ ) )is the zero map. If such a lift ˜ α exists, then we can find an element τ ∈ Aut ( Y ′ ) ( X ′ ) such that Q × (n + ( X ′ ) ) ˜ α → n + ( X ′ ) id + τ → n + ( X ′ )
29s a group action. We take a lift ˜ y ∈ ( X ′ ) of y ∈ ( Y ′ ) ). Define a map i ˜ y : Q → Q × ( n + ( X ′ ) ) by i ˜ y ( q ) = ( q, y ). Then we have a commutative diagram Q β y −−−→ Spin ( m ) × (n + ( X ′ ) ) i ˜ y y pr y Q × ( n + ( X ′ ) ) α −−−→ n + ( Y ′ ) (16)Correspondingly we have a commutative diagrams of fundamental groups π ( Q ) ob = ( β y ) ∗ −−−−−−→ π ( Spin ( m ) × ( n + ( X ′ ) ) ∼ = y ∼ = y π ( Q × ( n + ( X ′ ) ) α ∗ −−−→ π ( n + ( Y ′ ) ) (17)The vertical maps are isomorphisms because π (( X ′ ) ) = { } and π ( Spin ( m )) = { } . Therefore ob is the obstruction map to a lift.We shall prove that ob is not the zero map. Let us consider the composite¯ β y : Q β y → Spin ( m ) × ( n + ( Y ′ ) ) pr → n + ( Y ′ ) pr → ( Y ′ ) . If we restrict ¯ β y to U ⊂ Q , then it is a constant map sending all elements g ∈ U to y ∈ ( Y ′ ) . By the homotopy exact sequence π ( U ) → π ( Q ) → π ( L ) → , we see that ¯ β y induces homomorphisms π ( L ) → π ( Spin ( m ) × ( n + ( Y ′ ) )) → π ( n + ( Y ′ ) ) → π (( Y ′ ) ) . Here the last two maps are both isomorphisms because π ( Spin ( m )) = 1 and π ( n ) = 1.Since the L -action on ( Y ′ ) does not lift to ( X ′ ) , the map π ( L ) → π (( Y ′ ) ) is not thezero map. This means that ob = ( β y ) ∗ is not the zero map.Return to the original homotopy exact sequence. Then we have π ( Spin ( m ) × ( n +( Y ′ ) ) = Z / Z . In our case the Q -action on n + ( Y ′ ) does not lift to a Q -action onn + ( X ′ ) . This means that π ( Spin ( m ) × Q ( n + ( Y ′ ) ) = { } . (cid:3) We are now in a position to consider a rather odd partition p . Let p = [ d r d , ( d − d r − , ..., r , r ] be a partition of m such that(i) r i is even for each even i .Let i and j be different members of p . Then i and j are called adjacent if there are nomembers of p between i and j except themselves. We consider the following conditionsfor p :(iii) For any couple ( i, j ) of adjacent members, | i − j | ≤
4. Moreover, | i − j | = 4occurs only when i and j are both odd. The smallest member of p is smaller than 4.For example, both [11 , , ,
3] and [7 , , ] satisfy the conditions (i) and (iii). thenwe have 30 roposition (3.6) . Assume that a partition p satisfies the conditions (i) and (iii).Moreover, assume that p is rather odd. Let π : X → ¯ O d be the finite covering associatedwith the universal covering of O p ⊂ so ( m ) . Then Codim X Sing( X ) ≥ . In particular, X has Q -factorial terminal singularities. X is Q -factorial by Proposition (3.1), (2). If Codim X Sing( X ) ≥
4, then X hasterminal singularities by Corollary (0.3). We will prove Proposition (3.6) after somepreliminaries. Assume that p is a rather odd partition with (i) and (iii). Recall that agap member i of p is a member of p such that i > r i − = 0. For each gap member i , we will take an orbit O ¯ p ⊂ ¯ O p and look at the singularity of ¯ O p along O ¯ p by using[K-P, 3.4]. Let us say that a gap member i is exceptional if i is an odd number with i ≥ r i − = r j − = r j − = 0. In this case r i = 1 and r i − = 1 by the rather oddness andthe condition (iii). Otherwise we say that i is ordinary . For an exceptional gap member i , one can find a nilpotent orbit O ¯ p ⊂ ¯ O p with¯ p = [ d r d , ..., ( i + 1) r i +1 , ( i − , ( i − r i − , ..., r ] . Then Codim O p O ¯ p = 2 and the transverse slice for O ¯ p ⊂ ¯ O p is an A -surface singularity.If i is an even gap member, then there are two cases. The first case is when r i − = r i − = 0 and r j − = 0. The second case is when r i − = 0, but r i − = 0 (the case i = 2being in this case). In the first case, r i is a non-zero even number by the condition (i);hence, r i ≥
2. Then one can find a nilpotent orbit O ¯ p ⊂ ¯ O p with¯ p = [ d r d , ..., i r i − , ( i − , ( i − r i − − , ..., r ] . In the second case, r i and r i − are both nonzero even numbers; hence r i , r i − ≥
2. Thenone can find a nilpotent orbit O ¯ p ⊂ ¯ O p with¯ p = [ d d r , ..., i r i − , ( i − , ( i − r i − − , ..., r ] . In both cases, Codim O p O ¯ p = 2. In the first case, the transverse slice S for O ¯ p ⊂ ¯ O p isan A -surface singularity. Next let us consider the second case. If p is not very even,then S is isomorphic to a union of two A -surface singularities intersecting each otherin the singular point. In particular, ¯ O p is not normal along O ¯ p . If p is very even, S isan A -surface singularity. This can be understood more conceptually. In fact, there aretwo orbits O ± p with Jordan type p . The transverse slice for O ¯ p ⊂ ¯ O + p ∪ ¯ O − p is then isomorphic to the union of two A -surface singularities intersecting each other inthe singular point. Since we denote by O p one of the two orbits, the transverse slice for O ¯ p ⊂ ¯ O p is an A -surface singularity.If i is an ordinary, odd gap member, then there are two cases. The first case is when r i − = r i − = 0 and r i − = 0. The second case is when r i − = 0, but r j − = 0. In thefirst case, r i − is a nonzero even number by the condition (i); hence r i − ≥
2. Then onecan find a nilpotent orbit O ¯ p ⊂ ¯ O p with¯ p = [ d r d , ..., i r i − , ( i − , ( i − r i − − , ..., r ] .
