From conjugacy classes in the Weyl group to unipotent classes, II
aa r X i v : . [ m a t h . R T ] A p r FROM CONJUGACY CLASSES IN THEWEYL GROUP TO UNIPOTENT CLASSES, II
G. LusztigIntroduction
Let G be a connected reductive algebraic group over an algebraically closedfield k of characteristic p ≥
0. Let G be the set of unipotent conjugacy classesin G . Let W be the set of conjugacy classes in the Weyl group W of G . LetΦ : W −→ G be the (surjective) map defined in [L8]. For C ∈ W we denote by m C the dimension of the fixed point space of w : V −→ V where w ∈ C and V is thereflection representation of the Coxeter group W . The following result provides aone sided inverse for Φ. Theorem 0.2.
For any γ ∈ G the function Φ − ( γ ) −→ N , C m C reaches itsminimum at a unique element C ∈ Φ − ( γ ) . Thus we have a well defined map Ψ : G −→ W , γ C such that ΦΨ : G −→ G is the identity map. It is likely that when k = C , the map Ψ coincides with the map G −→ W defined in [KL, Section 9], which we will denote here by Ψ ′ (note that Ψ ′ has notbeen computed explicitly in all cases). It is enough to prove the theorem in thecase where G is almost simple; moreover in that case it is enough to consider onegroup in each isogeny class. When G has type A , the theorem is trivial (Φ is abijection). For the remaining types the proof is given in Sections 1,2. Let G be a connected reductive algebraic group over C of the same typeas G ; we identify the Weyl group of G with W . Let G be the set of unipotentclasses of G . Let Φ : W −→ G , Ψ : G −→ W be the maps defined like Φ , Ψ(for G instead of G ). Theorem 0.4. (a) There is a unique (necessarily surjective) map ρ : G −→ G such that Φ = ρ Φ . We have ρ = Φ Ψ .(b) There is a unique (necessarily injective) map π : G −→ G such that Ψ =Ψ π . We have π = ΦΨ . The second sentence in 0.4(a) (resp. 0.4(b)) follows from the first since ΦΨ = 1.The uniqueness of ρ (resp. π ) follows from the surjectivity of Φ (resp. injectivity Supported in part by the National Science Foundation Typeset by
AMS -TEX G. LUSZTIG of Ψ). The surjectivity of ρ (if it exists) follows from the surjectivity of Φ . Theinjectivity of π (if it exists) follows from the injectivity of Ψ . The existence of ρ (resp. π ), which is equivalent to the identity Φ = Φ ΨΦ (resp. Ψ = ΨΦΨ ) isproved in Section 3, where various other characterizations of ρ and π are given. Let ˆ W be the set of irreducible representations of W (over Q ) up to isomor-phism. Let S W be the subset of ˆ W introduced in [L1]; it consists of representationsof W which were later called ”special representations”. For any E ∈ ˆ W let [ E ] bethe unique object of S W such that E, [ E ] are in the same two-sided cell of W ; thus E [ E ] is a surjective map ˆ W −→ S W . For any γ ∈ G let E γ ∈ ˆ W the Springerrepresentation corresponding to γ . We define ˜Φ : W −→ S W by C [ E Φ ( C ) ].Let G ♠ = { γ ∈ G ; E γ ∈ S W } . It is known that γ E γ is a bijection G ♠ ∼ −→ S W .This, together with the surjectivity of Φ , shows that ˜Φ is surjective. We nowdefine ˜Ψ : S W −→ W by ˜Ψ ( E γ ) = Ψ ( γ ), γ ∈ G ♠ . Note that ˜Ψ is injective and˜Φ ˜Ψ = 1. We have the following result. Theorem 0.6. (a) The (surjective) map ˜Φ : W −→ S W depends only on theCoxeter group structure of W . In particular, ˜Φ for G of type B n coincides with ˜Φ for G of type C n ( n ≥ ).(b) The (injective) map ˜Ψ : S W −→ W depends only on the Coxeter groupstructure of W . In particular, ˜Ψ for G of type B n coincides with ˜Ψ for G oftype C n ( n ≥ ). The proof is given in Section 4. Note that (a) was conjectured in [L7, 1.4].Let W ♠ be the image of ˜Ψ (a subset of W ). We say that W ♠ is the set of special conjugacy classes of W . Note that ˜Φ defines a bijection W ♠ ∼ −→ S W .Thus, the set of special conjugacy classes in W is in natural bijection with the setof special representations of W . From Theorem 0.6 we see that there is a naturalretraction W −→ W ♠ , C ˜Ψ ( ˜Φ ( C )) which depends only on the Coxeter groupstructure of W . The paper is organized as follows. In Section 1 (resp. Section 2) we describeexplicitly the map Φ in the case where G is almost simple of classical (resp. ex-ceptional) type and prove Theorem 0.2. In Section 3 we prove Theorem 0.4. InSection 4 we prove Theorem 0.6. In Section 5 we describe explicitly for each simpletype the bijection W ♠ ∼ −→ S W defined by ˜Φ . For γ, γ ′ ∈ G (or γ, γ ′ ∈ G ) we write γ ≤ γ ′ if γ is contained inthe closure of γ ′ .
1. Isometry groups
Let P be the set of sequences p ∗ = ( p ≥ p ≥ · · · ≥ p σ ) in Z > . For p ∗ ∈ P we set | p ∗ | = p + p + · · · + p σ , τ p ∗ = σ , µ j ( p ∗ ) = |{ k ∈ [1 , σ ]; p k = j }| ( j ∈ Z > ). Let P = { p ∗ ∈ P ; τ p ∗ = even } . For N ∈ N , κ ∈ { , } let ONJUGACY CLASSES IN THE WEYL GROUP AND UNIPOTENT CLASSES, II 3 P κN = { p ∗ ∈ P κ ; | p ∗ | = N } . Let ˜ P = { p ∗ ∈ P ; p = p , p = p , . . . } . Let S be the set of r ∗ ∈ P such that r k is even for all k . Let S = S ∩ P . For N ∈ N , κ ∈ { , } let S κN = S κ ∩ P κN .We fix n ∈ N and we define n ∈ N , κ ∈ { , } by n = 2 n + κ . We set A k n = { ( r ∗ , p ∗ ) ∈ S κ × ˜ P ; | r ∗ | + | p ∗ | = 2 n } . In the remainder of this section we fix V , a k -vector space of dimension n = 2 n + κ (see 1.1) with a fixed bilinear form ( , ) : V × V −→ k and a fixedquadratic form Q : V −→ k such that (i) or (ii) below holds:(i) Q = 0, ( x, x ) = 0 for all x ∈ V , V ⊥ = 0;(ii) Q = 0, ( x, y ) = Q ( x + y ) − Q ( x ) − Q ( y ) for x, y ∈ V , Q : V ⊥ −→ k isinjective.Here, for any subspace V ′ of V we set V ′⊥ = { x ∈ V ; ( x, V ′ ) = 0 } . In case (ii) itfollows that V ⊥ = 0 unless κ = 1 and p = 2 in which case dim V ⊥ = 1. Let Is ( V )be the subgroup of GL ( V ) consisting of all g ∈ GL ( V ) such that ( gx, gy ) = ( x, y )for all x, y ∈ V and Q ( gx ) = Q ( x ) for all x ∈ V . In this section we assume that G is the identity component of Is ( V ). In this subsection we assume that n ≥
