aa r X i v : . [ m a t h . R T ] S e p Springer correspondence for symmetric spaces
Toshiaki Shoji
Abstract.
This is a survey on the Springer correspondence for symmetric spaces.We discuss various generalization of the theory of the Springer correspondence forreductive groups to symmetric spaces and exotic symmetric spaces associated toclassical groups. §
1. Introduction
The Springer correspondence is the canonical correspondence between the unipo-tent classes of a connected reductive group G and irreducible representations of theWeyl group W of G , established by Springer [Sp] in 1976. In 1981, Lusztig [L2]found a way to reformulate the Springer’s theory by means of the theory of per-verse sheaves. In the same paper, he gave a geometric interpretation of Kosktapolynomials in terms of the intersection cohomology associated to unipotent classesof GL n . The theory of Springer correspondence was generalized by Lusztig [L3] in1984 to the theory of generalized Springer correspondence, which became a basis ofhis theory of character sheaves [L4].It is an interesting problem to extend the theory of character sheaves to a moregeneral setting, such as a variety on which G acts. Already in 1989, Ginzburg [Gi]introduced the character sheaves on the symmetric space G/H , and Grojnowski [Gr]and Henderson [H] studied the case where
G/H = GL n /Sp n extensively. Recently,by Achar-Henderson [AH], Finkelberg-Ginzburg-Travkin [FGT], Kato [K1], Shoji-Sorlin [SS], Shoji [S3], different types of examples, such as the enhanced variety andthe exotic symmetric spaces associated to symplectic groups, are found. Those ex-amples enjoy the satisfied theory of the Springer correspondence, as an analogue ofthe Springer correspondence for GL n . In turn, in Chen-Vilonen-Xue [CVX], Shoji-Yang [SY], the case of the symmetric spaces associated to orthogonal groups werestudied. In this case, an analogue of the generalized Springer correspondence ap-pears, rather than the Springer correspondence. We note that the theory of perversesheaves associated to (Lie algebra version of) the symmetric spaces of general typewas studied by Lusztig-Yun [LY]. But in this case, the Springer correspondence doesnot hold in general, in the strict sense.This paper is a survey on the (generalized) Springer correspondence for sym-metric spaces (mainly associated to classical groups), based on the talks at theconference ALTReT2019 in Ito, Japan. Contents §
1. Introduction §
2. Springer correspondene for GL n §
3. Springer correspondence for reductive groups §
4. The interpretation via perverse sheaves §
5. The generalized Springer correspondence §
6. Geometric realization of Kostka polynomials §
7. The enhanced variety GL ( V ) × V §
8. Springer correspondence for the enhanced variety §
9. Double Kostka polynomials §
10. Symmetric spaces in algebraic setting §
11. Symmetric spaces associated to classical groups §
12. Unipotent orbits in H and in G ιθ uni §
13. Exotic symmetric space associated to symplectic groups §
14. Springer correspondence for G ιθ uni , the case H = Sp N §
15. Springer correspondence for G ιθ uni , the case H = SO n +1 §
16. Generalized Springer correspondence for G ιθ uni , the case H = SO n +1 §
17. Exotic symmetric spaces of higher level, the case H = Sp N §
18. Exotic symmetric spaces, the case H = SO n +1 §
19. Symmetric spaces in characteristic 2 §
2. Springer correspondence for GL n Throughout the paper, we assume that k is an algebraically closed field of char-acteristic p ≥
0, and we consider ¯ Q l -sheaves, where ¯ Q l is an algebraic closure of the l -adic number field Q l with l = p . Note that if p = 0, one can replace k by the com-plex number field C , and ¯ Q l -sheaves by ordinary C -sheaves. All the representationsof finite groups are considered over ¯ Q l ≃ C . For a finite group Γ , we denote by Γ ∧ the set of isomorphism classes of irreducible representations of Γ .We will start from the simplest example. Let G = GL n ( k ), B a Borel subgroupof G containing a maximal torus T . Let W = N G ( T ) /T be the Weyl group of G .Hence W is isomorphic to the symmetric group S n of degree n . Let P n be the setof partitions λ = ( λ , . . . , λ k ) such that P i λ i = n . It is well-known that there is abijection(2.1) S ∧ n ≃ P n , ( V λ ↔ λ )where we normalize this so that V λ is the identity (resp. the sign) representation if λ = ( n ) (resp. λ = (1 n )).On the other hand, let G uni be set of unipotent elements in G , called the unipo-tent variety of G . G acts on G uni by the conjugation, and G uni is a union of unipo-tent classes of G . We denote by G uni / ∼ G the set of unipotent classes in G . It is alsowell-known, via the Jordan normal form, that(2.2) G uni / ∼ G ≃ P n , ( O λ ↔ λ ) . It follows from (2.1) and (2.2) that there exists a bijection S ∧ n ≃ G uni / ∼ G by V λ ↔ O λ . But this is nothing more than the parametrization of two sets, S ∧ n and G uni / ∼ G , coincides each other, by accident. It would be more interesting, form amathematical point of view, to show that there exists a “canonical” bijection between S ∧ n and G uni / ∼ G , independent from the parametrization. Actually, this assertionwas achieved by the discovery of the Springer correspondence, as explained below.Let B = G/B be the flag variety of G . Consider the variety (2.3) e G uni = { ( x, gB ) ∈ G uni × B | g − xg ∈ B } and a map π : e G uni → G uni , ( x, gB ) x . Then e G uni is smooth, irreducible, and π is proper. In fact, π gives a resolution of singularities of G uni , and is called the Springer resolution of G uni . For x ∈ G uni , we define a closed set B x of B by(2.4) π − ( x ) ≃ B x = { gB ∈ B | g − xg ∈ B } . B x is called the Springer fibre of x . B x is not smooth, nor irreducible, butit is an interesting variety. We consider the cohomology group H i ( B x , ¯ Q l ). Then H i ( B x , ¯ Q l ) = 0 if i > d x , where d x = dim B x . Thus H d x ( B x , ¯ Q l ) is the topcohomology. The following result holds. Theorem 2.1 (Springer) . Let x ∈ G uni . (i) H i ( B x , ¯ Q l ) has a structure of S n -module, called the Springer representa-tion of S n . (ii) H d x ( B x , ¯ Q l ) is an irreducible S n -module. (iii) The map x H d x ( B x , ¯ Q l ) induces a “canonical” bijection G uni / ∼ G ∼−→ S ∧ n . (iv) For x ∈ O λ , H d x ( B x , ¯ Q l ) ≃ V λ as S n -modules. Hence we obtain the corre-spondence O λ ↔ V λ .The bijective correspondence in (iii) is called the Springer correspondence . §
3. Springer correspondence for reductive groups
The Springer correspondence for G = GL n gives a bijective correspondencebetween unipotent classes of G and irreducible representations of the Weyl group W of G . But this does not hold for reductive groups in general, and needs somemodification. Let G be a connected reductive group. B, T, W, G uni , etc. are definedsimilarly as in Section 2. The Springer resolution π : e G uni → G uni is defined similarly.(Note in the case where p >
0, we need to assume that G is simply connected forobtaining the resolution of singularities. But we ignore this point, and use the“Springer resolution” for reductive groups in general.) The Springer fibre B x isdefined similarly, and the representation of W on H i ( B x , ¯ Q l ), namely the Springerrepresentation of W , was constructed by Springer [Sp]. But the top cohomology H d x ( B x , ¯ Q l ) is not necessarily irreducible.For x ∈ G uni , put A G ( x ) = Z G ( x ) /Z G ( x ). Then A G ( x ) is a finite group. Z G ( x )acts on B x by z : gB zgB , which induces an action of Z G ( x ) on H i ( B x , ¯ Q l ).Since Z G ( x ) acts trivially on H i ( B x , ¯ Q l ), we have an action of A G ( x ) on H i ( B x , ¯ Q l ).It is proved that the action of W on H i ( B x , ¯ Q l ) commutes with A G ( x ), thus we havean action of W × A G ( x ) on H i ( B x , ¯ Q l ). Remark 3.1.