31n the second case, one can find a nilpotent orbit O ¯ p ⊂ ¯ O p with¯ p = [ d r d , ..., i r i − , ( i − , ( i − r i − − , ..., r ] . In this last case, ¯ p may possibly be very even. In such a case, we will have two orbits O ± ¯ p . In both cases, Codim O p O ¯ p = 2 and the transverse slices for O ¯ p ⊂ ¯ O p are A -surfacesingularities.As a consequence, for an ordinary gap member i of p , the transverse slice for O ¯ p ⊂ ¯ O p is an A -surface singularity or the union of two A -surface singularities. In the first casewe call i an ordinary gap member of type A . In the latter case we call i an ordinary gapmember of type A ∪ A .Let { i , i , ..., i k } be the set of all gap members of p . As defined above, for these gapmembers, we have nilpotent orbits O ¯ p , ..., O ¯ p k . More exactly, when ¯ p j is very even, we have two different orbits with Jordan type ¯ p j .In such a case we understand that O ¯ p j is both of them.To prove that Codim X Sing( X ) ≥
4, we only have to show that X is smooth along π − ( O ¯ p j ) for each 1 ≤ j ≤ k . Let S j be the transversal slices for O ¯ p j ⊂ ¯ O p . Moreexactly, when ¯ p j is very even, we must consider two slices S ± j for O ± ¯ p j . The claim is thenequivalent to showing that π − ( S j ) are disjoint union of finite copies of ( C , S j is an A -surface singularity if i j is exceptional, and S j is an A -surface singularityor of type A ∪ A if i j is ordinary.The gap member is closely related to the notion of induced orbits .(Type I) Let i be a gap member of p . Let O p ⊂ so ( m ) be a nilpotent orbit withJordan type. Put r := r d + ... + r i and let ¯ Q ⊂ SO ( m ) be a parabolic subgroup offlag type ( r, m − r, r ), and put Q := ρ − m ( ¯ Q ) ⊂ Spin ( m ). Take a Levi decomposition q = n ⊕ l . There is a nilpotent orbit O p ′ of l with Jordan type p ′ = [( d − r d , ..., ( i − r i +1 , ( i − r i + r i − , ( i − r i − , ..., r ]such that O p = Ind gl ( O p ′ ). Claim (3.6.1) (cf. [He, Theorem 7.1, (d)]).
The generalized Springer map µ : Spin ( m ) × Q ( n + ¯ O p ′ ) → ¯ O p is a birational map.Proof . First notice that Spin ( m ) × Q ( n + ¯ O p ′ ) = SO ( m ) × ¯ Q ( n + ¯ O p ′ ). It is enoughto prove the claim by replacing Spin ( m ) by SO ( m ). Then it is completely analogous tothe corresponding statement for Spin (2 n ) in the previous section.For x ∈ O p , we take a basis { e(i,j) } ( i,j ) ∈ Y ( p ) of C m so that(a) { e ( i, j ) } is a Jordan basis for x , i.e. x · e ( i, j ) = e ( i − , j ) for i > x · e (1 , j ) = 0.(b) h e ( i, j ) , e ( p, q ) i 6 = 0 if and only if p = d j − i + 1 and q = β ( j ). Here β is apermutation of { , , ..., s } such that β = id , d β ( j ) = λ j , and β ( j ) = j if d j is even (cf.[S-S], p.259, see also [C-M], 5.1). 32ut F := P ≤ j ≤ r C e (1 , j ). Then F ⊂ F ⊥ is an isotropic flag such that x · F = 0 and x · C m ⊂ F ⊥ and x is an endomorphism of F ⊥ /F with Jordan type p ′ . This is actuallya unique isotropic flag of type ( r, m − r, r ) satisfying these properties. Hence µ − ( x )consists of one element. (cid:3) For later convenience, we also introduce an induction of type II.(Type II) Let i be an odd member of p with r i = 2. Put r := r d + ... + r i − + 1 and let¯ Q ⊂ SO ( m ) be a parabolic subgroup of flag type ( r, m − r, r ) and put Q := ρ − m ( ¯ Q ) ⊂ Spin ( m ). Take a Levi decomposition q = n ⊕ l . Notice that l = gl ( r ) ⊕ so ( m − r ).There is a nilpotent orbit O p ′ of so ( m − r ) with Jordan type p ′ = [( d − r d , ..., i r i +2 , ( i − r i +1 +2+ r i − , ( i − r i − , ..., r ]such that O p = Ind gl ( O p ′ ) . Claim (3.6.2) (cf. [He, Theorem 7.1, (d)]).
The generalized Springer map µ : Spin ( m ) × Q ( n + ¯ O p ′ ) → ¯ O p is a birational map if one of the following holds :(a) p ′ is very even, or (b) m = 2 r . Otherwise µ is a generically finite map of degree .Proof . Since Spin ( m ) × Q ( n + ¯ O p ′ ) = SO ( m ) × ¯ Q ( n + ¯ O p ′ ), it is enough to prove theclaim by replacing Spin ( m ) by SO ( m ). For x ∈ O p , we take the same basis { e ( i, j ) } of C m as the previous claim. We put F := P ≤ j ≤ r C e (1 , j ). Then F ⊂ F ⊥ is an isotropicflag such that x · F = 0 and x · C m ⊂ F ⊥ and x is an endomorphism of F ⊥ /F withJordan type p ′ . On the other hand, put F ′ := P ≤ j ≤ r +1 , j = r C e (1 , j ). Then F ′ ⊂ ( F ′ ) ⊥ is an isotropic flag such that x · F ′ = 0 and x · C m ⊂ ( F ′ ) ⊥ and x is an endomorphismof ( F ′ ) ⊥ /F ′ with Jordan type p ′ . The isotropic flags of type ( r, m − r, r ) with theseproperties are exactly two flags above.Assume that ¯ Q is the parabolic subgroup of SO ( m ) stabilizing the flag F ⊂ F ⊥ . Wehave an SO ( m )-equivariant (locally closed) immersion ι : SO ( m ) × ¯ Q ( n + O p ′ ) ⊂ SO ( m ) / ¯ Q × so ( m ) , [ g, y ] → ( g ¯ Q, Ad g ( y )) . Consider ( F ⊂ F ⊥ , x ) and ( F ′ ⊂ ( F ′ ) ⊥ , x ) as elements of SO ( m ) / ¯ Q × so (2 n ). Note that ι ([1 , x ]) = ( ¯ Q, x ) = ( F ⊂ F ⊥ , x ). Let ¯ Q ′ be the parabolic subgroup of SO ( m ) stabilizingthe flag F ′ ⊂ ( F ′ ) ⊥ . If m = 2 r , then ¯ Q and ¯ Q ′ are conjugate parabolic subgroups.Moreover, if p ′ is not very even, a nilpotent orbit of so ( m − r ) with Jordan type p ′ is unique. Hence, if neither (a) nor (b) holds, we have ( ¯ Q ′ , x ) ∈ Im( ι ) by the sameargument as in Sp (2 n ). This means that µ − ( x ) consists of two elements.If (b) holds and r = 1, then Q ′ is not conjugate to Q . In particular, ( ¯ Q ′ , x ) / ∈ Im( ι ).This means that µ − ( x ) consists of one element. When m = 2 and r = 1. In this case,two flags F ⊂ F ⊥ and F ′ ⊂ ( F ′ ) ⊥ are different, but ¯ Q = ¯ Q ′ . Therefore µ − ( x ) consistsof one element.Finally, if (a) holds, we have one more nilpotent orbit O − p ′ of so ( m − r ) differentfrom O p ′ with Jordan type p ′ . We have one more SO ( m )-equivariant immersion ι − : SO ( m ) × ¯ Q ( n + O − p ′ ) ⊂ SO ( m ) / ¯ Q × so ( m ) .