3. Let W be the group of permutationsof [1 , n ] that commute with the involution χ : i n − i + 1. If Q = 0 or if κ = 1we identify (as in [L8, 1.4, 1.5]) W with W . If Q = 0 and κ = 0 we identify (asin [L8, 1.4, 1.5]) W with the group W ′ of even permutations of [1 , n ] commutingwith χ ; in this case let W (resp. W ) be the set of conjugacy classes in W whichare not conjugacy classes of W (resp. form a single conjugacy class in W ) and wedenote by ˜ W the set of W -conjugay classes in W ′ .If Q = 0 we identify W with A n by associating to the conjugacy class of w ∈ W the pair ( r ∗ , p ∗ ) where r ∗ is the multiset consisting of the sizes of cycles of w whichare χ -stable and p ∗ is the multiset consisting of the sizes of the cycles of w whichare not χ -stable. If Q = 0 we identify W = A n (if κ = 1) and ˜ W = A n (if κ = 0) by associating to the conjugacy class of w in W the pair ( r ∗ , p ∗ ) where r ∗ is the multiset consisting of the sizes of cycles of w (other than fixed points) whichare χ -stable and p ∗ is the multiset consisting of the sizes of cycles of w which arenot χ -stable. Let T n be the set of all c ∗ ∈ P n such that µ j ( c ∗ ) is even for any odd j .Let T (2)2 n be the set of all pairs ( c ∗ , ǫ ) where c ∗ ∈ T n and j ǫ ( j ) ∈ { , } is afunction defined on the set { j ∈ { , , , . . . } ; µ j ( c ∗ ) ∈ { , , , . . . }} .Let ˜ T (2)2 n = { ( c ∗ , ǫ ) ∈ T (2)2 n ; τ c ∗ is even } .Let Q be the set of all c ∗ ∈ P such that µ j ( c ∗ ) is even for any even j . For N ∈ N let Q N = Q ∩ P N .If Q = 0 , p = 2 we identify G = T n by associating to γ ∈ G the multisetconsisting of the sizes of the Jordan blocks of an element of γ .If Q = 0 , p = 2 we identify G = T (2)2 n by associating to γ ∈ G the pair ( c ∗ , ǫ )where c ∗ is the multiset consisting of the sizes of the Jordan blocks of an element G. LUSZTIG of γ ; ǫ ( j ) is equal to 0 if (( g − j − ( x ) , x ) = 0 for all x ∈ ker( g − j ( g ∈ γ ) and ǫ ( j ) = 1 otherwise (see [S1]).If Q = 0 , κ = 1 , p = 2 we identify G = Q n by associating to γ ∈ G the multisetconsisting of the sizes of the Jordan blocks of an element of γ .If Q = 0 , κ = 1 , p = 2 we identify G = T (2)2 n by associating to γ ∈ G thepair ( c ∗ , ǫ ) corresponding as above to the image of γ under the obvious bijectivehomomorphism from G to a symplectic group of an n − Q = 0 , κ = 0 we denote by G (0) (resp. G (1) ) the set of unipotent classes in G which are not conjugacy classes in Is ( V ) (resp. are also conjugacy classes in Is ( V )); let ˜ G be the set of Is ( V )-conjugacy classes of unipotent elements of G .We have an obvious imbedding G (1) ⊂ ˜ G .If Q = 0 , κ = 0 , p = 2 we identify ˜ G = Q n by associating to γ ∈ ˜ G the multisetconsisting of the sizes of the Jordan blocks of an element of γ .If Q = 0 , κ = 0 , p = 2 we identify ˜ G = ˜ T (2)2 n by associating to γ ∈ ˜ G thepair ( c ∗ , ǫ ) ∈ T (2)2 n corresponding as above to the image of γ under the obviousimbedding of Is ( V ) into the symplectic group of V, ( , ).We define ι : A n −→ T n by ( r ∗ , p ∗ ) c ∗ where the multiset of entries of c ∗ is the union of the multiset of entries of r ∗ with the multiset of entries of p ∗ . Wedefine ι (2) : A n −→ T (2)2 n by ( r ∗ , p ∗ ) ( c ∗ , ǫ ) where c ∗ = ι ( r ∗ , p ∗ ) and for any j ∈ { , , , . . . } such that µ j ( c ∗ ) ∈ { , , , . . . } , we have ǫ ( j ) = 1 if j = r i forsome i and ǫ ( j ) = 0, otherwise. When κ = 0 we define ˜ ι (2) : A n −→ ˜ T (2)2 n to be therestriction of ι (2) : A n −→ T (2)2 n .In the remainder of this section we assume that n ≥
3. We define Ξ : S κ −→ P by ( r ≥ r ≥ · · · ≥ r σ ) ( r + ψ (1) ≥ r + ψ (2) ≥ · · · ≥ r σ + ψ ( σ ))if σ + κ is even,( r ≥ r ≥ · · · ≥ r σ ) ( r + ψ (1) ≥ r + ψ (2) ≥ · · · ≥ r σ + ψ ( σ ) ≥ σ + κ is odd, where ψ : [1 , σ ] −→ {− , , } is as follows:if t ∈ [1 , σ ] is odd and r t < r x for any x ∈ [1 , t −
1] then ψ ( t ) = 1;if t ∈ [1 , σ ] is even and r x < r t for any x ∈ [ t + 1 , σ ], then ψ ( t ) = − t ∈ [1 , σ ] we have ψ ( t ) = 0.We define ι ′ : A κ n −→ Q n by ( r ∗ , p ∗ ) c ∗ where the multiset of entries of isthe union of the multiset of entries of Ξ( r ∗ ) with the multiset of entries of p ∗ . (Wewill see below that ι ′ is well defined.)With the identifications above and those in 1.3, we see that the map Φ : W −→ G becomes:(a) ι : A n −→ T n if Q = 0 , p = 2, see [L8, 3.7, 1.1];(b) ι ′ : A n −→ Q n if Q = 0 , κ = 1 , p = 2, see [L8, 3.8, 1.1];(c) ι (2) : A n −→ T (2)2 n if Q = 0 , p = 2, see [L8, 4.6, 1.1]; ONJUGACY CLASSES IN THE WEYL GROUP AND UNIPOTENT CLASSES, II 5 (d) ι (2) : A n −→ T (2)2 n if Q = 0 , κ = 1 , p = 2, see [L8, 4.6, 1.1];and that (when Q = 0 , κ = 0) the map ˜Φ : ˜ W −→ ˜ G induced by Φ becomes:(e) ι ′ : A n −→ Q n if Q = 0 , κ = 0 , p = 2, see [L8, 3.9, 1.1];(f) ˜ ι (2) : A n −→ ˜ T (2)2 n if Q = 0 , κ = 0 , p = 2, see [L8, 4.6, 1.1].In particular, ι ′ is well defined.We see that to prove 0.2 it is enough to prove the following statement:(g) In each of the cases (a)-(f ), the function ( r ∗ , p ∗ ) τ p ∗ / on any fibre ofthe map described in that case, reaches its minimum value at exactly one elementin that fibre. We have used that in the case where Q = 0 , κ = 0, the fibre of Φ over any elementin G (0) has exactly one element (necessarily in W ) and the fibre of Φ over anyelement in G (1) is contained in W and is the same as the fibre of ˜Φ over thatelement. We prove 1.4(g) in the case 1.4(a). Let c ∗ ∈ T n . Let ( r ∗ , p ∗ ) ∈ ι − ( c ∗ ). Let M e = µ e ( r ∗ ), N e = µ e ( p ∗ ), Q e = µ e ( c ∗ ) so that M e + N e = Q e . If e is odd then M e = 0 hence N e = Q e . Thus P e N e ≥ P e odd Q e . We