In the case where G = GL n , Z G ( x ) is connected, hence A G ( x ) = { } .For G = Sp N , SO N , A G ( x ) ≃ ( Z / Z ) c for some c . In turn, for G of type G , F , E , there exist a unique class O in G such that A G ( x ) ≃ S , S and S for x ∈ O ,respectively.The following theorem is the original form of the Springer correspondence dueto Springer. Theorem 3.2 (Springer [Sp]) . Let G be reductive, and x ∈ G uni . (i) Consider the decomposition of H d x ( B x , ¯ Q l ) by A G ( x ) × W -modules, H d x ( B x , ¯ Q l ) ≃ M τ ∈ A G ( x ) ∧ τ ⊗ V ( x,τ ) . Then V ( x,τ ) is an irreducible W -module if it is non-zero. (ii) All the irreducible W -modules are realized in this way uniquely. In particular,we have an injective map W ∧ ֒ → N G := { ( x, τ ) | x ∈ G uni / ∼ G , τ ∈ A G ( x ) ∧ } . Remarks 3.3. (i) Except the case where G = GL n , the above map in (ii) is notsurjective. In fact, the difference δ = | N G | − | W ∧ | = 1 if G is of type G , F or E ,while δ = 0 if G is of type E or E . In the former case, the missing one is the pair( x, τ ), where x is the unique class such that A G ( x ) ≃ S , S , S (see Remark 3.1)and τ is the sign representation of S , S , S , respectively.(ii) In the case of classical groups, such as G = Sp N or SO N , δ tends to ∞ if N → ∞ . §
4. The interpretation via perverse sheaves
In [L2] Lusztig reconstructed Springer representations of W on H i ( B x , ¯ Q l ) interms of the intersection cohomology. Based on Lusztig’s construction, Borho-MacPherson [BM] reformulated Springer’s theorem (Theorem 3.2) in the frameworkof the theory of perverse sheaves, which I will explain below.Let N G be the set of pairs ( O , E ), where O is a unipotent class in G uni and E is a G -equivariant simple local system on O . If we choose x ∈ O , the stalk E x of E at x has a structure of a simple A G ( x )-module, say τ ∈ A G ( x ) ∧ , and τ characterizes a G -equivariant simple local system E , which we denote by E τ . Thenby the correspondence ( O , E ) ↔ ( x, τ ), our set N G can be identified with the set N G appeared in Theorem 3.2 (ii). For each ( O , E ) ∈ N G , we consider the intersectioncohomology complex IC( O , E )[dim O ], which gives a G -equivariant simple perversesheaf on G uni . It is known by [L1] that the number of unipotent classes in G uni isfinite. Thus by a general theory, we have(4.1) The set { IC( O , E )[dim O ] | ( O , E ) ∈ N G } gives a complete set of isomorphismclasses of G -equivariant simple local systems on G uni . Theorem 4.1 (Borho-MacPherson [BM]) . Let π : e G uni → G uni be as in Section 3,and consider the constant sheaf ¯ Q l on e G uni . Then π ∗ ¯ Q l [dim G uni ] is a semisimpleperverse sheaf on G uni , equipped with W -action, and is decomposed as π ∗ ¯ Q l [dim G uni ] ≃ M ( O , E ) ∈ N G V ( O , E ) ⊗ IC( O , E )[dim O ] , where V ( O , E ) is an irreducible W -module if it is non-zero. Note that if K = π ∗ ¯ Q l [dim G uni ] is a G -equivariant semisimple perverse sheaf on G uni , it is a direct sum of various IC( O , E )[dim O ] by (4.1). If K is equipped with W -action, the multiplicity space V ( O , E ) has a structure of W -module. The theoremasserts that this W -module is irreducible. Also note that the stalk H ix K of thecohomology sheaf H i K at x ∈ G uni is isomorphic to H i − dim G uni ( B x , ¯ Q l ). ThusTheorem 3.2 is obtained as a corollary of Theorem 4.1. §
5. The generalized Springer correspondence
The map W ∧ ֒ → N G is not necessarily surjective. It is an interesting problem tounderstand the pair ( O , E ) which is not contained in the image of W ∧ (for example,the unique missing pair ( O , E ) in the case of type G , F or E in Remarks 3.3).As an analogue of the Harish-Chandra theory of the representations of reductivegroups, Lusztig extended this map to a bijection to N G . His ingredients are asfollows; • The notion of a cuspidal pair for ( O , E ) ∈ N G , • The notion of an induction ind GP for a parabolic subgroup P of G and itsLevi subgroup L ,ind GP : { L -equiv. perverse sheaves on L } → { semisimple complexes on G } . Note that K is called a semisimple complex if it is a direct sum of various A [ i ],where A [ i ] is a degree shift of a semisimple perverse sheaf A . The induction functorind GP is defined as follows. We consider the diagram L α ←−−− b X P ψ −−−→ e X P π −−−→ G, where b X P = { ( x, g ) ∈ G × G | g − xg ∈ P } , e X P = { ( x, gP ) ∈ G × G/P | g − xg ∈ P } , and the maps are defined by α : ( x, g ) η P ( g − xg ) , ψ : ( x, g ) ( x, gP ) , π : ( x, gP ) x. ( η : P → L ≃ P/U P is the natural projection.) Here α is a smooth morphism withconnected fibre, and ψ is a principal P -bundle. Let K be an L -equivariant perversesheaf on L . Then α ∗ K [ a ] is a P -equivariant perverse sheaf on b X ( a is the dimensionof the fibre of α ), and there exists a unique perverse sheaf e K on e X such that α ∗ K [ a ] ≃ ψ ∗ e K [ b ] with b = dim P . Since e X P is smooth, irreducible, and π is proper, π ∗ e K is a semisim-ple complex on G by the decomposition theorem of Deligne-Gabber. We defineind GP K = π ∗ e K . Note that unless P = B , the map e X P → L is not defined directly,so we need to consider b X P for defining e K .Let S G be the set of triples ( L, O , E ), up to the natural action of G , where L is a Levi subgroup of a parabolic subgroup of G , and ( O , E ) ∈ N L is a cuspidal pair.Put W L = N G ( L ) /L . In general, W L is not a Coxeter group, but in a very specialsituation that N L has a cuspidal pair, it turns out that W L is a Coxeter group. For ξ = ( L, O , E ) ∈ S G , put K ξ = IC( O , E )[dim O ]. Then K ξ is an L -equivariantperverse sheaf on L , and one can consider the complex ind GP K ξ on G . The followingresult was proved by Lusztig. Theorem 5.1 (Lusztig [L3]) . (i) For ξ = ( L, O , E ) ∈ S G , K ξ is a semisimpleperverse sheaf on G , equipped with W L -action, and is decomposed as ind GP K ξ ≃ M ( O , E ) ∈ N G V ξ ( O , E ) ⊗ IC( O , E )[dim O ] , where V ξ ( O , E ) is an irreducible W L -module if it is non-zero. (ii) For any ξ ∈ S G , we have a bijection N ( ξ ) G := { ( O , E ) ∈ N G | V ξ ( O , E ) = 0 } ∼−→ W ∧ L . (iii) We have a partition N G = ` ξ ∈ S G N ( ξ ) G . In particular, there exists a naturalbijection N G ∼−→ a ξ ∈ S G W ∧ L , which is called the generalized Springer correspondence.Remarks 5.2. (i) In general, cuspidal pairs occur very rarely. Let G = Sp N or SO N , and δ N the number of cuspidal pairs in G . Then we have • G = Sp N : δ N = ( N = d ( d −