33e have Im( ι ) ∩ Im( ι − ) = ∅ . Then ( F ′ ⊂ ( F ′ ) ⊥ , x ) ∈ Im( ι − ). This implies that µ − ( x )consists of one element. (cid:3) For each gap member i j , consider an induction of type I: p ′ j = [( d − r d , ..., ( i j − r ij +1 , ( i j − r ij + r ij − , ( i j − r ij − , ..., r ] . We then have a generalized Springer map µ j : Spin ( m ) × Q j ( n + ¯ O p ′ j ) → ¯ O p , which is a birational map. Lemma (3.7) .(1)
Assume that i j is an ordinary gap member of p . If S j is an A -surface singularity,then µ − j ( S j ) is the minimal resolution of S j . If S j is of type A ∪ A , then µ − j ( S j ) isthe minimal resolution of the normalization of S j . On the other hand, µ − j ( S l ) → S l isan isomorphism for each l = j .(2) Assume that i j is an exceptional gap member of p . Then µ − j ( S j ) is a crepantpartial resolution of S j with one exceptional curve C j ∼ = P . µ − j ( S j ) has A -surfacesingularities at two points p ± j ∈ C j . On the other hand, µ − j ( S l ) → S l is an isomorphismfor each l = j Proof . (1) Assume that i j is ordinary. Then the gap members of p ′ j are i − , ..., i j − − , i j +1 , ..., i k . For the later convenience we put i ′ := i − , ..., i ′ j − := i j − − , i ′ j +1 := i j +1 , ..., i ′ k := i k . Corresponding to these gap members, we get nilpotent orbits O ( p ′ j ) , ..., O ( p ′ j ) j − , O ( p ′ j ) j +1 ,..., O ( p ′ j ) k in ¯ O p ′ j . These are irreducible components of Sing( ¯ O p ′ j ) which have codimension2 in ¯ O p ′ j . Take l so that 1 ≤ l ≤ k and l = j . (¯ p ′ j ) l is very even if and only if ¯ p l is veryeven. If i l is ordinary gap member of p of type A (resp. of type A ∪ A ), then i ′ l is alsoan ordinary gap member of p ′ j of type A (resp. of type A ∪ A ). If i l is an exceptionalgap member of p , then i ′ l is an exceptional gap member of p ′ j . For each l , we have anatural embedding Spin ( m ) × Q j ( n j + ¯ O ( p ′ j ) l ) ⊂ Spin ( m ) × Q j ( n j + ¯ O p ′ j ) . One can check that µ j ( Spin ( m ) × Q j ( n j + ¯ O ( p ′ j ) l ) ) = ¯ O ¯ p l . This implies the last statement of (1). There are no singularities of
Spin ( m ) × Q j ( n j + ¯ O p ′ j )lying over O ¯ p j . This implies the first and the second statements of (1).(2) Assume that i j is exceptional. Then i j − p ′ j . Hence thegap members of p ′ j are i − , ..., i j − − , i j − , i j +1 , ..., i k .
34e put i ′ := i − , ..., i ′ j − := i j − − , i ′ j := i j − , i ′ j +1 := i j +1 , ..., i ′ k := i k . Corresponding to these gap members, we get nilpotent orbits O ( p ′ j ) , ..., O ( p ′ j ) j , ..., O ( p ′ j ) k .Note that we have an additional orbit O ( p ′ j ) j in the exceptional case. Take l so that1 ≤ l ≤ k . For each l , we have a natural embedding Spin ( m ) × Q j ( n j + ¯ O ( p ′ j ) l ) ⊂ Spin ( m ) × Q j ( n j + ¯ O p ′ j ) . One can check that µ j ( Spin ( m ) × Q j ( n j + ¯ O ( p ′ j ) l ) ) = ¯ O ¯ p l . For l = j , we apply the same argument as in (1), and we see that the last statement of(2) holds true. Let us consider the case l = j . We have¯ p j = [ d r d , ..., ( i j + 1) r ij +1 , ( i j − , ( i j − r ij − , ..., r ] , p ′ j = [( d − r d , ..., ( i j − r ij +1 , i j − , i j − , ( i j − r ij − , ..., r ] , ( ¯ p ′ j ) j = [( d − r d , ..., ( i j − r ij +1 , ( i j − , ( i j − r ij − , ..., r ] . We then see that ( ¯ p ′ j ) j → ¯ p j is an induction of type II. When ¯ p ′ j is not very even, thegeneralized Springer map Spin ( m ) × Q j ( n j + ¯ O ( p ′ j ) j ) → ¯ O ¯ p j is a generically finite map of degree 2. Moreover, it is an etale double cover over O ¯ p j .Since ¯ O p ′ j has an A -surface singularity along O ¯ p ′ j , Spin ( m ) × Q j ( n j + ¯ O ( p ′ j ) j ) has an A -surface singularity along Spin ( m ) × Q j ( n j + O ¯ p ′ j ). This implies the first and secondstatements of (2). Assume that ¯ p ′ j is very even. Then there are two orbits O ± ¯ p ′ j . Theneach generalized Springer map Spin ( m ) × Q j ( n j + ¯ O ± ( p ′ j ) j ) → ¯ O ¯ p j is a birational map. Moreover, it is an isomorphism over O ¯ p j . Spin ( m ) × Q j ( n j + ¯ O ( p ′ j ) j )has an A -surface singularity along two disjoint subvarieties Spin ( m ) × Q j ( n j + O +¯ p ′ j ) and Spin ( m ) × Q j ( n j + O − ¯ p ′ j ). Therefore the first and the second statements of (2) still holdsin this case. (cid:3) In the above, we associate a nilpotent orbit O ¯ p ⊂ ¯ O p for a gap member i of p .Assume that p is not very even and i is an even gap member with r i − = 0. We alreadyremarked that the transverse slice S for O ¯ p ⊂ ¯ O p is of type A ∪ A . In this case, O p isnot normal. So let us consider the normalization map ν : ˜ O p → ¯ O p . Before going to theproof of Proposition (3.6), we will give a description of ˜ O p . Lemma (3.8) . ν − ( O ¯ p ) is a connected, etale double cover of O ¯ p . roof . By the description of S , the statement is clear except that ν − ( O ¯ p ) is con-nected.We apply inductions to p repeatedly by using the gap members different from i andfinally get a partition p ′ such that p ′ has a unique gap member i ′ , which comes fromthe originally fixed gap member i . For example, start with p = [10 , , , , ] and i = 4. Then 10, 7, 2 are gap members different from 4. Take an induction for 2 to getnew partition p = [8 , , , ]. The partition p has gap members 8, 5 except 2, whichcomes from the originally fixed gap member 4. We next take an induction for 5 to get p = [6 , , ]. Finally we get p ′ = [4 , , ] by taking a induction for 6.Write the partition p ′ as p ′ = [ d ′ r d ′ , ..., ( i ′ + 1) r ′ i ′ +1 , i ′ r ′ i ′ , ( i ′ − r ′ i ′− , ( i ′ − r ′ i ′− , ..., r ′ ] . By the construction, r ′ d ′ , ..., r ′ i ′ +1 , r ′ i ′ , r ′ i ′ − , r ′ i ′ − , ... r ′ are all nonzero. There is ageneralized Springer map µ ′ : Spin ( m ) × Q ′ ( n ′ + ¯ O p ′ ) → ¯ O p . Put ¯ p ′ = [ d ′ r ′ d ′ , ..., i ′ r ′ i ′ − , ( i ′ − , ( i ′ − r ′ i ′− − , ..., r ′ ] . Then ¯ O p ′ has A ∪ A -singularity along O ¯ p ′ . Let ν ′ : ˜ O