see that the minimumvalue of the function 1.4(g) on ι − ( c ∗ ) is reached when M e = Q e , N e = 0 for e even and M e = 0 , N e = Q e for e odd. This proves 1.4(g) in our case. We prove 1.4(g) in the case 1.4(c). Let ( c ∗ , ǫ ) ∈ T (2)2 n . Let ( r ∗ , p ∗ ) ∈ ( ι (2) ) − ( c ∗ , ǫ ). Let M e = µ e ( r ∗ ), N e = µ e ( p ∗ ), Q e = µ e ( c ∗ ) so that M e + N e = Q e .If e is odd then M e = 0 hence N e = Q e . If e is even, Q e is even ≥ ǫ ( e ) = 0then M e = 0. Thus X e N e ≥ X e odd Q e + X e even ,Q e even , ≥ ,ǫ ( e )=0 Q e . We see that the minimum value of the function 1.4(g) on ( ι (2) ) − ( c ∗ , ǫ )) is reachedwhen M e = 0 , N e = Q e for e odd and for e even with Q e even , ≥ , ǫ ( e ) = 0 and M e = Q e , N e = 0 for all other e . This proves 1.4(g) in our case. The same proofyields 1.4(g) in the case 1.4(d); a similar proof yields 1.4(g) in the case 1.4(f). In this subsection we prove 1.4(g) in the cases 1.4(b) and 1.4(e).Let R be the set of all r ∗ = ( r ≥ r ≥ · · · ≥ r τ ) ∈ Q such that the followingconditions are satisfied. Let J r ∗ = { k ∈ [1 , τ ]; r k is odd } . We write the multiset { r k ; k ∈ J r ∗ } as a sequence r ≥ r ≥ · · · ≥ r s . (We have necessarily τ = s mod 2.) We require that:-if τ = 0 then 1 ∈ J r ∗ ;-if τ = 0 is even then τ ∈ J r ∗ ;-if u ∈ [1 , s −
1] is odd then r u > r u +1 ;-if u ∈ [1 , s −
1] is even then there is no k ′ ∈ [1 , τ ] such that r u > r k ′ > r u +1 .For N ∈ N we set R N = R ∩ Q N . G. LUSZTIG
Note that for any even N ∈ N , Ξ in 1.1 restricts to a bijection S κN −→ R N + κ with inverse map R N + κ −→ S κN given by( r ≥ r ≥ · · · ≥ r τ ) ( r + ζ (1) ≥ r + ζ (2) ≥ · · · ≥ r τ + ζ ( τ ))if r τ > κ ,( r ≥ r ≥ · · · ≥ r τ ) ( r + ζ (1) ≥ r + ζ (2) ≥ · · · ≥ r τ − + ζ ( τ − r τ = κ , where ζ : [1 , τ ] −→ {− , , } is given by ζ ( k ) = ( − k (1 − ( − r k ) / ζ ( k ) = ( − k if r k is odd and ζ ( k ) = 0 if r k is even. We have r k + ζ ( k ) ∈ N for any k and r k + ζ ( k ) ≥ r k +1 + ζ ( k + 1) for k ∈ [1 , τ − A n be the set of all pairs ( r ∗ , p ∗ ) ∈ R × ˜ P such that | r ∗ | + | p ∗ | = n . Wedefine ˜ ι : A n −→ Q n by ( r ∗ , p ∗ ) c ∗ where the multiset of entries of c ∗ is theunion of the multiset of entries of r ∗ with the multiset of entries of p ∗ . In viewof the bijection S κN −→ R N + κ defined by the restriction of Ξ we see that to prove1.4(g) in our case it is enough to prove the following statement.(a) For any c ∗ ∈ Q n there is exactly one element ( r ∗ , p ∗ ) ∈ ˜ ι − ( c ∗ ) such thatthe number of entries of p ∗ is minimal. Let c ∗ = ( c ≥ c ≥ · · · ≥ c τ ) ∈ Q n . Let K = { k ∈ [1 , τ ]; c k is odd } . We writethe multiset { c k ; k ∈ K } as a sequence c ≥ c ≥ · · · ≥ c t . (We have necessarily τ = n = t mod 2.) We associate to c ∗ an element ( r ∗ , p ∗ ) ∈ Q × ˜ P by specifying M e = µ e ( r ∗ ), N e = µ e ( p ∗ ) for e ≥
1. Let Q e = µ e ( c ∗ ).(i) If e ∈ N + 1 and Q e = 2 g + 1, then M e = 1, N e = 2 g .(ii) If e ∈ N + 1 and Q e = 2 g , so that c d = c d +1 = · · · = c d +2 g − = e with d even, then M e = 2, N e = 2 g − g >
0) and M e = N e = 0 (if g = 0).(iii) If e ∈ N + 1 and Q e = 2 g so that c d = c d +1 = · · · = c d +2 g − = e with d odd, then M e = 0, N e = 2 g .Thus the odd entries of r ∗ are defined. We write them in a sequence r ≥ r ≥· · · ≥ r s .(iv) If e ∈ N + 2, Q e = 2 g and if( ∗ ) r v > e > r v +1 for some v , or e > r , or r s > e (with s even),then M e = 0 , N e = 2 g .(v) If e ∈ N + 2, Q e = 2 g and if ( ∗ ) does not hold, then M e = 2 g, N e = 0.Now r ∗ ∈ Q , p ∗ ∈ ˜ P are defined and | r ∗ | + | p ∗ | = n .Assume that | r ∗ | >
0; then from (iv) we see that the largest entry of r ∗ is odd.Assume that | r ∗ | > n is even; then from (iv) we see that the smallest entryof r ∗ is odd. If u ∈ [1 , s −
1] and r u = r u +1 then from (i),(ii),(iii) we see that u iseven. If u ∈ [1 , s −
1] and there is k ′ ∈ [1 , τ ] such that r u > r k ′ > r u +1 , then r k ′ is even and e = r k ′ is as in (v) and u must be odd. We see that r ∗ ∈ R .We see that c ∗ ( r ∗ , p ∗ ) is a well defined map ˜ ι ′ : Q n −→ A n ; moreover,˜ ι ˜ ι ′ : Q n −→ Q n is the identity map.We preserve the notation for c ∗ , r ∗ , p ∗ as above (so that ( r ∗ , p ∗ ) ∈ ˜ ι − ( c ∗ )) andwe assume that ( r ′∗ , p ′∗ ) ∈ ˜ ι − ( c ∗ ). We write the odd entries of r ′∗ in a sequence r ′ ≥ r ′ ≥ · · · ≥ r ′ s ′ . ONJUGACY CLASSES IN THE WEYL GROUP AND UNIPOTENT CLASSES, II 7
Let M ′ e = µ e ( r ′∗ ), N ′ e = µ e ( p ′∗ ) for e ≥
1. Note that M ′ e + N ′ e = M e + N e .In the setup of (i) we have M ′ e = 1 , N ′ e = N e . (Indeed, M ′ e + N ′ e is odd, N ′ e iseven hence M ′ e is odd. Since M ′ e is 0 , g > M ′ e = 2 , N ′ e = N e or M ′ e = 0 , N ′ e = N e + 2. (Indeed, M ′ e + N ′ e is even, N ′ e is even hence M ′ e is even.Since M ′ e is 0 , g = 0 we have M ′ e = N ′ e = 0.In the setup of (iii) we have M ′ e = 0 , N ′ e = N e . (Indeed, M ′ e + N ′ e is even, N ′ e is even hence M ′ e is even. Since M ′ e is 0 , M ′ e = 2. Then e = r ′ u = r ′ u +1 with u even in [1 , s ′ − c d = c d +1 = · · · = c d +2 g − = e with d odd. From the definitions we see that u = d mod 2 and we have a contradiction. Thus, M ′ e = 0.)Now the sequence r ′ ≥ r ′ ≥ · · · ≥ r ′ s ′ is obtained from the sequence r ≥ r ≥ · · · ≥ r s by deleting some pairs of the form r h = r h +1 . Hence in the setupof (iv) we have r v = r ′ v ′ > e > r ′ v ′ +1 = r v +1 for some v ′ or e > r ′ or r ′ s ′ > e (with s, s ′ even) and we see that M ′ e = 0 so that N ′ e = 2 g = N e .In the setup of (v) we have N ′ e ≥ N ′ e ≥ N e . It follows that P e N ′ e ≥ P e N e (and the equality implies that N ′ e = N e for all e hence ( r ′∗ , p ′∗ ) = ( r ∗ , p ∗ )). Thisproves (a) and completes the proof of 1.4(g) in all cases hence the proof of 0.2 for G almost simple of type B, C or D .