1) for some d ∈ Z , • G = SO N : δ N = ( N = d for some d ∈ Z , N = 2 , , , , , . . . ( G : Sp N ) ,N = 1 , , , , , . . . ( G : SO N ) . (ii) If ξ = ( L, O , E ), where L = T is a maximal torus, O = { } is theidentity class, and E is the constant sheaf ¯ Q l , then ind GP K ξ gives the originalSpringer correspondence in Theorem 3.2 and Theorem 4.1. In this case, • W ↔ IC( O , ¯ Q l ) : O : the regular unipotent class in G uni . • ε W ↔ IC( O , ¯ Q l ) : O = { } ; the identity class. §
6. Geometric realization of Kostka polynomials
In this section, we will discuss the relationship between Kostka polynomials andthe theory of the Springer correspondence in the case of GL n . In [L2], Lusztig gave ageometric realization of Kostka polynomials in terms of the intersection cohomologyassociated to unipotent classes in GL n . First we review the definition of Kostkapolynomials.Consider x = ( x , x , . . . , ) infinitely many variables, and t another parameter.Let Λ ( x ) = Λ = L n ≥ Λ n be the ring of symmetric functions, where Λ n is the n -thhomogeneous part, and consider Λ [ t ] = Λ ⊗ Z Z [ t ], the ring over Z [ t ]. For λ ∈ P n ,one can consider the Schur function s λ ( x ) ∈ Λ n and the Hall-Littlewood function P λ ( x ; t ) ∈ Λ n [ t ] = Λ n ⊗ Z Z [ t ]. { s λ ( x ) | λ ∈ P n } and { P λ ( x ; t ) | λ ∈ P n } give Z [ t ]-bases of Λ n [ t ]. Thus one can define the Kostka polynomial K λ,µ ( t ) ∈ Z [ t ] bythe condition that(6.1) s λ ( x ) = X µ ∈ P n K λ,µ ( t ) P µ ( x ; t ) . Here we introduce two combinatorial objects. For λ, µ ∈ P n , write them as λ = ( λ , . . . , λ k ) , µ = ( µ , · · · , µ k ) with λ i , µ i ≥
0. We define a partial order λ ≤ µ on P n , called the dominance order, by the condition that, for any 1 ≤ j ≤ k ,(6.2) j X i =1 λ i ≤ j X i =1 µ i . On the other hand, for λ ∈ P n , we define an n -function n : P n → Z ≥ by(6.3) n ( λ ) = k X i =1 ( i − λ i . Then it is known that K λ,µ ( t ) is a monic polynomial with deg K λ,µ ( t ) = n ( µ ) − n ( λ ).It follows that one can define a modified Kostka polynomial e K λ,µ ( t ) ∈ Z [ t ] by(6.4) e K λ,µ ( t ) = t n ( µ ) K λ,µ ( t − ) . Let O λ be the unipotent class in G uni corresponding to λ ∈ P n for G = GL n as in Section 2. The following result shows that the closure relations of unipotentclasses in GL n can be described in terms of the dominance order on P n .(6.5.) O λ = a µ ≤ λ O µ The following result gives a geometric realization of Kostka polynomials.
Theorem 6.1 (Lusztig [L2]) . For λ ∈ P n , put A λ = IC( O λ , ¯ Q l ) . Then (i) H i A λ = 0 for odd i . (ii) For x ∈ O µ ⊂ O λ , we have (6.6) e K λ,µ ( t ) = t n ( λ ) X i ≥ (dim H ix A λ ) t i . In particular, K λ,µ ( t ) ∈ Z ≥ [ t ] . In the case of GL n , Borho-MacPherson’s theorem gives a decomposition(6.7) π ∗ ¯ Q l [dim G uni ] ≃ M λ ∈ P n V λ ⊗ A λ [dim O λ ] . By taking the stalk of the i -th cohomology on both sides of (6.7), and by comparingthem with Theorem 6.1 (ii), we obtain the interpretation of Ksotka polynomials interms of the Springer representations of S n . Corollary 6.2.
For x ∈ O µ , we have e K λ,µ ( t ) = t n ( λ ) X i ≥ h H i ( B x , ¯ Q l ) , V λ i S n t i , where h , i S n is the inner product of characters of S n . Remark 6.3.
In the case of the Springer correspondence, only Springer represen-tations of the top cohomology H d x ( B x , ¯ Q l ) are involved. While, for the descriptionof the Kostka polynomials, all the Springer representations H i ( B x , ¯ Q l ) are used. §
7. The enhanced variety GL ( V ) × V Later we discuss the theory of the Springer correspondence for the exotic sym-metric space associated to symplectic groups. The enhanced variety is an extensionof GL n , and is also regarded as a degenerate form of the exotic symmetric space, andwas studied extensively by Achar-Henderson [AH] and Finkelberg-Ginzburg-Travkin[FGT]. So we will start from the exposition on the enhanced variety.Let G = GL ( V ) ≃ GL n with dim V = n . We consider the direct product X = G × V , on which G acts as g : ( x, v ) ( gxg − , gv ), where gv is the naturalaction of G on V . The role of G uni is played by X uni = G uni × V , which is a G -invariantclosed subset of X . The varieties X , X uni are called the enhanced varieties .Let P n, be the set of double partitions λ = ( λ ′ , λ ′′ ) such that | λ | = | λ ′ | + | λ ′′ | = n . The following result is known by Achar-Henderson and Travkin. The finitenessof the G -orbits is crucial for later discussion. Lemma 7.1.
Let X uni / ∼ G be the set of G -orbits in X uni . Then we have X uni / ∼ G ≃ P n, . The correspondence is given as follows; take z = ( x, v ) ∈ G uni × V . Put E x = { y ∈ End( v ) | xy = yx } . Then E x is a subalgebra of End( V ), and x ∈ E x . Put V x = E x v ⊂ V . Then V x is an x -stable subspace of V . Let λ ′ be the Jordantype of x | V x and λ ′′ the Jordan type of x | V/V x . Then λ = ( λ ′ , λ ′′ ) ∈ P n, , and thecorrespondence z λ gives a bijection X uni / ∼ G ∼−→ P n, . We shall define an analogue of the dominance order on P n, as follows; for λ = ( λ ′ , λ ′′ ) ∈ P n, , write it as λ ′ = ( λ ′ , . . . , λ ′ m ) , λ ′′ = ( λ ′′ , . . . , λ ′′ m ) with common m , and define c ( λ ) ∈ Z m ≥ by(7.1) c ( λ ) = ( λ ′ , λ ′′ , λ ′ , λ ′′ , . . . , λ ′ m , λ ′′ m ) . Now define a dominance order λ ≤ µ on P n, by the condition that c ( λ ) ≤ c ( µ ) in Z m ≥ . Note that the definition of the dominance order on P n makes sense even for Z m ≥ . The following is known by [AH]. Lemma 7.2.