p ′ → ¯ O p ′ be the normalization map.The subvariety Spin ( m ) × Q ′ ( n ′ + ¯ O ¯ p ′ ) ⊂ Spin ( m ) × Q ′ ( n ′ + ¯ O p ′ ) is birationally mappedto ¯ O ¯ p by µ ′ . Put ¯ µ ′ := µ ′ | Spin ( m ) × Q ′ ( n ′ + ¯ O ¯ p ′ ) . Then (¯ µ ′ ) − ( O ¯ p ) ⊂ Spin ( m ) × Q ′ ( n ′ + O ¯ p ′ )and (¯ µ ′ ) − ( O ¯ p ) → O ¯ p is an isomorphism. Moreover, µ ′ induces a birational map of thenormalizations of both sides: µ ′ n : Spin ( m ) × Q ′ ( n ′ + ˜ O p ′ ) → ˜ O p , which induces a dominating map Spin ( m ) × Q ′ ( n ′ + ( ν ′ ) − ( ¯ O ¯ p ′ )) → ν − ( ¯ O ¯ p ) . Therefore it suffices to show that ( ν ′ ) − ( O ¯ p ′ ) is connected in order to show that ν − ( O ¯ p )is connected. In fact, if ( ν ′ ) − ( O ¯ p ′ ) is connected, then ( ν ′ ) − ( ¯ O ¯ p ′ ) is irreducible. Then ν − ( ¯ O ¯ p ) is also irreducible by the dominating map. This means that ν − ( O ¯ p ) is con-nected. By the argument above, we may assume that p contains all numbers from 1 to d except i −
1. Apply the induction for the unique gap member i of p . We get a partition p full := [( d − r d , ..., ( i − r i +1 , ( i − r i + r i − , ( i − r i − , ..., r ] . Notice that p full has full members, i.e. all numbers from 1 to d −
2. Put r := r d + ... + r i and let Q r,m − r,r be a parabolic subgroup of SO ( m ) with flag type ( r, m − r, r ). Put Q := ρ − m ( Q r,m − r,r ) for ρ m : Spin ( m ) → SO ( m ).Then the Levi part l of q contains so ( m − r ) as a direct factor. In particular, the nilpotent orbit O p full is contained in l .By [Na 1, Corollary (1.4.3)], the normalization ˜ O p full of ¯ O p full has Q -factorial terminalsingularities except when p full = [2 ( m − r − / , ].36herefore the generalized Springer map µ : Spin ( m ) × Q ( n + ˜ O p full ) → ¯ O p gives a Q -factorial terminalization of ¯ O p except when(a) p full = [2 ( m − r − / , ].In case (a), ˜ O p full is not yet Q -factorial. To construct a Q -factorial terminal-ization of ¯ O p , we take a parabolic subgroup Q r, ( m − r ) / , ( m − r ) / ,r of SO ( m ) and put˜ Q := ρ m ( Q r, ( m − r ) / , ( m − r ) / ,r ).Then˜ µ : T ∗ ( Spin ( m ) / ˜ Q ) = Spin ( m ) × ˜ Q ˜ n → ¯ O p gives a crepant resolution of ¯ O p .Assume that (a) does not occur. Suppose that ν − ( O ¯ p ) is not connected, Then wehave two different µ -exceptional divisors E ± which respectively dominate the closures oftwo connected components of ν − ( O ¯ p ). We consider the following two cases separately(Case I) p full = [1 ](Case II) otherwise.When Case (II) occurs, Q is a maximal parabolic subgroup. Hence, the relativePicard number ρ ( µ ) of µ must be one. This contradicts that Exc( µ ) contains 2 irreducibledivisors. We next consider Case (I). In this case Q is not maximal and ρ ( µ ) = 2. It iseasily checked that Case (I) happens only when p = [3 , ]. In this case π ( O p ) = { } .Moreover, since the odd member 3 has multiplicity 2, ˜ O p is not Q -factorial. On theother hand, since ρ ( µ ) = 2, [ E + ] and [ E − ] span NS( µ ) ⊗ Q , which means that ˜ O p is Q -factorial by [Na 1, Lemma (1.1.1)]. This is a contradiction.Finally assume that (a) occurs. In this case ρ (˜ µ ) = 2. It is easily checked that(a) happens only when p = [4 , , ]. Then π ( O p ) = { } . Moreover, since the oddmember 3 has multiplicity 2, ˜ O p is not Q -factorial. On the other hand, since ρ (˜ µ ) = 2,[ E + ] and [ E − ] span NS(˜ µ ) ⊗ Q , which means that ˜ O p by [ibid, Lemma (1.1.1)]. is Q -factorial. This is a contradiction. (cid:3) Proof of Proposition (3.6) .Let { i , ..., i k } be the set of gap members of p . For 1 ≤ j ≤ k , we take a transverseslice S j for O ¯ p j ⊂ ¯ O p and show that π − ( S j ) is a disjoint union of finite number of thecopies of ( C , Case 1 : Assume that i j is an odd, ordinary gap member of p such that r i j − = 0.By the assumption we have a ( p ) >
0. Let τ : Y → ¯ O p be a finite covering determinedby the surjection π ( O p ) → ( Z / Z ) ⊕ a ( p ) in Proposition (3.1), (1). Then π factors through Y : X ρ → Y τ → ¯ O p . Let O ¯ p j be a nilpotent orbit with Jordan type ¯ p j Note that when ¯ p j is very even, thereare two such orbits. The transverse slice S j is an A -surface singularity. By applying thefollowing lemma, we see that τ − ( S j ) is a disjoint union of the copies of ( C , ρ is etale over an open open neighborhood of τ − ( S j ) ⊂ Y ; hence π − ( S j ) is a disjointunion of the copies of ( C , emma (3.9) . Let p = [ d r d , ..., ( i + 1) r i +1 , i, i − , ( i − r i − , ..., r ] be a rather odd partition satisfying the conditions (i) and (iii) with an ordinary, oddgap member i . Let τ : Y → ¯ O p be the finite covering determined by the surjection π ( O p ) → ( Z / Z ) ⊕ max( a ( p ) − , in Proposition (3.1), (2). Put ¯ p := [ d r d , ..., ( i + 1) r i +1 , ( i − , ( i − r i − , ..., r ] and let S be a transverse slice for O ¯ p ⊂ ¯ O p . Then τ − ( S ) is a disjoint union of somecopies of ( C , .Proof . Note that π ( O p ) has order 2 a ( p ) . We put p ′ := [( d − r d , ..., ( i − r i +1 , ( i − , ( i − r i − , ..., r ] . Then p ′ is a partition of m ′ := m − r d + ... + r i +1 + 1), which is not rather oddbecause the odd member i − O p ′ ⊂ so ( m ′ ). Since a ( p ′ ) = a ( p ) − p ′ is not rather odd, π ( O p ′ ) has order2 a ( p ) − . Let Q r d + ... + r i +1 +1 ,m ′ ,r d + ... + r i +1 +1 ⊂ SO ( m ) be a parabolic subgroup of flag type( r d + ... + r i +1 + 1 , m ′ , r d + ... + r i +1 + 1) and put Q := ρ − m ( Q r d + ... + r i +1 +1 ,m ′ ,r d + ... + r i +1 +1 )for the double covering ρ m : Spin ( m ) → SO ( m ). We have a generalized Springer map µ : Spin ( m ) × Q ( n + ¯ O p ′ ) → ¯ O p . Assume that ¯ p is not very even. We can write Sing( ¯ O p ) as a union of ¯ O ¯ p and finitenumber of other nilpotent orbit closures:Sing( ¯ O p ) = ¯ O ¯ p ∪ ¯ O ∪ ... ∪ ¯ O k ∪ ¯ O k +1 ∪ ... ∪ ¯ O k + l Here Codim ¯ O p O j = 2 for 1 ≤ j ≤ k and Codim ¯ O p O j ≥ k + 1 ≤ j ≤ k + l . ByLemma (3.7), (1), the map µ is an isomorphism over O p ∪ O ∪ ... ∪ O k , and µ inducesa crepant resolution of an open neighborhood of O ¯ p ⊂ ¯ O p . We calculate π ( O p ∪ O ¯ p ).There is an isomorphism µ ∗ : π ( µ − ( O p ∪ O ¯ p )) → π ( O p ∪ O ¯ p ) . By Coroolary (0.2) µ − ( O p ∪ O ¯ p ) is obtained from the smooth variety Spin ( m ) × Q ( n + O p ′ ) by removing a closed subset of codimension at least 2. Therefore we have π ( µ − ( O p ∪ O ¯ p )) ∼ = π ( Spin ( m ) × Q ( n + O p ′ )) . By the exact sequence π ( n + O p ′ ) → π ( Spin ( m ) × Q ( n + O p ′ )) → π ( Spin ( m ) /Q ) → π ( Spin ( m ) × Q ( n + O p ′ )) has order at most 2 a ( p ) − because π ( O p ′ ) has order2 a ( p ) − . Let us consider the finite covering τ . By definition, deg( τ ) = 2 a ( p ) − . Since S