2. Exceptional groups
In 2.2-2.6 we describe explicitly the map Φ : W −→ G in the case where G isa simple exceptional group in the form of tables. Each table consists of lines of theform [ a, b, . . . , r ] γ where γ ∈ G is specified by its name in [S2] and a, b, . . . , r are the elements of W which are mapped by Φ to γ (here a, b, . . . , r are specifiedby their name in [Ca]); by inspection we see that 0.2 holds in each case and in factΨ( γ ) = a is the first element of W in the list a, b, . . . , r . The tables are obtainedfrom the results in [L8]. G . If p = 3 we have[ A ] A [ A ] A [ A + ˜ A , ˜ A ] ˜ A [ A ] G ( a )[ G ] G .When p = 3 the line [ A + ˜ A , ˜ A ] ˜ A should be replaced by [ A + ˜ A ] ˜ A ,[ ˜ A ] ( ˜ A ) . F . If p = 2 we have[ A ] A [ A ] A [2 A , ˜ A ] ˜ A G. LUSZTIG [4 A , A , A + ˜ A , A + ˜ A ] A + ˜ A [ A ] A [ ˜ A ] ˜ A [ A + ˜ A ] A + ˜ A [ A + ˜ A , ˜ A + A ] ˜ A + A [ A , B ] B [ A + ˜ A , B + A ] C ( a )[ D ( a )] F ( a )[ D , B ] B [ C + A , C ] C [ F ( a )] F ( a )[ B ] F ( a )[ F ] F .When p = 2 the lines [2 A , ˜ A ] ˜ A , [ A + ˜ A , ˜ A + A ] ˜ A + A , [ A , B ] B ,[ A + ˜ A , B + A ] C ( a ), should be replaced by[2 A ] ˜ A , [ ˜ A ] ( ˜ A ) [ A + ˜ A ] ˜ A + A , [ ˜ A + A ] ( ˜ A + A ) [ A ] B , [ B ] ( B ) [ A + ˜ A ] C ( a ), [ B + A ] ( C ( a )) respectively. E . We have[ A ] A [ A ] A [2 A ] A [4 A , A ] A [ A ] A [ A + A ] A + A [2 A ] A [ A + 2 A ] A + 2 A [ A ] A [3 A , A + A ] A + A [ A + 2 A , A + A ] A + A [ D ( a )] D ( a )[ A ] A [ D ] D [ A + A ] A + A [ A + A , A ] A [ D ( a )] D ( a )[ E ( a )] A + A [ D ] D [ E ( a )] E ( a )[ E ] E . ONJUGACY CLASSES IN THE WEYL GROUP AND UNIPOTENT CLASSES, II 9 E . If p = 2 we have[ A ] A [ A ] A [2 A ] A [(3 A ) ′ ] (3 A ) ′′ [(4 A ) ′′ , (3 A ) ′′ ] (3 A ) ′ [ A ] A [7 A , A , A , (4 A ) ′ ] A [ A + A ] A + A [ A + 2 A ] A + 2 A [ A ] A [2 A ] A [ A + 3 A ] A + 3 A [( A + A ) ′ ] ( A + A ) ′′ [3 A , A + A ] A + A [( A + 2 A ) ′′ , ( A + A ) ′′ ] ( A + A ) ′ [ D ( a )] D ( a )[ A + 3 A , ( A + 2 A ) ′ ] A + 2 A [ D ] D [ D ( a ) + A ] D ( a ) + A [ D ( a ) + 2 A , A + A ] A + A [2 A + A , A + A + A ] A + A + A [ A ] A [ D + 3 A , D + 2 A , D + A ] D + A [ A ′ ] A ′′ [ A + A ] A + A [ D ( a )] D ( a )[ A + A ] A + A [( A + A ) ′′ , A ′′ ] A ′ [ A + A , ( A + A ) ′ ] ( A + A ) ′′ [ D ( a ) + A ] D ( a ) + A [ E ( a )] ( A + A ) ′ [ D ( a ) + A , D ( a )] D ( a )[ E ( a )] D ( a ) + A [ D ] D [ A ] A [ D + A ] D + A [ D ( a )] D ( a )[ A ] D ( a ) + A [ E ( a )] E ( a )[ D + A , D ] D [ E ] E [ E ( a )] D + A [ E ( a )] E ( a )[ E ( a )] E ( a )[ E ] E .If p = 2, the line [ D ( a ) + 2 A , A + A ] A + A should be replaced by[ D ( a ) + 2 A ] A + A , [ A + A ] ( A + A ) . E . If p = 2 , A ] A [ A ] A [2 A ] A [(4 A ) ′ , A ] A [ A ] A [8 A , A , A , A , (4 A ) ′′ ] A [ A + A ] A + A [ A + 2 A ] A + 2 A [ A ] A [ A + 4 A , A + 3 A ] A + 3 A [2 A ] A [3 A , A + A ] A + A [( A + 2 A ) ′ , A + A ] A + A [ D ( a )] D ( a )[4 A , A + A , A + 2 A ] A + 2 A [ D ] D [ A + 4 A , A + 3 A , ( A + 2 A ) ′′ ] A + 2 A [ D ( a ) + A ] D ( a ) + A [(2 A ) ′ , A + A A + A [ A ] A [2 A + 2 A , A + A + 2 A , A + A , A + A + A ] A + A + A [ D ( a ) + A ] D ( a ) + A [ D + 4 A , D + 3 A , D + 2 A , D + A ] D + A [2 D ( a ) , D ( a ) + A , (2 A ) ′′ ] A [ A + A ] A + A [ D ( a )] D ( a )[ A + 2 A ] A + 2 A [ A + A ] A + A [ A + A + A ] A + A + A [ D ( a ) + A ] D ( a ) + A [( A + A ) ′ , A ] A [ D + A , D + A ] D + A [ E ( a )] ( A + A ) ′′ [2 A , A + A ] A + A [ D ] D [ D ( a ) + A , D ( a ) + A ] D ( a ) + A [ A + A + A , A + A , A + 2 A , ( A + A ) ′′ ] ( A + A ) ′ ONJUGACY CLASSES IN THE WEYL GROUP AND UNIPOTENT CLASSES, II 11 [ E ( a ) + A , E ( a ) + A ] A + 2 A [2 D , D ( a ) + A , D ( a )] D ( a )[ E ( a ) + A , E ( a )] A + A [ D + 2 A , D + A ] D + A [ E ( a )] A [ D ( a ) D ( a )][ A ] A [ A + A ] A + A [ A ′ ] D ( a ) + A [ A + A , D + A ] D + A [ E ( a )] E ( a )[ D + 2 A , D + A , D ] D [ D ( a )] D ( a )[ E ] E [ D ( a ) , A ′′ ] A [ E ( a ) + A ] E ( a ) + A [ E ( a )] D + A [ A ] D ( a )[ D ( a ) , D ( a )] D ( a )[ E + A , E + A ] E + A [ E ( a ) + A , E ( a )] E ( a )[ E ( a )] A [ E ( a )] E ( a ) + A [ D ( a ) , D ] D [ E ( a )] D ( a )[ E ( a )] E ( a )[ D ] E ( a ) + A [ E ( a )] D [ E + A , E ] E [ E ( a )] E + A [ E ( a )] E ( a )[ E ( a )] E ( a )[ E ] E .If p = 3 the line [ D ( a ) , A ′′ ] A should be replaced by [ D ( a )] A , [ A ′′ ] ( A ) . If p = 2 the lines [(2 A ) ′ , A + A ] A + A , [ D + A , D + A ] D + A ,[ A + A , D + A ] D + A , [ D ( a ) , D ( a )] D ( a ) should be replaced by[(2 A ) ′ ] A + A , [ A + A ] ( A + A ) [ D + A ] D + A , [ D + A ] ( D + A ) [ A + A ] D + A , [ D + A ] ( D + A ) [ D ( a )] D ( a ), [ D ( a )] ( D ( a )) respectively. From the tables above we see that (assuming that G is almost simple ofexceptional type) the following holds.