For each λ ∈ P n, , we have O λ = ` µ ≤ λ O µ . Remark 7.3.
For z = ( x, v ) ∈ G uni × V , the stabilizer Z G ( z ) in G is connected.Hence the G -equivariant simple local system on a G -orbit O in X uni is a constantsheaf ¯ Q l , and we are in a quite similar situation as in the case of GL n . §
8. Springer correspondence for the enhanced variety
We fix a B -stable flag M ⊂ M ⊂ · · · ⊂ M n = V , with dim M i = i . For aninteger 0 ≤ m ≤ n , we define varieties e X m = { ( x, v, gB ) ∈ G uni × V × B | g − xg ∈ B, g − v ∈ M m } , (8.1) X m = [ g ∈ G g ( U × M m ) , where U is the unipotent radical of B , and define a map π m : e X m → X m by( x, v, gB ) ( x, v ). Then e X m is smooth, irreducible, and π is proper.For 0 ≤ m ≤ n , put m = ( m, n − m ) and S m = S m × S n − m . Then S m is a Weylsubgroup of S n . Put(8.2) P ( m ) = { λ = ( λ ′ , λ ′′ ) ∈ P n, | | λ ′ | = m, | λ ′′ | = n − m } . Then it is known that(8.3) S ∧ m ≃ P ( m ) , V λ = V λ ′ ⊠ V λ ′′ ↔ λ = ( λ ′ , λ ′′ ) . The following result is an analogue of Borho-MacPherson’s theorem to the caseof the enhanced variety.
Theorem 8.1 ([SS]) . Put d m = dim X m . (i) ( π m ) ∗ ¯ Q l [ d m ] is a semisimple perverse sheaf on X m , equipped with S m -action,and is decomposed as ( π m ) ∗ ¯ Q l [ d m ] ≃ M λ ∈ P ( m ) V λ ⊗ IC( O λ , ¯ Q l )[dim O λ ] . (ii) The Springer correspondence is given by a ≤ m ≤ n ( S m × S n − m ) ∧ ≃ a ≤ m ≤ n { O λ | λ ∈ P ( m ) } = X uni / ∼ G . §
9. Double Kostka polynomials
Kostka polynomials K λ,µ ( t ) ( λ, µ ∈ P n ) was generalized to double Kostka poly-nomials K λ , µ ( t ) ( λ , µ ∈ P n, ). (See [S1, S2], where they are defined for any r -partitions. Here we only consider the special case where r = 2). Recall the dis-cussion in Section 6. Here we prepare two types of variables x ′ = ( x ′ , x ′ , . . . ), x ′′ = ( x ′′ , x ′′ , . . . ) and consider Ξ = L n ≥ Λ ( x ′ ) ⊗ Λ ( x ′′ ), the ring of symmetric func-tions with respect to x = ( x ′ , x ′′ ). Put Ξ[ t ] = Ξ ⊗ Z Z [ t ]. For λ = ( λ ′ , λ ′′ ) ∈ P n, ,one can define the Schur function s λ ( x ) = s λ ′ ( x ′ ) s λ ′′ ( x ′′ ) and Hall-Littlewood func-tions P λ ( x ; t ). (Note that the definition of P λ ( x ; t ) is rather complicated.) Then { s λ ( x ) | λ ∈ P n, } , { P λ ( x ; t ) | λ ∈ P n, } give two Z [ t ]-bases of Ξ n [ t ] = Ξ n ⊗ Z Z [ t ].We define the double Kostka polynomial P λ , µ ( t ) ∈ Z [ t ] by(9.1) s λ ( x ) = X µ ∈ P n, K λ , µ ( t ) P µ ( x ; t ) . For λ = ( λ ′ , λ ′′ ) ∈ P n, , define an a -function a : P n, → Z ≥ by(9.2) a ( λ ) = 2( n ( λ ′ ) + n ( λ ′′ )) + | λ ′′ | . The a -function has a role of n -function in the case of P n, . It is known that K λ , µ ( t )is a monic of degree a ( µ ) − a ( λ ). As an analogue of (6.4), we define a modifieddouble Kostka polynomial e K λ , µ ( t ) by(9.3) e K λ , µ ( t ) = t a ( µ ) K λ , µ ( t − ) . The following result was proved by [AH], which is an analogue of Theorem 6.1,and gives a geometric realization of double Kostka polynomials.
Theorem 9.1 ([AH]) . For λ ∈ P n, , put A λ = IC( O λ , ¯ Q l ) . Then we have (i) H i A λ = 0 for odd i . (ii) Take λ , µ ∈ P n, such that µ ≤ λ . For z ∈ O µ ⊂ O λ , we have (9.4) e K λ , µ ( t ) = t a ( λ ) X i ≥ (dim H iz A λ ) t i . Remark 9.2
If we compare the formula (9.4) with the formula (6.6) in Theorem6.1, we notice the discrepancy of the relations between t i and cohomology sheaves. In fact, in (6.6) t i corresponds to H ix A λ , while in (9.4) t i corresponds to H iz A λ .Later, we also consider the geometric realization of double Kostka polynomials bymeans of exotic symmetric space. In that case, this discrepancy is removed (seeTheorem 13.4). §
10. Symmetric spaces in algebraic setting
Historically, the symmetric space
G/K , where G is a connected Lie group and K is a compact subgroup such that ( G θ ) ⊂ K ⊂ G θ for an involutive automorphism θ of G , has been studied extensively from a point of view of the Riemannian geometry.However, here we are interested in its algebraic structure, namely, we consider G, K as algebraic groups and
G/K as an algebraic variety over k of any characteristic.The basis of the algebraic study of symmetric spaces was achieved by Vust [V] andRichardson [R], which will be summarized below.Let G be a connected reductive group over k with ch k ≥
0, and θ : G → G anautomorphism such that θ = 1. Consider the fixed point subgroup G θ = { g ∈ G | θ ( g ) = g } of G , and put H = ( G θ ) . (Here we use the notation H rather than K ,since we often use K as complexes of sheaves.) Put G ιθ = { g ∈ G | θ ( g ) = g − } , where ι : G → G, g g − is an anti-automorphism, so we regard G ιθ as the subsetof G consisting of ιθ -fixed elements. Put G ιθ = { gθ ( g ) − | g ∈ G } . It is known that G ιθ is a connected component of G ιθ , and there exists an isomorphism G ιθ ∼−→ G/G θ , gθ ( g ) − ↔ gG θ .G θ acts on G ιθ and G ιθ by the conjugation action, which corresponds to the leftmultiplication of G θ on G/G θ via the above isomorphism. Thus one can identify G ιθ with the symmetric space G/G θ , and from now on, we consider G ιθ as a symmetricspace in an algebraic setting. Remark 10.1
Assume that p = 2. Let θ : G → G be as above. Let g be the Liealgebra of G . Then θ induces an involutive automorphism on g , which we denoteby the same symbol as θ : g → g . We have the decomposition g ≃ g θ ⊕ g − θ , where g ± θ are ± θ . We note that g θ ≃ Lie G θ and g − θ is a G θ -stablesubspace of g such that g − θ ≃ T e ( G ιθ ). g − θ is usually referred as the symmetricspace with respect to G θ . Thus G ιθ is regarded as a global analogue of g − θ .Let G ιθ uni = G ιθ ∩ G uni be the set of unipotent elements in G ιθ . Then G ιθ uni is an H -stable closed subset of G ιθ , which has a role of the unipotent variety G uni in the caseof symmetric spaces. The following important result was proved by Richardson [R].In fact, Richadson reduced the problem to a similar problem for reductive groups,which certainly holds by Lusztig [L1]. Proposition 10.2.