38s an A -surface singularity, there are two possibilities. The first case is when τ − ( S ) isa disjoint union of 2 a ( p ) − copies of an A -surface singularity. The second case is when τ − ( S ) is a disjoint union of 2 a ( p ) − copies of C , τ induces an etale coveringof O p ∪ O ¯ p . Since deg( τ ) = 2 a ( p ) − , this implies that | π ( O p ∪ O ¯ p ) | = 2 a ( p ) − . This is acontradiction.Next asuume that ¯ p is very even. In this case, there are two nilpotent orbits O ± ¯ p with Jordan type ¯ p . We can writeSing( ¯ O p ) = ¯ O +¯ p ∪ ¯ O − ¯ p ∪ ¯ O ∪ ... ∪ ¯ O k ∪ ¯ O k +1 ∪ ... ∪ ¯ O k + l Here Codim ¯ O p O j = 2 for 1 ≤ j ≤ k and Codim ¯ O p O j ≥ k + 1 ≤ j ≤ k + l . By thesame reasoning as the case when ¯ p is not very even, we see that π ( O p ∪ O +¯ p ∪ O − ¯ p ) hasorder at most 2 a ( p ) − . Let S ± be respectively transverse slices for O ± ¯ p ⊂ ¯ O p . The adjointaction of O ( m ) interchanges O +¯ p and O − ¯ p . Since τ is O ( m )-equivariant by Lemma (3.3), τ − ( S + ) and τ − ( S − ) have the same splitting type. Therefore, there are two possibilities.The first case is when τ − ( S ± ) are both disjoint unions of 2 a ( p ) − copies of an A -surfacesingularity. The second case is when τ − ( S ± ) are both disjoint unions of 2 a ( p ) − copiesof ( C , p is not very even. (cid:3) . Case 2 : Assume that i j is an odd, ordinary gap member of p such that r i j − = 0 or i j is an even, ordinary gap member of p .Then p ′ j is also rather odd. Moreover, a ( p ′ j ) = a ( p ). When p is very even, p ′ j isalso very even. In this case there are two orbits with Jordan type p ′ j . But there is aunique nilpotent orbit O p ′ j with Jordan type p ′ j such that O p is induced from O p ′ j . Let X p ′ j → ¯ O p ′ j be the finite covering associated with the universal covering of O p ′ j . Then Q j -action on n j + ¯ O p ′ j does not lifts to a Q j -action on n j + X p ′ j . Instead, we take acover τ j : Y p ′ j → ¯ O p ′ j corresponding to the surjection π ( O p ′ j ) → ( Z / Z ) ⊕ max( a ( p ′ j ) − , inProposition (3.1), (1). Then the Q j -action on n j + ¯ O p ′ j lifts to a Q j -action on n j + Y p ′ j .Therefore we have a commutative diagram Spin ( m ) × Q j ( n j + Y p ′ j ) µ ′ j −−−→ Y jπ ′ j y π j y Spin ( m ) × Q j ( n j + ¯ O p ′ j ) µ j −−−→ ¯ O p (18)Here Y j is the Stein factorization of µ j ◦ π ′ j . π factorizes as X ρ j → Y j π j → ¯ O p , where deg( ρ j ) = 2 and deg( π j ) = 2 max( a ( p ) − , . A transverse slice S j for O ¯ p j ⊂ ¯ O p is an A -surface singularity or a union of two A -surface singularities. Case 2-(i) : Assume that i j is an odd, ordinary gap member of p with r i j − = 0,or i j is an even, ordinary gap member of p with r i j − = 0. Then S j is an A -surface39ingularity. By Lemma (3.7), (i) µ j induces a crepant resolution of an open neighborhoodof O ¯ p j ⊂ ¯ O p . Put E := µ − j ( O ¯ p j ). Then E is a divisor and Spin ( m ) × Q j ( n j + ¯ O p ′ j )is smooth around E . Since µ ′ j is a crepant partial resolution, π ′ j must be unramified at E . This means that π − j ( S j ) is a disjoint union of 2 max( a ( p ) − , copies of an A -surfacesingularity. Since π is a Galois cover, there are two possibilities for π − ( S j ). The first caseis when π − ( S j ) is a disjoint union of 2 max( a ( p ) − , copies of ( C , π − ( S j ) is a disjoint union of 2 max( a ( p ) − , copies of an A -surface singularity. Wewill prove that the second case does not occur. If the second case occurs, then ρ j inducesan etale cover of π − j ( O p ∪ O ¯ p j ) of degree 2. In particular, π ( π − j ( O p ∪ O ¯ p j )) = { } .The birational map ( µ ′ j ) − ( π − j ( O p ∪ O ¯ p j )) → π − j ( O p ∪ O ¯ p j )induces an isomorphism of the fundamental groups of both sides. We calculate π (( µ ′ j ) − ( π − j ( O p ∪ O ¯ p j ))). For l = j , µ j is an isomorphism over an open neighborhood of O ¯ p l ⊂ ¯ O p byLemma (3.7), (1). By Zariski’s Main Theorem µ ′ j is also an isomorphism over an openneighborhood of π − j ( O ¯ p l ) ⊂ Y j . Put Y p ′ j := τ − j ( O p ′ j ). This means that( µ ′ j ) − ( π − j ( O p ∪ O ¯ p j )) ⊂ Spin ( m ) × Q j ( n j + Y p ′ j )and the complement of the inclusion is contained in ( µ ′ j ) − ( π − j ( F )), where F is theunion of all nilpotent orbits contained O ′ in ¯ O p with Codim ¯ O p O ′ ≥
4. By Corollary (0.2)( µ ′ j ) − ( F ) has codimension ≥ Spin ( m ) × Q j ( n j + Y p ′ j ). In particular, ( µ ′ j ) − ( π − j ( O p ∪ O ¯ p j ) is obtained from a smooth variety Spin ( m ) × Q j ( n j + Y p ′ j ) by removing a closedsubset of codimension ≥