(a) If γ is a distinguished unipotent class in G then Φ − ( γ ) consists of a singleconjugacy class in W . In fact (a) is also valid without any assumption on G . Indeed, assume that C ∈ Φ − ( γ ). From the arguments in [L8, 1.1] we see that if C is not elliptic then Φ( C )is not distinguished. Thus C must be elliptic. Then the desired result follows fromthe injectivity of the restriction of Φ to elliptic conjugacy classes in W , see [L8,0.6]. We have the following result (for general G ).(a) If C is an elliptic conjugacy class in W , then C = Ψ(Φ( C )) . In particular, C is in the image of Ψ : G −→ W . Since C ∈ Φ − (Φ( C )), (a) follows immediately from Theorem 0.2.
3. Proof of Theorem 0.4
There is a well defined (injective) map π ′ : G −→ G , γ π ′ ( γ ), where π ′ ( γ ) is the unique unipotent class in G which has the same Springer represen-tation of W as γ . One can show that π ′ coincides with the order preserving anddimension preserving imbedding defined in [S1, III,5.2].To prove Theorem 0.4(a) we can assume that G, G are almost simple. It is alsoenough to prove the theorem for a single G in each isogeny class of almost simplegroups. We can assume that p is a bad prime for G (if p is not a bad prime, theresult is obvious). Now G, G cannot be of type A since p is a bad prime for G .In the case where G is of exceptional type, the theorem follows by inspection ofthe tables in Section 2. In the case where G is of type B, C or D so that p = 2,we define ˜ ρ : G −→ G by ˜ ρ ( γ ) = γ where γ ∈ G is in the ”unipotent piece” of G indexed by γ ∈ G (in the sense of [L6]). It is enough to prove that(a) ˜ ρ Φ = Φ . (This would prove the existence of ρ in 0.4 and that ρ = ˜ ρ .) If G, G are of type C n ( n ≥
2) then Φ , Φ may be identified with ι : A n −→ T n , ι (2) : A n −→ T (2)2 n (see 1.4) and by [L4], ˜ ρ may be identified with T (2)2 n −→ T n , ( c ∗ , ǫ ) c ∗ (notationof 1.4). Then the identity (a) is obvious. The proof of (a) for G of type B and D is given in 3.5.Similarly to prove Theorem 1.4(b) it is enough to prove that(b) Ψ π ′ = Ψ . (This would prove the existence of π in 0.4 and that π = π ′ .) Again it is enoughto prove (b) in the case where G is of type B, C or D so that p = 2. The proof isgiven in 3.9. ONJUGACY CLASSES IN THE WEYL GROUP AND UNIPOTENT CLASSES, II 13
In this subsection we assume that all simple factors of G are of type A, B, C or D . Then ˜ ρ : G −→ G is defined as in 3.1. We show:(a) Φ ( C ) = ˜ ρ Φ( C ) for any elliptic conjugacy class C in W . From the definitions we see that(b) ˜ ρπ ′ = 1.Now for C as in (a) we have by definition Φ( C ) = π ′ Φ ( C ). To prove (a) it isenough to show that Φ ( C ) = ˜ ρπ Φ ( C ). But this follows from (b). In this subsection we assume that
V, Q, ( , ) , n , Is ( V ) are as in 1.2 with p = 2and that either Q = 0 or V = 0. Let SO ( V ) be the identity component of Is ( V ).Let U V be the set of unipotent elements in SO ( V ).We say that u ∈ U V is split if the corresponding pair ( c ∗ , ǫ ) (see 1.4) satisfies µ j ( c ∗ ) =even for all j and ǫ ( j ) = 0 for all even j such that µ j ( c ∗ ) > µ j as in1.1).For u ∈ U V let e = e u be the smallest integer ≥ u − e = 0.(When u = 1 we have e = 1 if n > e = 0 if n = 0. When u = 1 we have e ≥ u = 1 we define λ = λ u : V −→ k by λ ( x ) = p ( x, ( u − e − x ), a linearform on V ; in this case we define L = L u as follows: L = ( u − e − V if λ = 0; L = (ker λ ) ⊥ if λ = 0 , n = even; L = { x ∈ (ker λ ) ⊥ ; Q ( x ) = 0 if λ = 0 , n = odd . Note that L = 0.As in [L5, 2.5], for any u ∈ U V we define subspaces V a = V au ( a ∈ Z ) of V as follows. If u = 1 we set V a = V for a ≤ V a = 0 for a ≥
1. If u = 1 sothat e = e u ≥