Assume that p = 2 . Then the number of H -orbits in G ιθ isfinite. Remark 10.3.
In the Lie algebra case, put g − θ nil = g − θ ∩ g nil , where g nil is the set ofnilpotent elements in g . Then g − θ nil is an H -stable closed subset of g − θ . The finitenessproperty also holds in the Lie algebra case. But this does not hold for G ιθ uni if p = 2. §
11. Symmetric spaces associated to classical groups
From now on, we concentrate ourselves to the special type of symmetric spaces,namely the symmetric spaces associated to classical groups.Let V be an N -dimensional vector space over k with ch k = 2, and put G = GL N ≃ GL ( V ). We define an involutive automorphism θ : G → G by θ ( g ) = J − ( t g − ) J , where J = J or J , J = n n if N = 2 n + 1, J = (cid:18) n n (cid:19) if N = 2 n , J = (cid:18) n − n (cid:19) for N = 2 n .If J = J , then G θ = O N and H = SO N . While if J = J , then G θ = H = Sp N . Inboth cases, the following identity holds.(11.1) G ιθ = G ιθ . For later applications, we consider a generalization of the symmetric space G ιθ of the following type; for an integer r ≥
1, consider the direct product G ιθ uni × V r − ,on which H acts diagonally. G ιθ uni × V r − is called the exotic symmetric space oflevel r . In the following discussion, we are interested in extending the theory of theSpringer correspondence to the case of exotic symmetric spaces.Let T ⊂ B be the pair of θ -stable maximal torus and θ -stable Borel subgroup of G . Then the unipotent radical U of B is θ -stable. Put B H = ( B θ ) and T H = ( T θ ) .Then B H is a Borel subgroup of H and T H is a maximal torus of H contained in B H .We consider the flag variety B H = H/B H of H . Let M ⊂ · · · ⊂ M n be the isotropicflag in V such that the stabilizer of ( M i ) i in H is B H . Consider the varieties f X = { ( x, v , gB H ) ∈ G ιθ uni × V r − × B H | g − xg ∈ B ιθ , g − v ∈ M r − n } , X = [ g ∈ H g ( U ιθ × M r − n ) . and define a map π : f X → X by ( x, v , gB H ) ( x, v ).The map π : f X → X is an analogue of the Springer resolution π : e G uni → G uni ,and we want to consider an analogue of the Borho-MacPherson’s theorem. But inorder to apply the previous discussion, we need to verify two crucial properties,namely (i) the map π gives a resolution of singularities, (ii) the number of H -orbitsis finite. (Actually, (i) is too strong. For the theory of the Springer correspondence, enough to show that π is “semi-small”, which implies that (i ′ ) dim f X = dim X ).Thus we will verify those two properties (i ′ ) and (ii). Put δ = dim f X − dim X .The following holds. • H = Sp N r r = 1 r = 2 r ≥ δ δ > δ = 0 δ = 0number of orbits < ∞ < ∞ ∞• H = SO N r r = 1 r ≥ δ δ = 0 δ = 0number of orbits < ∞ ∞ From those tables, we see that the most suitable situation for the Springercorrespondence is that X = G ιθ uni × V for the case H = Sp N , and X = G ιθ uni for thecase H = SO N . This gives a reason why considering the exotic symmetric space,rather then the symmetric space itself, is important. In the case where H = Sp N , itis more natural to consider G ιθ uni × V than G ιθ uni . In the following sections, we considerthose standard cases, separately. However, we consider the other cases also, sincesome modified theory of the Springer correspondence holds in those cases, and theyhave own interests. §
12. Unipotent orbits in H and in G ιθ uni For later applications, in this section we describe the H -orbits in G ιθ uni in con-nection with the unipotent classes in G θ uni .In the following discussion, we only consider H = SO N ( N : odd) or H = Sp N ,namely H is of type B n or C n . Similar results hold in the case where H is of type D n ,but since the description becomes more complicated, we omit this, for simplicity.Let λ = (1 m , m , . . . ) ∈ P N be a partition of N . Let O λ be the unipotentclass in G = GL N corresponding to λ . The following result is well-known. Proposition 12.1 ( group case). Under the notation above, we have (i) O λ ∩ H = ∅ ⇐⇒ m i is even for odd i ( resp. for even i ) if H = Sp N ( resp. H = SO N ) . (ii) If O λ ∩ H = ∅ , then O λ = O λ ∩ H is a single class in H . (iii) For x ∈ O λ , Z G θ ( x ) ≃ ((cid:0)Q i :odd Sp m i × Q i :even O m i (cid:1) ⋉ U ( Sp N -case ) , (cid:0)Q i :even Sp m i × Q i :odd O m i (cid:1) ⋉ U ( SO N -case ) ,where U : connected unipotent normal group of Z G θ ( x ) . The case of H -orbits in G ιθ uni is given as follows. Proposition 12.2 ( symmetric space case). Let H = Sp N . (i) O λ ∩ G ιθ = ∅ ⇐⇒ m i is even for all i . (ii) If O λ ∩ G ιθ = ∅ , then O λ = O λ ∩ G ιθ is a single H -orbit. (iii) For x ∈ O λ , Z G θ ( x ) ≃ Y i Sp m i ⋉ U . Proposition 12.3 ( symmetric space case). Let H = SO N ( N : odd ) . (i) O λ ∩ G ιθ is always non-empty. (ii) O λ = O λ ∩ G ιθ is a single H -orbit. (iii) For x ∈ O λ , Z G θ ( x ) ≃ Y i O m i ⋉ U , Remark 12.4
By comparing Proposition 12.2 and Propositions 12.3, 12.4, we findvery interesting phenomena. In the group case, Z G θ ( x ) involves the subgroups oftype Sp and SO . Hence in the study of unipotent classes in H , the case of Sp andof SO cannot be separated. While in the case of symmetric spaces, the structureof Z G θ ( x ) is completely separated, namely in the case of H = Sp N , Z G θ ( x ) involvesonly subgroups of type Sp , and similarly for H = SO N . All the properties of Z G θ ( x )in the case of symmetric spaces are inherited from similar properties in the groupcase, but they produce two extreme phenomena to the opposite directions.As corollaries of above results, we can determine the structure of the componentgroup A H ( x ) as follows. Corollary 12.5 ( group case). Assume x ∈ O λ ⊂ H uni . Put A H ( x ) = Z H ( x ) /Z H ( x ) .Then A H ( x ) ≃ ( Z / Z ) a ( λ ) , where a ( λ ) = ( ♯ { i | i : even, m i = 0 } if H = Sp N , ♯ { i | i : odd, m i = 0 } − or if H = SO N . Corollary 12.6 ( symmetric space case). Assume x ∈ O λ ⊂ G ιθ uni . Put A H ( x ) = Z H ( x ) /Z H ( x ) . Then A H ( x ) ≃ ( Z / Z ) b ( λ ) , where b ( λ ) = ( if H = Sp N , ♯ { i | m i = 0 } − or if H = SO N . §
13. Exotic symmetric space associated to symplectic groups
We now consider the case of exotic symmetric space associated to symplecticgroups. So assume that H = Sp N and X = G ιθ uni × V . The fact that the set of H -orbits in X is parametrized by P n, was first proved by Kato [K1]. The followingis a reformulation of Kato’s result due to [AH].Let M n be the maximal isotropic subspace in V , stable by B H . There exists a θ -stable Levi subgroup L ⊂ G such that L θ ≃ GL ( M n ). Then we have embeddings(13.1) L ιθ uni × M n ⊂ G ιθ uni × V ⊂ G uni × V. Note that the left hand side and the right hand side of (13.1) are enhanced varietiesdiscussed in Section 7; GL ( M n ) acts on L ιθ uni × M n diagonally, and their orbits are parametrized by P n by Lemma 7.1. In turn, G acts on G uni × V diagonally, andtheir orbits are parametrized by P n by Lemma 7.1. We have Lemma 13.1 ([K1], [AH]) . Let λ ∈ P n, , and λ ∪ λ ∈ P n, . Let O L λ be the GL ( M n ) -orbit corresponding to λ , and O λ ∪ λ the G -orbit corresponding to λ ∪ λ .Then O λ is the unique H -orbit in X such that O L λ ⊂ O λ ⊂ O λ ∪ λ . The analogue of Lemma 7.2 and Remark 7.3 also hold for G ιθ uni × V . Proposition 13.2.