2. Therefore we have an isomorphism π (( µ ′ j ) − ( π − j ( O p ∪ O ¯ p j ))) ∼ = π ( Spin ( m ) × Q j ( n j + Y p ′ j )) . By Claim (3.5.2), the right hand side is trivial. This is a contradiction. As a consequence, π − ( S j ) is a disjoint union of 2 max( a ( p ) − , copies of ( C , Case 2-(ii) : Assume that i j is an even, ordinary gap member of p with r i j − = 0. If p is very even, S j is an A -surface singularity. By the same argument as in Case 2-(i),we see that π − ( S j ) is a disjoint union of copies of ( C , p is not very even. Then S j is a union of two A -surface singularities. Let ν : ˜ O p → ¯ O p be the normalization. SO ( m ) naturally acts on ˜ O p . Then ν − ( O ¯ p j ) → O ¯ p j is an etale double cover. Moreover, ν − ( O ¯ p j ) is connected by Lemma (3.8). Hence SO ( m ) acts on ν − ( O ¯ p j ) transitively. π i induces a finite covering ˜ π j : Y j → ˜ O p . Then˜ S j := ν − ( S j ) is a disjoint union of two A -surface singularities: ˜ S j = ˜ S + j ⊔ ˜ S − j . We mayassume that ˜ S + j and ˜ S − j are interchanged each other by a suitable element of SO ( m ).Let us consider π j − ( S j ). Notice that π j − ( S j ) = ˜ π − j ( ˜ S + j ) ⊔ ˜ π − j ( ˜ S − j ). By Lemma(3.7), (i), µ j induces a crepant resolution of an open neighborhood of O ¯ p j ⊂ ¯ O p . Put E := µ − j ( O ¯ p j ). Then E is a divisor and Spin ( m ) × Q j ( n j + ¯ O p ′ j ) is smooth around E .Since µ ′ j is a crepant partial resolution, π ′ j must be unramified at E . This means that˜ π − j ( ˜ S + j ) and ˜ π − j ( ˜ S − j ) are respectively disjoint unions of 2 max( a ( p ) − , copies of an A -surface singularity. Let us look at the double cover ρ j . Since ρ j is SO ( m )-equivariant,40here are two possibilities. The first case is when (˜ π j ◦ ρ j ) − ( ˜ S ± ) are both disjoint unionsof 2 max( a ( p ) − , copies of ( C , π j ◦ ρ j ) − ( ˜ S ± ) are bothdisjoint unions of 2 max( a ( p ) − , copies of A -surface singularity. By the same argumentas in Case 2-(i), the second case does not occur. As a consequence, π − ( S j ) is a disjointunion of 2 max( a ( p ) − , copies of ( C , Case 3 : Assume that i j is an exceptional gap member of p .Then p ′ j is also rather odd. Moreover, a ( p ′ j ) = a ( p ). Let X p ′ j → ¯ O p ′ j be the finitecovering associated with the universal covering of O p ′ j . Then Q j -action on n j + ¯ O p ′ j does not lifts to a Q j -action on n j + X p ′ j . Instead, we have a cover τ j : Y p ′ j → ¯ O p ′ j corresponding to the surjection π ( O p ′ j ) → ( Z / Z ) ⊕ max( a ( p ′ j ) − , in Proposition (3.1),(1). Then the Q j -action on n j + ¯ O p ′ j lifts to a Q j -action on n j + Y p ′ j . Therefore we havea commutative diagram Spin ( m ) × Q j ( n j + Y p ′ j ) µ ′ j −−−→ Y jπ ′ j y π j y Spin ( m ) × Q j ( n j + ¯ O p ′ j ) µ j −−−→ ¯ O p (19)Here Y j is the Stein factorization of µ j ◦ π ′ j . π factorizes as X ρ j → Y j π j → ¯ O p , where deg( ρ j ) = 2 and deg( π j ) = 2 a ( p ) − . A transverse slice S j for O ¯ p j ⊂ ¯ O p is an A -surface singularity. By Lemma (3.7), µ − j ( S j ) is a crepant partial resolution of S j with one exceptional curve C j ∼ = P . µ − j ( S j ) has two A -surface singularities at twopoints p ± j ∈ C j . We prove that π ′ j − ( µ − j ( S j )) is a disjoint union of 2 a ( p ) − copies ofthe minimal resolution of an A -surface singularity. This would mean that π − j ( S j ) is adisjoint union of 2 a ( p ) − copies of A -surface singularity. For this purpose we must lookat τ j : Y p ′ j → ¯ O p ′ j . Recall that p ′ j = [( d − r d , ..., ( i j − r ij +1 , i j − , i j − , ( i j − r ij − , ..., r ] . The nilpotent orbit closure ¯ O p ′ j contains nilpotent orbits of Jordan type ( ¯ p ′ j ) j . Bydefinition ( ¯ p ′ j ) j = [( d − r d , ..., ( i j − r ij +1 , ( i j − , ( i j − r ij − , ..., r ] . The partition ( ¯ p ′ j ) j may possibly be very even. In such a case, there are two orbitswith Jordan type ( ¯ p ′ j ) j . If ( ¯ p ′ j ) j is not very even, such an orbit is unique. In any case,let O ( ¯ p ′ j ) j be one of such an orbit, and let S be a transverse slice for O ( ¯ p ′ j ) j ⊂ ¯ O p ′ j . S is an A -surface singularity. By applying Lemma (3.9) to τ j , we see that τ − j ( S ) isa disjoint union of some copies of ( C , π ′ j − ( µ − j ( S j )) → µ − j ( S j )induced by π ′ j . Recall that there are two points p ± ∈ C j such that µ − j ( S j ) has A -surfacesingularities at these points. By the observation above, this map is ramified at p ± . This41eans that π ′ j − ( µ − j ( S j )) is a disjoint union of 2 a ( p ) − copies of the minimal resolutionof an A -surface singularity.We have seen that π − j ( S j ) is a disjoint union of 2 a ( p ) − copies of an A -surfacesingularity. The final task is to show that π − ( S j ) = ρ − j ( π − j ( S j )) is a disjoint union of2 a ( p ) − copies of ( C , (cid:3) .Let p be a partition of m with Condition (i). Let O p ⊂ so ( m ) be a nilpotent orbitwith Jordan type p , and let X → ¯ O p be the finite covering associated with the universalcovering of O p . We are now in a position to construct a Q -factorial terminalization of X . The case when p is rather odd .We will use the following induction step.(Double Induction of type I) Let i be a gap member of p . When i is odd, we assumethat i ≥ r i − = ... = r i − = 0. When i is even, we assume that i ≥ r i − = ... = r i − = 0. We remark that such a gap member exists if p does not satisfiedthe condition (iii) (which is defined just before Proposition (3.6)). Put p ′ = [( d − r d , ..., ( i − r i +1 , ( i − r i + r i − , ( i − r i − , ..., r ] . Then p ′ satisfies (i) and is still rather odd. If we put s := r d + ... + r i , then p ′ is a partitionof m − s . Let Q (2 s,m − s, s ⊂ SO ( m ) be a parabolic subgroup of flag type (2 s, m − s, s )and put Q := ρ − m ( Q (2 s,m − s, s ) for the double cover ρ m : Spin ( m ) → SO ( m ). We writethe Levi decomposition of q as q = l ⊕ n . We have l = gl (2 s ) ⊕ so ( m − s ). One can takea nilpotent orbit O [2 s ] × O p ′ in gl ( sr ) ⊕ so ( m − s ) so that O p = Ind so ( m l ( O [2 s ] × O p ′ ).The generalized Springer map µ : Spin ( m ) × Q ( n + ¯ O [2 s ] × ¯ O p ′ ) → ¯ O p is a birational map. We indicate this process simply by p Type I ← p ′ . Recall that a ( p ) is the number of distinct odd members of p . Now assume that p does not satisfy the condition (iii). Then we can repeat double induction step of typeI to p so that a does not change, and finally get a partition p ′ with conditions (i) and(iii). p Type I ← p ← ... Type I ← p k = p ′ In each step, we record the number s i and put q := m − P s i .Put the m × m matrix J m = ...