2, and λ = λ u , L = L u are defined, we have L ⊂ L ⊥ , Q | L = 0(see [L5,2.2(a),(e)]) and we set ¯ V = L ⊥ /L ; now ¯ V has an induced nondegeneratequadratic form and there is an induced unipotent element ¯ u ∈ SO ( ¯ V ); let r : L ⊥ −→ L ⊥ /L = ¯ V be the canonical map. Since dim ¯ V < dim V , we can assume byinduction that ¯ V a = ¯ V a ¯ u are defined for a ∈ Z . We set V a = V if a ≤ − e, V a = r − ( ¯ V a ) if 2 − e ≤ a ≤ e − , V a = 0 if a ≥ e when λ = 0 and V a = V if a ≤ − e, V a = r − ( ¯ V a ) if 1 − e ≤ a ≤ e, V a = 0 if a ≥ e when λ = 0. This completes the inductive definition of V a = V au .Note that if u ∈ U V is split and u = 1 then ¯ u (as above) is split.Now assume that V = V ′ ⊕ V ′′ where V ′ , V ′′ are subspaces of V such that( V ′ , V ′′ ) = 0 so that Q | V ′ , Q | V ′ are nondegenerate and let u ∈ U V be such that V ′ , V ′′ are u -stable; we set u ′ = u | V ′ , u ′′ = u | V ′′ . We assume that u ′ ∈ U V ′ , u ′′ ∈ U V ′′ . For a ∈ Z we set V a = V au , V ′ a = V ′ au ′ , V ′′ a = V ′′ au ′′ . We show: (a) if u ′′ is split, then V a = V ′ a ⊕ V ′′ a for all a . If u = 1, then (a) is trivial. So we can assume that u = 1. We can also assumethat (a) is true when V is replaced by a vector space of smaller dimension. Let e = e u , e ′ = e u ′ , e ′′ = e u ′′ . Let λ = λ u , L = L u . Let r : L ⊥ −→ L ⊥ /L = ¯ V beas above. If u ′ = 1 (resp. u ′′ = 1) we define λ ′ , L ′ , r ′ , ¯ V ′ (resp. λ ′′ , L ′′ , r ′′ , ¯ V ′′ )in terms of u ′ , Q V ′ (resp. u ′′ , Q V ′′ ) in the same way as λ, L, r, ¯ V were defined interms of u, Q . Note that λ ′′ = 0 (when u ′′ = 1) since u ′′ is split.Assume first that e ′′ > e ′ . We have e = e ′′ hence u ′′ = 1. Moreover, L = 0 ⊕ L ′′ ,¯ V = V ′ ⊕ ¯ V ′′ . By the induction hypothesis we have ¯ V a = V ′ a ⊕ ¯ V ′′ a for all a . If a ≤ − e then V a = V = V ′ ⊕ V ′′ = V ′ a ⊕ V ′′ a . (We use that V ′ = V ′ a ; if e ′ ≥ a ≤ − e ′ ; if e ′ ≤ a ≤ − e ≤ a ≤ e − V a = (0 ⊕ r ′′ ) − ( ¯ V a ) = (0 ⊕ r ′′ ) − ( V ′ a ⊕ ¯ V ′′ a ) = V ′ a ⊕ V ′′ a (we use that e ′′ = e ); if a ≥ e then V a = 0 = V ′ a ⊕ V ′′ a . (We use that V ′ a = 0; if e ′ ≥ a ≥ e ′ + 1; if e ′ ≤ a ≥ e ′ = e ′′ (hence both are equal to e ≥
2) and that λ = 0(hence λ ′ = 0). Then L = L ′ ⊕ L ′′ , ¯ V = ¯ V ′ ⊕ ¯ V ′′ . By the induction hypothesis wehave ¯ V a = ¯ V ′ a ⊕ ¯ V ′′ a for all a . If a ≤ − e then V a = V = V ′ ⊕ V ′′ = V ′ a ⊕ V ′′ a .If 2 − e ≤ a ≤ e −
1, we have V a = ( r ′ ⊕ r ′′ ) − )( ¯ V a ) = ( r ′ ⊕ r ′′ ) − )( ¯ V ′ a ⊕ ¯ V ′′ a ) = V ′ a ⊕ V ′′ a . If a ≥ e then V a = 0 = V ′ a ⊕ V ′′ a .Next we assume that e ′ > e ′′ (hence e ′ = e ≥
2) and λ = 0 (hence λ ′ = 0). Then L = L ′ ⊕
0, ¯ V = ¯ V ′ ⊕ V ′′ . By the induction hypothesis we have ¯ V a = ¯ V ′ a ⊕ V ′′ a for all a . If a ≤ − e then V a = V = V ′ ⊕ V ′′ = V ′ a ⊕ V ′′ a . (We use that a ≤ − e ′′ .) If 2 − e ≤ a ≤ e − V a = ( r ′ ⊕ − ( ¯ V a ) = ( r ′ ⊕ V ′ a ⊕ V ′′ a ) = V ′ a ⊕ V ′′ a . If a ≥ e then V a = 0 = V ′ a ⊕ V ′′ a . (We use that a ≥ e ′′ .)Finally we assume that e ′ ≥ e ′′ (hence e ′ = e ≥
2) and λ = 0 (hence λ ′ = 0).Then L = L ′ ⊕
0, ¯ V = ¯ V ′ ⊕ V ′′ . By the induction hypothesis we have ¯ V a =¯ V ′ a ⊕ V ′′ a for all a . If a ≤ − e then V a = V = V ′ ⊕ V ′′ = V ′ a ⊕ V ′′ a . (We usethat a ≤ − e ′′ .) If 1 − e ≤ a ≤ e , we have V a = ( r ′ + 0) − ( ¯ V a ) = ( r ′ + 0) − ( ¯ V ′ a ⊕ V ′′ a ) = V ′ a ⊕ V ′′ a . If a ≥ e + 1 then V a = 0 = V ′ a ⊕ V ′′ a . (We use that a ≥ e ′′ .)This completes the proof of (a). ONJUGACY CLASSES IN THE WEYL GROUP AND UNIPOTENT CLASSES, II 15
In this subsection we assume that
V, Q, ( , ) , n , Is ( V ) are as in 1.2 with p = 2and n ≥
3. Let SO ( V ) be the identity component of Is ( V ). Let V be a C -vectorspace of dimension n with a fixed nondegenerate symmetric bilinear form ( , ). Let SO ( V ) be the corresponding special orthogonal group. Let U V be the set ofunipotent elements in SO ( V ). For any u ∈ U V we define subspaces V a = ( V ) au ( a ∈ Z ) of V in the same way as V au were defined in 3.5 for u ∈ U V except thatwe now take λ to be always zero (compare [L5, 3.3]).Assume that we are given a direct sum decomposition V = V ′ ⊕ V ′′ ⊕ V ′′ ⊕ . . . ⊕ V ′′ k − ⊕ V ′′ k where V ′ , V ′′ i are subspaces of V such that Q | V ′′ i = 0, ( V ′ , V ′′ i ) = 0 for i ∈ [1 , k ],( V ′′ i , V ′′ j ) = 0 if i + j = 2 k + 1. Let V ′′ = V ′′ ⊕ V ′′ ⊕ . . . ⊕ V ′′ k − ⊕ V ′′ k . Let V = V ′ ⊕ V ′′ ⊕ V ′′ ⊕ . . . ⊕ V ′′ k − , ⊕ V ′′ k, be a direct sum decomposition where V ′ , V ′′ i are subspaces of V such that ( , ) is0 on V ′′ i , ( V ′ , V ′′ i ) = 0 for i ∈ [1 , k ], ( V ′′ i , V ′′ j ) = 0 if i + j = 2 k + 1. Let V ′′ = V ′′ ⊕ V ′′ ⊕ . . . ⊕ V ′′ k − , ⊕ V ′′ k, . Let L (resp. L ) be the simultaneous stabilizer in SO ( V ) (resp. SO ( V )) of thesubspaces V ′ , V ′′ i (resp. V ′ , V ′′ i ), i ∈ [1 , k ].Let u ∈ SO ( V ) be such that uV ′ = V ′ , uV ′′ i = V ′′ i for i ∈ [1 , k ] and suchthat the SO ( V )-conjugacy class γ of u is also an Is ( V )-conjugacy class. Then uV ′′ = V ′′ . Let u ′ ∈ SO ( V ′ ), u ′′ ∈ SO ( V ′′ ) be the restrictions of u . Note that u ′′ is split. Let γ be the conjugacy