Let O λ be an H -orbit in G ιθ uni × V corresponding to λ ∈ P n, . (i) For z = ( x, v ) ∈ G ιθ uni × V , Z H ( z ) is connected. (ii) O λ = ` µ ≤ λ O µ , namely the closure relations are described by the dominanceorder on P n, . In this case, the map π : f X → X in Section 11 is given as follows; f X = { ( x, v, gB H ) ∈ G ιθ uni × V × B H | g − xg ∈ B ιθ , g − v ∈ M n } , X = [ g ∈ H g ( U ιθ × V ) = G ιθ uni × V, and π : ( x, v, gB H ) ( x, v ).Let W n be the Weyl group of H = Sp N , namely the Weyl group of type C n . Itis known that W ∧ n ≃ P n, , we denote by e V λ irreducible representation of W n cor-responding to λ ∈ P n, . (Note the difference from V λ , which was defined in (8.3).)The following theorem was first proved by Kato [K1] by applying the Ginzburg the-ory of Hecke algebras, and then reproved by [SS, I] in the framework of Lusztig’stheory of the generalized Springer correspondence. Theorem 13.3. π ∗ ¯ Q l [dim X ] is a semisimple perverse sheaf on X , equipped with W n -action, and is decomposed as π ∗ ¯ Q l [dim X ] ≃ M λ ∈ P n, e V λ ⊗ IC( O λ , ¯ Q l )[dim O λ ] . The geometric realization of double Kostka polynomials in terms of G ιθ uni × V was conjectured in [AH], and was proved by [K3], [SS, II], independently. In [SS,II],it is proved by making use of the discussion on character sheaves due to Lusztig[L4]. Theorem 13.4.
For λ ∈ P n, , put A λ = IC( O λ , ¯ Q l ) . Then we have (i) H i A λ = 0 unless i ≡ . (ii) For z ∈ O µ ⊂ O λ , (13.2) e K λ , µ ( t ) = t a ( λ ) X i ≥ (dim H iz A λ ) t i . Compare the formula (13.2) with the formula (9.4) in the enhanced case. Thecorrespondence H i ↔ t i in (13.2) is more natural than the correspondence H i ↔ t i in (9.4). Note that the modulo 4 vanishing of the cohomology of A λ was firstnoticed by Grojnowski [Gr].For z = ( x, v ) ∈ G ιθ uni × V , consider the Springer fibre π − ( z ) ≃ B Hz = { gB H ∈ B H | g − xg ∈ B ιθ , g − v ∈ M n } . Then H i ( B Hz , ¯ Q l ) has a structure of W n -module, which is an analogue of the Springerrepresentations. By using the Springer representation of W n , we obtain an expressionof double Kostka polynomials, which is an analogue of Corollary 6.2. Corollary 13.5.
Assume that z ∈ O µ . Then H i ( B Hz , ¯ Q l ) = 0 unless i ≡ ,and we have e K λ , µ ( t ) = t a ( λ ) X i ≥ h H i ( B Hz , ¯ Q l ) , e V λ i W n t i . §
14. Springer correspondence for G ιθ uni , the case H = Sp N As pointed out in Section 11, G ιθ uni for H = Sp N does not satisfy the conditionexplained there, so the Springer correspondence does not hold in the strict sense.However some modified theory of the Springer correspondence still holds for G ιθ uni ,as discussed in Henderson [H], which I will explain below.First we prepare some notation which is common for H = Sp N and SO N .Assume that H = Sp N or SO N ( N : odd), and consider G ιθ uni . Let N G ιθ be the setof pairs ( O , E ), where O is an H -orbit in G ιθ uni , and E is an H -equivariant simplelocal system on O . Thus as in Section 4, N G ιθ can be expressed as follows;(14.1) N G ιθ ≃ { ( x, τ ) | x ∈ G ιθ uni / ∼ H , τ ∈ A H ( x ) ∧ } . We consider the variety(14.2) e G ιθ uni = { ( x, gB H ) ∈ G ιθ uni × B H | g − xg ∈ B ιθ } and define a map π : e G ιθ uni → G ιθ uni by ( x, gB H ) x . Then π is proper, surjective, and e G ιθ uni is smooth, irreducible. If we put δ = dim e G ιθ uni − dim G ιθ uni , δ = 0 for H = SO N ,while δ > H = Sp N . In fact, in the latter case, π gives rise to a ( P ) n -bundleon its open dense part, and δ = n . Let W n = N H ( T H ) /T H be the Weyl group of H ,thus W n is of type C n or B n according to the case where H = Sp N or SO N . Weconsider S n as a subgroup of W n .Now assume that H = Sp N . By Proposition 12.2, the set G ιθ uni / ∼ H is parametrizedby P n , under the correspondence(14.3) O λ = O λ ∩ G ιθ uni ↔ λ ∈ P n , and by Corollary 12.6, Z H ( x ) is always connected for x ∈ G ιθ uni . Hence the localsystem E on O is the constant sheaf ¯ Q l , and we have (14.4) N G ιθ = { ( O , ¯ Q l ) } ≃ P n . The following result was first proved by [H] for g − θ nil , without the explicit corre-spondence, and was proved for G ιθ uni in [SS, I] in the following form. Theorem 14.1.
Let H = Sp N . Then π ∗ ¯ Q l [dim G ιθ uni ] is a semisimple complex,equipped with S n -action, and is decomposed as π ∗ ¯ Q l [dim G ιθ uni ] ≃ H • ( P n ) ⊗ M λ ∈ P n V λ ⊗ IC( O λ , ¯ Q l )[dim O λ ] , where H • ( P n ) = L i ≥ H i ( P n , ¯ Q l ) is a complex of vector spaces, and V λ is theirreducible S n -module corresponding to λ ∈ P n . Note that the occurrence of the factor H • ( P n ) depends on the fact that π isgenerically a P n -bundle. Remark 14.2.
If we consider the symmetric space G ιθ uni of general type, one canexpect that a similar phenomenon as in the case of H = Sp N occurs. The casewhere H = SO N seems to be rather special, and we could not find other examplessuch that the Springer correspondence holds in the strict sense. Thus in order todiscuss the (generalized) Springer correspondence for the general case, we nned toconsider a correction factor such as H • ( P n ) in the case of Sp N . In the Lie algebracase, Lusztig-Yun [LY] discusses a related problem. §
15. Springer correspondence for G ιθ uni , the case H = SO n +1 We now assume that H = SO n +1 . Then by Proposition 12.3, the set of H -orbitsin G ιθ uni is parametrized by P n +1 ;(15.1) O λ = O λ ∩ G ιθ uni ↔ λ ∈ P n +1 . We define a map Γ : P n → P n +1 by(15.2) Γ : µ = ( µ , . . . , µ k ) (2 µ + 1 , µ , . . . , µ k ) . The following result gives the Springer correspondence for G ιθ uni . It was first provedby Chen-Vilonen-Xue [CVX] for g − θ nil with k = C . The group case G ιθ uni is due to[SY]. Theorem 15.2.