10 0 ... ... ... ... ... ... ... ... . SO ( m ) = { A ∈ SL ( m ) | A t J A = J } . Now let us consider the isotropic flag of type (2 s , s , ..., s k , q, s k , ..., s , s ):0 ⊂ h e , ..., e s i ⊂ h e , ...e s +2 s i ... ⊂ ... h e , ..., e P k s i i ⊂ h e , ..., e P k s i + q i⊂ h e , ..., e P k s i + q +2 s k i ..., ⊂ h e , ..., e P k s i + q + P k s i i = C m Let Q ′ be the parabolic subgroup of SO ( m ) stabilizing this flag. One has a Levi decom-position Q ′ = U ′ · L ′ with L ′ = { A ... ... ... ... ... A ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... A k ... ... ... ... ... B ... ... ... ... ... A ′ k ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... A ′ ... ... ... ... ... ... A ′ | A i ∈ GL (2 s i ) , A ′ i = J s i ( A ti ) − J s i , B ∈ SO ( q ) } . In particular, L ′ ∼ = Q GL (2 s i ) × SO ( q ). We define a parabolic subgroup Q of Spin ( m ) by Q := ρ − m ( Q ′ ) for ρ m : Spin ( m ) → SO ( m ). Then the Levi part L of Q is a double coverof L , which is described as follows. There is a natural map SL (2 s i ) × C ∗ → GL (2 s i )defined by ( X, λ ) → λX . The kernel of this map is the cyclic group µ s i := { ( λ − I s i , λ ) | λ s i = 1 } . Let µ s i be the subgroup of µ s i of order s i . Then the covering map SL (2 s i ) × C ∗ /µ s i → GL (2 s i ) has degree 2. We then have a covering map( SL (2 s ) × C ∗ ) /µ s × ... × ( SL (2 s k ) × C ∗ ) /µ s k × Spin ( q ) → GL (2 s ) × ... × GL (2 s k ) × SO ( q ) . The Galois group of this covering is ( Z / Z ) ⊕ k +1 . Put H := Ker[( Z / Z ) ⊕ k +1 P → Z / Z ],where P is defined by ( x , ..., x k +1 ) → P x i . By Claim (3.5.1) we have L = { ( SL (2 s ) × C ∗ ) /µ s × ... × ( SL (2 s k ) × C ∗ ) /µ s k × Spin ( q ) } / H. The Levi part L has the same Lie algebra of that of L ′ . Hence we have l = ⊕ gl (2 s i ) ⊕ so ( q ). We consider the nilpotent orbit Q O [2 si ] × O p ′ in l . By the construction O p = Ind so ( m ) l ( Y O [2 si ] × O p ′ ) . Let X [2 si ] → ¯ O [2 si ] be the double covering associated with the universal covering of O [2 si ] .By Proposition (1.8), (2) and Proposition (1.9), X [2 si ] has only Q -factorial terminalsingularities. Let X p ′ → ¯ O p ′ be the finite covering associated with the universal coveringof O p ′ . By Proposition (3.6) X p ′ has only Q -factorial terminal singularities. ( SL (2 s ) × ∗ ) /µ s × ... × ( SL (2 s k ) × C ∗ ) /µ s k × Spin ( q ) acts on Q X [2 si ] × X p ′ because each SL (2 s i ) × C ∗ /µ s i acts on X [2 si ] , and Spin ( q ) acts on X p ′ . By Proposition (3.1), (1) we have anexact sequence 1 → Z / Z → π ( O p ′ ) → ( Z / Z ) ⊕ max( a ( p ′ ) − , → . Then the generator of Z / Z induces a covering involution − X p ′ → ¯ O p ′ . Hence( Z / Z ) ⊕ k +1 acts on Q X [2 si ] × X p ′ . Let H be the subgroup of ( Z / Z ) ⊕ k +1 defined asabove. The quotient space Q X [2 si ] × X p ′ /H has only terminal singularities becausethe fixed locus of every nonzero element of H has codimension ≥
4. Moreover, since Q X [2 si ] × X p ′ is Q -factorial, Q X [2 si ] × X p ′ /H is also Q -factorial by Lemma (1.4). TheLevi part L acts on ( Q X [2 si ] × X p ′ ) /H . This action is a lifting of the adjoint L -action on Q O [2 si ] × O p ′ . Let us consider the Levi decomposition q = l ⊕ n . Then the observationabove means that n + ( Q X [2 si ] × X p ′ ) /H is a Q -space. There is a finite cover π ′ : Spin ( m ) × Q ( n + ( Y X [2 si ] × X p ′ ) /H ) → Spin ( m ) × Q ( n + Y ¯ O [2 si ] × ¯ O p ′ )and deg( π ) = deg( π ′ ). We now have a commutative diagram Spin ( m ) × Q ( n + ( Q X [2 si ] × X p ′ ) /H ) µ ′ −−−→ X π ′ y π y Spin ( m ) × Q ( n + Q ¯ O [2 si ] × ¯ O p ′ ) µ −−−→ ¯ O p (20)Here X coincides with the Stein factorization of µ ◦ π ′ . The map µ ′ gives a Q -factorialterminalization of X . The case when p is not rather odd
We will use two induction steps of type (I) and of type (II).By using the inductions of type (I) repeatedly for p , we can finally find a parabolicsubgroup Q of Spin ( m ) and a nilpotent orbit O p ′ of a Levi part l of q such that(a) O p = Ind so ( m ) l ( O p ′ ) . (b) a ( p ′ ) = a ( p ), and(c) p ′ satisfies the condition (ii).For the partition p ′ thus obtained, we let e be the number of odd members i of p ′ such that r i = 2. By using the inductions of type (II) repeatedly for p ′ , we can finallyfind a parabolic subgroup Q ′ of Spin ( m ) and a nilpotent orbit O p ′′ of the Levi part l ′ of q ′ such that(a’) O p ′ = Ind sp (2 n ′ ) l ′ ( O p ′′ ) , (b’) a ( p ′′ ) = a ( p ′ ) − e ,(c’) p ′′ satisfies the condition (ii) and(d’) the multiplicity