class of u in L . Let γ ′ = ˜ ρ ( γ ), a unipotentclass in L ; here ˜ ρ : L −→ L is defined like ˜ ρ in 3.1 but in terms of L, L insteadof G, G . Let u ∈ γ ′ . Let u ′ , u ′′ be the restrictions of u to V ′ , V ′′ . By results in[L5, 2.9], we havedim V ′ ,u ′ a = dim V ′ u ′ a , dim V ′′ ,u ′′ a = dim V ′′ u ′′ a for all a .Let γ be the conjugacy class of u in SO ( V ). From 3.3(a) and the analogousresult for V = V ′ ⊕ V ′′ we see that dim V ,u a = dim V au for all a . From this wededuce, using the definitions and the fact that γ is an Is ( V )-conjugacy class, that γ = ˜ ρ ( γ ). We now prove 3.1(a) assuming that G = SO ( V ) , G = SO ( V ) are as in3.4. Let C ∈ W be such that C is a W -conjugacy class. We can find a standardparabolic subgroup W ′ of W and an elliptic conjugacy class C ′ of W ′ such that C ′ ⊂ C . Let P (resp. P ) be a parabolic subgroup of G (resp. G ) of the sametype as W ′ . Let L be a Levi subgroup of P and let L be a Levi subgroup of P .We can assume that L, L are as in 3.4. Let Φ L , Φ L be the maps analogous toΦ , Φ with G , G replaced by L , L . Let γ = Φ L ( C ′ ). By definition γ := Φ( C )is the unipotent class in G that contains γ ; note that γ is an Is ( V )-conjugacyclass. Let γ ′ = ˜ ρ ( γ ) (notation of 3.4), a unipotent class in L . Using 3.2(a) for L, L , C ′ instead of G, G , C we see that γ ′ = ˜ ρ Φ L ( C ′ ) = Φ L ( C ′ ). By definition, γ := Φ ( C ) is the unipotent class in G that contains γ ′ . From 3.4 we have γ = ˜ ρ ( γ ) that is Φ ( C ) = ˜ ρ Φ( C ). This completes the proof of 3.1(a) and that ofTheorem 0.4. The equality ρ = ˜ ρ in 3.1 provides an explicit computation of the map ˜ ρ forspecial orthogonal groups (since the maps Φ , Ψ are described in each case explicitlyin Sections 1,2). The first explicit computation of ˜ ρ in this case was given in [Xue]in terms of Springer representations instead of the maps Φ , Ψ. According to [S1,III,5.4(b)] there is a well defined map ρ ′ : G −→ G , γ γ such that: γ ≤ π ′ ( γ ) ( γ ∈ G ); if γ ≤ π ′ ( γ ′ ) ( γ ′ ∈ G ) then γ ≤ γ ′ . We notethe following result (see [Xue]):(a) If G is almost simple of type B, C or D then ˜ ρ = ρ ′ . Next we note:(b)
For any G we have ρ = ρ ′ . We can assume that G is almost simple and that p is a bad prime for G . If G is ofexceptional type, then ρ can be computed from the tables in Section 2 and ρ ′ canbe computed from the tables in [S1, IV]; the result follows. If G is of type B, C or D then as we have seen earlier we have ˜ ρ = ρ and the result follows from (a). The results in this subsection are not used elsewhere in this paper. We definea map ρ ′′ : G −→ G as follows. Let γ ∈ G . We can find a Levi subgroup L of aparabolic subgroup P of G and γ ∈ L such that γ ⊂ γ and γ is ”distinguished”in L (that is, any torus in the centralizer in L of an element in γ is contained inthe centre of L ). Let L be a Levi subgroup of a parabolic subgroup P of G ofthe same type as L . We have γ = π L ( γ ′ ) for a well defined γ ′ ∈ L where π L is the map analogous to π in 3.1 but for L, L instead of G, G . Let ρ ′′ ( γ ) be theunique unipotent class in G which contains γ ′ . This is independent of the choicesand γ ρ ′′ ( γ ) defines the map ρ ′′ . We show:(a) ρ ′′ = ρ .Let γ, L, L , P, P , γ , γ ′ be as above. Let W ′ be a standard parabolic subgroupof W of the same type as P, P . We can find an elliptic conjugacy class C ′ of W ′ such that γ = Φ L ( C ′ ), γ ′ = Φ L ( C ′ ), where Φ L , Φ L are defined like Φ in termsof L, L instead of G . Let C be the conjugacy class of W that contains W ′ . From[L8, 1.1, 4.5] we see that Φ( C ) is the unique unipotent class in G that contains γ ONJUGACY CLASSES IN THE WEYL GROUP AND UNIPOTENT CLASSES, II 17 and Φ ( C ) is the unique unipotent class in G that contains γ ′ . Thus Φ( C ) = γ and Φ ( C ) = ρ ( γ ). We see that ρ ′′ ( γ ) = ρ ′′ (Φ( C )) = Φ ( C ) = ρ Φ( C ) = ρ ( γ ) . This proves (a).
In this subsection we assume that G is of type B, C or D and p = 2. Theidentity Ψ = Ψ π ′ follows from the explicit combinatorial description of Ψ , Ψgiven in section 1 and the explicit combinatorial description of π ′ given in [S1,III,6.1,7.2,8.2]. This completes the proof of 3.1(b) hence also that 0.4(b).The fact that Ψ ′ (see 0.2) and π ′ might be described by the same combinatoricswas noticed by the author in 1987 who proposed it as a problem to Spaltenstein;he proved it in [S3] (see [S3, p.193]). Combining this with the (simple) descriptionof Ψ given in Section 1, we deduce that Ψ ′ = Ψ π ′ . In particular we have Ψ = Ψ ′ .