Assume that H = SO n +1 . Then π ∗ ¯ Q l [dim G ιθ uni ] is a semisimpleperverse sheaf on G ιθ uni , equipped with S n -action, and is decomposed as (15.3) π ∗ ¯ Q l [dim G ιθ uni ] ≃ M µ ∈ P n V µ ⊗ IC( O Γ ( µ ) , ¯ Q l )[dim O Γ ( µ ) ] , where V µ is the irreducible S n -module corresponding to µ ∈ P n . Remarks 15.2. (i) The formula (15.3) looks very similar to the formula in thecase of GL n . But the pattern of the Springer correspondence is quite different fromthat of GL n (see Remarks 5.2 (ii)). In fact, the Springer correspondence for G ιθ uni isgiven as follows; • S n = V µ with µ = ( n ) ↔ IC( O Γ ( µ ) , ¯ Q l ), where Γ ( µ ) = (2 n + 1) ∈ P n +1 , • ε S n = V µ with µ = (1 n ) ↔ IC( O Γ ( µ ) , ¯ Q l ), where Γ ( µ ) = (3 , n − ) ∈ P n +1 .If λ = (2 n + 1), O λ is the open dense orbit in G ιθ uni , hence it is similar to O inRemarks 5.2 (ii). However if λ = (3 , n − ), the orbit O λ is not like O in [loc. cit.], O λ is much bigger than the unit orbit O λ ′ with λ ′ = (1 n +1 ).(ii) By Corollary 12.6, N G ιθ contains lots of pairs ( O , E ) such that E is not theconstant sheaf ¯ Q l . The formula (15.3) only involves the pairs such that E = ¯ Q l ,the constant sheaf. Thus the Springer correspondence is not enough to cover all thepairs in N G ιθ , and we need to consider the generalized Springer correspondence. §
16. Generalized Springer correspondence for G ιθ uni , the case H = SO n +1 In this section, we consider the generalized Springer correspondence for G ιθ uni with H = SO N ( N = 2 n + 1). First we will give a combinatorial description of N G ιθ . Let λ =( λ , . . . , λ N ) ∈ P N be a partition with λ N ≥
0. We define a symbol τ = ( τ , . . . , τ N )of type λ as follows;(i) τ i = ±
1, and τ i = 1 if λ i = 0.(ii) τ i = τ j if λ i = λ j .(iii) τ k = 1 if λ k is the largest odd number among λ , . . . , λ N .Then by Proposition 12.3, the set of symbols of type λ is in bijection with theset A H ( x ) ∧ for x ∈ O λ . Note that the condition (iii) is related to the differencebetween A G θ ( x ) and A H ( x ). Now the set N G ιθ can be expressed combinatorially bya set Ψ N as follows;(16.1) Ψ N := { ( λ, τ ) | λ ∈ P N , τ : type λ } ≃ N G ιθ . An element ( λ, τ ) ∈ Ψ N is called a cuspidal symbol if(i) λ i − λ i +1 ≤ i = 1 , . . . , N ( here we put λ N +1 = 0 ),(ii) If λ i − λ i +1 = 2, then τ i = τ i +1 .We denote by Ψ (0) N the set of cuspidal symbols in N G ιθ . Lusztig’s theory of the generalized Springer correspondence for reductivegroups can be extended to the case of symmetric spaces G ιθ uni , namely • The notion of a cuspidal pair for ( O , E ) ∈ N G ιθ , • The notion of induction ind GP for a θ -stable parabolic subgroup P and its θ -stable Levi subgroup L ,ind GP : { L H -equiv. perverse sheaves on L ιθ } → { semisimple complexes on G ιθ } The induction functor is defined by modifying the arguments in Section 5.The following result was proved in [SY].
Proposition 16.3.
Under the identification N G ιθ ≃ Ψ N , the pair ( O , E ) is cuspidalif and only if the corresponding symbol ( λ, τ ) ∈ Ψ N is cuspidal. Remark 16.4.
In general, θ -stable Levi subgroup L has the form L H ≃ GL a × G a × · · · × GL a r × SO N , where N = N − P i a i . Among them, L ιθ uni has a cuspidal pair ( O , E ) only when L H ≃ ( GL ) a × SO N for some 0 ≤ a ≤ n . We give here some example of cuspidal pairs. It would be interesting tocompare this with the case of reductive groups (Remark 5.2).(i) For λ = (2 a , N − a ), the pair ( O λ , E ) is always cuspidal, for any local system E .(ii) In particular, for O λ = { } with λ = (1 N ),the pair ( O λ , ¯ Q l ) is cuspidal(iii) The number | Ψ (0) N | is always large. For example, for N = 3 , ,
7, we have | Ψ (0) N | = 3 , , | Ψ (0) N | . Lemma 16.6.
We define a function q ( n ) for n ≥ by ∞ Y i =1 (1 + t i ) = X n ≥ q ( n ) t n Then we have | Ψ (0)2 n +1 | = q (2 n + 1) . We prepare some notations for formulating the generalized Springer cor-respondence. We define S G ιθ as the set of triples, ( L, O , E ), up to the conjugationaction of H , where L is a θ -stable Levi subgroup of a θ -stable parabolic subgroup P ,and ( O , E ) is a cuspidal pair in L ιθ uni . Since L H ≃ ( GL ) a × SO N with N = N − a (Remark 16.4), we have(16.2) S G ιθ ≃ C N := { ( N , ν, σ ) | N ≥ N : odd, ( ν, σ ) ∈ Ψ (0) N } Take ξ = ( N , ν, σ ) ∈ C N corresponding to ( L, O , E ) ∈ S G ιθ , and put K ξ =IC( O , E )[dim O ]. Since K ξ is an L H -equivariant perverse sheaf on L ιθ , one canconsider the complex ind GP K ξ on G ιθ uni .For each ξ = ( N , ν, σ ) ∈ C N , define a map Γ ξ : P a Ψ N as follows. Put e σ = ( σ , . . . , σ N ) = ( σ , . . . , σ N , , . . . ,
1) for σ = ( σ , . . . , σ N ). For µ ∈ P a , put λ = ν + 2 µ ∈ P N , and define(16.3) Γ ξ ( µ ) = ( λ, e σ ) ∈ Ψ N . The following result gives the generalized Springer correspondence for G ιθ uni ,which is an analogue of Theorem 5.1. Theorem 16.8 ([SY]) . (i) For ξ ∈ C N ≃ S G ιθ , ind GP K ξ is a semisimple per-verse sheaf on G ιθ uni , equipped with S a -action, and is decomposed as ind GP K ξ = M µ ∈ P a V µ ⊗ IC( O , E )[dim O ] , where Γ ξ ( µ ) ↔ ( O , E ) under the identification Ψ N ≃ N G ιθ (ii) For ξ ∈ S G ιθ , let N ( ξ ) G ιθ be the subset of N G ιθ corresponding to { Γ ξ ( µ ) | µ ∈ P a } ⊂ Ψ N . Then we have N G ιθ = a ξ ∈ S Gιθ N ( ξ ) G ιθ . (iii) The correspondence ( O , E ) ↔ Γ ξ ( µ ) ↔ V µ gives a bijection N G ιθ ≃ a ξ ∈ S Gιθ S ∧ a ( generalized Springer correspondence ) §