of any odd member of p ′′ is not 2.All together, we get a generalized Springer map µ ′′ : Spin ( m ) × Q ′′ ( n ′′ + ¯ O p ′′ ) → ¯ O p , where µ ′′ is generically finite of degree 2 a ( p ) − a ( p ′′ ) .44ereafter we consider two cases separately. Case 1 : r i ≥ i of p In this case, p ′′ is not rather odd. By the assumption a ( p ) ≥ e + 1. Let X ′′ → ¯ O p ′′ be the finite covering associated with the universal covering of O p ′′ . Then X ′′ has onlyterminal singularities by Proposition (3.4). On the other hand, X is Q -factorial by (d’).Since p ′′ is not rather odd, the Q ′′ -action on n ′′ + ¯ O p ′′ lifts to a Q ′′ -action on n ′′ + X ′′ .There is a finite cover π ′′ : Spin ( m ) × Q ′′ ( n ′′ + X ′′ ) → Spin ( m ) × Q ′′ ( n ′′ + ¯ O p ′′ ) . We have deg( µ ′′ ) = 2 e , deg( τ ′′ ) = 2 max( a ( p ′′ ) − , = 2 max( a ( p ) − e − , = 2 a ( p ) − e − and deg( π ) = 2 a ( p ) − . It can be checked that deg( π ) = deg( π ′′ ) · deg( µ ′′ ). Therefore wehave a commutative diagram Spin ( m ) × Q ′′ ( n ′′ + X ′′ ) −−−→ X π ′′ y π y Spin ( m ) × Q ′′ ( n ′′ + ¯ O p ′′ ) µ ′′ −−−→ ¯ O p (21)where X is obtained as the Stein factorization of the map µ ′′ ◦ π ′′ . The map Spin ( m ) × Q ′′ ( n ′′ + X ′′ ) → X here obtained is a Q -factorial terminalization of X . Case 2 : r i ≤ i of p In this case p ′′ is rather odd. If there is an odd i such that r i = 1, then a ( p ) ≥ e + 1.If r i = 2 for all odd i , then a ( p ) = e ≥
1. The last inequality holds because if e = 0,then p is rather odd and this is not our case. Then we have a short exact sequence1 → Z / Z → π ( O p ′′ ) → ( Z / Z ) ⊕ max( a ( p ′′ ) − , → . Let Y ′′ → ¯ O p ′′ be the finite covering determined by the surjection π ( O p ′′ ) → ( Z / Z ) ⊕ max( a ( p ′′ ) − , .Since p ′′ has only odd, ordinary gap members, we can apply Lemma (3.9) for each gapmember of p ′′ to conclude that Codim Y ′′ Sing( Y ′′ ) ≥
4. Since p ′′ is rather odd, Y ′′ is Q -factorial by Proposition (3.1), (2). Therefore Y ′′ has only Q -factorial terminal sin-gularities. Then the Q ′′ -action on n ′′ + ¯ O p ′′ lifts to a Q ′′ -action on n ′′ + Y ′′ . There is afinite cover τ ′′ : Spin ( m ) × Q ′′ ( n ′′ + Y ′′ ) → Spin ( m ) × Q ′′ ( n ′′ + ¯ O p ′′ ) . We have deg( µ ′′ ) = (cid:26) e ( r i = 1 for some odd i )2 e − ( r i = 2 for all odd i )deg( τ ′′ ) = 2 max( a ( p ′′ ) − , = 2 max( a ( p ) − e − , and deg( π ) = 2 max( a ( p ) − , . It can be checked that deg( π ) = deg( τ ′′ ) · deg( µ ′′ ) in anycase. Therefore we have a commutative diagram45 pin ( m ) × Q ′′ ( n ′′ + Y ′′ ) −−−→ X τ ′′ y π y Spin ( m ) × Q ′′ ( n ′′ + ¯ O p ′′ ) µ ′′ −−−→ ¯ O p (22)where X is obtained as the Stein factorization of the map µ ′′ ◦ τ ′′ . The map Spin ( m ) × Q ′′ ( n ′′ + Y ′′ ) → X here obtained is a Q -factorial terminalization of X . Examples (3.10) . (1) Let us construct a Q -factorial terminalization of a finitecovering X → ¯ O [15 , , ⊂ so (34) associated with the universal covering of O [15 , , . Bydouble inductions of type I[15 , , T ypeI ← [11 , , T ypeI ← [7 , , , ,
3] of 18 with conditions (i) and (iii). Let ¯ Q ⊂ SO (34) be aparabolic subgroup stabilizing an isotropic flag of type (2 , , , ,
2) and put Q := ρ − ( ¯ Q )for ρ : Spin (34) → SO (34). Let X [2] (resp. X [2 ] ) be a finite covering of ¯ O [2] ⊂ sl (2)(resp. ¯ O [2 ] ⊂ sl (6)) associated with the universal covering of O [2] (resp. O [2 ] ). Moreover,let X [7 , , be a finite covering of ¯ O [7 , , ⊂ so (18) associated with the universal coveringof O [7 , , . Then Spin (34) × Q ( n + ( X [2] × X [2 ] × X [7 , , ) /H )is a Q -factorial terminalization of X . Here H := Ker[ P : ( Z / Z ) ⊕ → Z / Z ].(2) Let X be a finite covering of ¯ O [11 , , ⊂ so (40) associated with the universalcovering of O [11 , , . By inductions of type I and of type II[11 , , T ypeI ← [9 , , T ypeI ← [7 , , T ypeI ← [5 , , T ypeII ← [3 , , , ,
1] of 14. Let ¯ Q ⊂ SO (40) be a parabolic subgroupstabilizing an isotropic flag of type (3 , , , , , , , ,
3) and put Q := ρ − ( ¯ Q ) for ρ : Spin (40) → SO (40). Let us consider the nilpotent orbit O [3 , , ⊂ so (14). Let X [3 , , → ¯ O [3 , , be a finite covering associated with the universal covering of O [3 , , .Then Spin (40) × Q ( n + X [3 , , )is a Q -factorial terminalization of X .(3) Let X be a finite covering of ¯ O [13 , , ⊂ so (30) associated with the universalcovering of O [13 , , . By inductions of type I and of type II[13 , , T ypeI ← [11 , , T ypeI ← [9 , , T ypeI ← [7 , , T ypeI ← [5 , , T ypeII ← [4 , , , ,
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