17. Exotic symmetric spaces of higher level, the case H = Sp N In this section, we consider the Springer correspondence for the exotic symmetricspaces of higher level associated to symplectic groups. As was remarked in Section11, the crucial difficulty in this case is that the number of H -orbits is not necessarilyfinite if r ≥ H = Sp N ≃ Sp ( V ), and consider G ιθ uni × V r − ( r ≥ H . Let M ⊂ · · · ⊂ M n ⊂ V be the total isotropic flag whose stabilizer in H is equal to B H . Consider the variety f X = { ( x, v , gB H ) ∈ G ιθ uni × V r − × B H | g − xg ∈ B ιθ , g − v ∈ M r − n } and define a map π : f X → G ιθ uni × V r − by ( x, v , gB H ) ( x, v ).Let Q n,r = { m = ( m , . . . , m r ) ∈ Z r ≥ | P m i = n } , and put Q n,r = { m ∈ Q n,r | m r = 0 } . For given m ∈ Q n,r , define p , . . . , p r − by p k = P ki =1 m i . Put, for m ∈ Q n,r , X m = [ g ∈ H g ( U ιθ × Y ≤ i 1. Note that if r = 2, this coincides withthe formula in Theorem 13.3. Also note that since X m has infinitely many H -orbits,we need to construct a suitable variety X λ instead of an H -orbit O λ . Theorem 17.1 ([S3]) . Assume m ∈ Q n,r , and put d m = dim X m . Then ( π m ) ∗ ¯ Q l [ d m ] is a semisimple perverse sheaf on X m , equipped with W n,r -action, and is decomposedas ( π m ) ∗ ¯ Q l [ d m ] ≃ M λ ∈ P ( m ) e V λ ⊗ IC( X λ , ¯ Q l )[dim X λ ] , where X λ is an H -stable, smooth, irreducible, locally closed sub-variety of G ιθ uni × V r − . If r ≥ , X λ is an infinite union of H -orbits. Assume m ∈ Q n,r , and take λ ∈ P ( m ) . Then X λ ⊂ X m . Put(17.2) d λ = (dim X m − dim X λ ) / . For z = ( x, v ) ∈ X m , define the Springer fibre B Hz ≃ π − ( z ) by(17.3) B Hz = { gB H ∈ B H | g − xg ∈ B ιθ , g − v ∈ M r − n } Then H i ( B Hz , ¯ Q l ) has a structure of W n,r -module, which is called the Springer rep-resentation of W n,r . As a corollary of Theorem 17.1, we obtain the following result,which is the Springer correspondence for the complex reflection group W n,r . Proposition 17.2. There exists an open dense subset X λ of X λ satisfying the fol-lowing. For z ∈ X λ , dim B Hz = d λ , and H d λ ( B Hz , ¯ Q l ) ≃ e V λ . In particular, themap X λ H d λ ( B Hz , ¯ Q l ) gives a bijective correspondence { X λ | λ ∈ P n,r } = a m ∈ Q n,r { X λ | λ ∈ P ( m ) } ≃ a m ∈ Q n,r { e V λ | λ ∈ P ( m ) } = W ∧ n,r . § 18. Exotic symmetric spaces, the case H = SO n +1 In this section, assume that H = SO n +1 . Consider the variety G ιθ uni × V r − and the isotropic flag ( M i ) i similarly as in Section 17. We fix ξ = ( N , ν, σ ) ↔ ( L, O , E ) ∈ C G ιθ , hence L H ≃ ( GL ) a × SO N with N = N − a . Consider thediagram(18.1) L ιθ α ←−−− c X ψ −−−→ f X π −−−→ G ιθ × V r − , where c X = { ( x, v , g ) ∈ G ιθ × V r − × H | g − xg ∈ P ιθ , g − v ∈ M r − a } , f X = { ( x, v , gP H ) ∈ G ιθ × V r − × H/P H | g − xg ∈ P ιθ , g − v ∈ M r − a } , and the maps are defined as ψ : ( x, v , g ) ( x, v , gP H ) , π : ( x, gP H ) x, α : ( x, v , g ) η P ( g − xg ) , where η P : P ιθ → L ιθ is the natural map induced from the projection P → L ≃ P/U P . Note that unless P = B , the map f X → L ιθ can not be defined directly.For m ∈ Q a,r , put X m = [ g ∈ H (cid:18) η − P ( L ιθ uni ) × Y ≤ i Let m ∈ Q a,r . For each λ ∈ P ( m ) , one can define a subvariety X λ of X m and an H -orbit O [ λ ] in G ιθ uni satisfying the following; (i) X λ is a locally closed, smooth, irreducible, H -stable subset of X m . (ii) The map ( x, v ) x gives a locally trivial fibration f λ : X λ → O [ λ ] ,whose fibre is isomorphic to an open dense subset of Q ≤ i The construction of ( π m ) ∗ e K ( m ) ξ is essentially the same as theconstruction of the induction ind GP K ξ in Section 16, which is a variant of the induc-tion functor of Lusztig. But in contrast to the cases of reductive groups or of G ιθ uni for H = SO n +1 , ( π m ) ∗ e K ( m ) ξ is not a priori a semisimple complex. So the previousdiscussion can not be applied directly, and we need a special care. § 19. Symmetric spaces in characteristic 2 The discussion in Section 11 makes sense even if p = 2, and one can define asymmetric space G ιθ uni associated to classical groups in characteristic 2. In this case,a different phenomenon appears since the fundamental properties given in Section10 do not hold. However, an analogue of the Springer correspondence still holds forthem. The Springer correspondence for G ιθ uni was studied in [DSY], which we brieflyexplain below.Let V ′ be an N -dimensional vector space over k with p = 2, and consider G = GL ( V ′ ). The involution θ : G → G is defined as in Section 11, and we consider G θ and G ιθ . Here G θ ≃ Sp ( V ), where V is an 2 n -dimensional subspace of V ′ if N = 2 n + 1 and V = V ′ if N = 2 n . Put H = Sp ( V ) and h = Lie H . Then h = sp ( V ): the symplectic Lie algebra with p = 2. θ induces an involution θ on g = Lie G , and we consider the subalgebra g θ = { x ∈ g | θ ( x ) = x } . The case where N is even, the situation is rather simple, namely we have Proposition 19.1. Assume that N = 2 n . Then (i) g θ ≃ Lie H = sp ( V ) : the symplectic Lie algebra over k . (ii) G ιθ uni ≃ g θ nil , compatible with the action of H . By the above result, considering the Springer correspondence for G ιθ uni is equiv-alent to considering it for sp ( V ) nil . The Springer correspondence for the nilpotentcone of the Lie algebras in characteristic 2 was established by T. Xue [X]. Thus theSpringer correspondence for G ιθ uni follows from her result. In particular, G ιθ uni / ∼ H isin bijection with P n, , via O λ ↔ λ , and the following formula holds.(19.1) π ∗ ¯ Q l [dim G ιθ uni ] ≃ M λ ∈ P n, e V λ ⊗ IC( O λ , ¯ Q l )[dim O λ ] , where e V λ is the irreducible representation of W n, corresponding to λ ∈ P n, . Next consider the case where N is odd. In this case, a quite different situationoccurs. We have Proposition 19.2.