4. Proof of Theorem 0.6
Let G ♠ be the image of G ♠ (see 0.5) under the imbedding π ′ : G −→ G (see3.1). The unipotent classes in G ♠ are said to be special. The following result canbe extracted from [S1, III].(a) There exists order preserving maps e : G −→ G , e : G −→ G such that theimage of e (resp. e ) is equal to G ♠ (resp. G ♠ ) and such that for any γ ∈ G ,(resp. γ ∈ G ), γ ≤ e ( γ ) (resp. γ ≤ e ( γ ) ). The map e (resp. e ) is unique.Moreover we have e = e , e = e , e = π ′ e ρ ′ where ρ ′ is as in 3.7 . Strictly speaking (a) does not appear in [S1, III] in the form stated above sincethe notion of special representations from [L1] and the related notion of specialunipotent class do not explicitly appear in [S1, III] (although they served as amotivation for Spaltenstein, see [S1, III,9.4]). Actually (a) is a reformulation ofresults in [S1, III] taking into account developments in the theory of Springerrepresentations which occured after [S1, III] was written.We now discuss (a) assuming that G is almost simple. Let d : G −→ G bea map as in [S1, III,1.4]. If G is of exceptional type we further require that theimage of d has a minimum number of elements (see [S1, III,9.4]). Let e = d .Then e is order preserving and γ ≤ e ( γ ) for any γ ∈ G . Moreover if weset e = π ′ e ρ ′ then e is order preserving and γ ≤ e ( γ ) for any γ ∈ G (see [S1,III,5.6]). If G is of type B, C or D then the map e is described combinatorially in[S1, III,3.10] hence its image is explicitly known; using the explicit description ofthe Springer correspondence given in these cases in [L2] we see that this image isexactly G ♠ . (See also [S1, III,3.11].) If G is of exceptional type then the image of e is described explicitly in the tables in [S1, p.247-250] and one can again checkthat it is exactly G ♠ . Then the image of e is automatically G ♠ . The uniquenessin (a) is discussed in 4.2. Let γ ∈ G . We show that(a) there is a unique element ˜ e ( γ ) ∈ G ♠ such that:( ∗ ) γ ≤ ˜ e ( γ ) ; if γ ≤ γ ′ , ( γ ′ ∈ G ♠ ) , then ˜ e ( γ ) ≤ γ ′ .Moreover we have ˜ e (˜ e ( γ )) = ˜ e ( γ ) . We show that ˜ e ( γ ) = e ( γ ) satisfies ( ∗ ). We have γ ≤ e ( γ ). If γ ′ ∈ G ♠ and γ ≤ γ ′ then e ( γ ) ≤ e ( γ ′ ) = γ ′ (since e = e ), as required. Conversely let ˜ e ( γ ) be as in( ∗ ). From γ ≤ ˜ e ( γ ) we deduce e ( γ ) ≤ e (˜ e ( γ )) hence e ( γ ) ≤ ˜ e ( γ ). On the otherhand taking γ ′ = e ( g ) in ( ∗ ) (which satisfies γ ≤ γ ′ ) we have ˜ e ( γ ) ≤ e ( γ ) hence˜ e ( γ ) = e ( γ ). This proves (a) and that(b) ˜ e ( γ ) = e ( γ ).Note that (a),(b) and the analogous statements for G establish the uniquenessstatement in 4.1(a) and the identity(c) ˜ e = π ′ ˜ e ρ ′ where ˜ e : G −→ G is the map analogous to ˜ e : G −→ G (for G instead of G ); wehave ˜ e = e . Note that another proof of (c) which does not rely on the results of[S1, III] is given in [Xue].According to [L3], for any γ ∈ G we have(d) E ˜ e ( γ ) = [ E γ ]; In this subsection we assume that p = 2, G, G are simple adjoint of type B n ( n ≥
3) and let G ∗ , G ∗ be almost simple simply connected groups of type C n defined over k , C respectively. Note that G, G ∗ have the same Weyl group W .Define ξ : G −→ S W by γ [ E γ ] (notation of 0.5). Define ξ : G −→ S W by γ ( Springer representation attached to ˜ e ( γ )). We write G ∗ , G ∗ , Φ ∗ , Φ ∗ , ρ ∗ , ρ ′∗ , ξ ∗ , ξ ∗ for the analogues of G, G , Φ , Φ , ρ, ρ ′ , ξ , ξ with G, G replaced by G ∗ , G ∗ . We show that(a) ξ ρ ′ ( γ ) = ξ ( γ )for any γ ∈ G . The right hand side is the Springer representation attached to˜ e ( γ ). The left hand side is [ E ρ ′ ( γ ) ] which by 4.2(d) is equal to E ˜ e ( ρ ′ ( γ )) ; by thedefinition of π ′ this equals the Springer representation attached to π ′ ˜ e ρ ′ ( γ ) = ˜ e ( γ )(see 4.2(c)), proving (a).Let α : G ∗ −→ G be the standard exceptional isogeny. Let α ′ : W −→ W , α ′′ : S W −→ S W be the induced bijections. ( α ′ , α ′′ are the identity in our case.) ONJUGACY CLASSES IN THE WEYL GROUP AND UNIPOTENT CLASSES, II 19
We consider the diagram W Φ −−−−→ G −−−−→ G ξ −−−−→ S W = x ρ x ρ ′ x = x W Φ −−−−→ G = −−−−→ G ξ −−−−→ S W α ′ y α y α y α ′′ y W Φ ∗ −−−−→ G ∗ = −−−−→ G ∗ ξ ∗ −−−−→ S W = y ρ ∗ y ρ ′∗ y = y W Φ ∗ −−−−→ G ∗ −−−−→ G ∗ ξ ∗ −−−−→ S W The top three squares are commutative by 0.4(a), 3.7(b) and (a) (from left toright). The bottom three squares are commutative by 0.4(a), 3.7(b) and (a) (fromleft to right) with G replaced by G ∗ . The middle three squares are commutativeby the definition of α . We see that the diagram above is commutative. It followsthat ξ Φ = ξ Φ = ξ ∗ Φ ∗ = ξ ∗ Φ ∗ . In particular, ξ Φ = ξ ∗ Φ ∗ . From the definitionwe have ˜Φ = ξ Φ ; similarly we have ˜Φ ∗ = ξ ∗ Φ ∗ . Hence we have ˜Φ = ˜Φ ∗ . Thisproves the last sentence in 0.6(a). (Note that in the case where C ∈ W is elliptic,the equality ˜Φ ( C ) = ˜Φ ∗ ( C ) follows also from [L9, 3.6].) We now repeat the arguments of 4.3 in the case where G = G ∗ , G = G ∗ aresimple of type F with p = 2 or of type G with p = 3 or of type B with p = 2 and α : G ∗ −→ G is the standard exceptional isogeny. In these cases α ′ : W −→ W isthe nontrivial involution of the Coxeter group W and α ′′ : S W −→ S W is inducedby α ′ . We have ˜Φ = ˜Φ ∗ . As in 4.3 we obtain that ˜Φ α ′ = α ′′ ˜Φ . This, togetherwith 4.3, implies the validity of Theorem 0.6(a). In the setup of 4.3 we define η : S W −→ G by E γ γ where γ ∈ G ♠ (notation of 0.5). Define η : S W −→ G by E γ π ′ ( γ ) where γ ∈ G ♠ . Let π ′∗ , η ∗ , η ∗ be the analogues of π ′ , η , η with G, G replaced by G ∗ , G ∗ . We consider the diagram W Ψ ←−−−− G η ←−−−− S W = y π ′ y = x W Ψ ←−−−− G η ←−−−− S W α ′ y α y α ′′ y W Ψ ∗ ←−−−− G ∗ η ∗ ←−−−− S W = x π ′∗ x = y W Φ ∗ ←−−−− G ∗ η ∗ ←−−−− S W The left top square is commutative by 0.4(b) and by the equality π = π ′ . Similarly,the left bottom square is commutative. The right top square and the right bottomsquare are commutative by definition. The middle two squares are commutativeby the definition of α . We see that the diagram above is commutative. It followsthat Ψ η = Ψ η = Ψ ∗ η ∗ = Ψ ∗ η ∗ . In particular, Ψ η = Ψ ∗ η ∗ . Hence we have˜Ψ = ˜Ψ ∗ . This proves the last sentence in 0.6(b). We now repeat the arguments of 4.5 in the case where G = G ∗ , G = G ∗ aresimple of type F with p = 2 or of type G with p = 3 or of type B with p = 2 and α : G ∗ −→ G is the standard exceptional isogeny. In these cases α ′ : W −→ W isthe nontrivial involution of the Coxeter group W and α ′′ : S W −→ S W is inducedby α ′ . We have ˜Ψ = ˜Ψ ∗ . As in 4.5 we obtain that ˜Ψ α ′′ = α ′ ˜Ψ . This, togetherwith 4.5, implies the validity of Theorem 0.6(b).
5. Explicit description of the set of special conjugacy classes in
W5.1.
In this section we assume that k = C . We will describe in each case, assumingthat G is simple, the set W ♠ of special conjugacy classes in W and the bijection τ : W ♠ ∼ −→ S W induced by ˜Φ in 0.5.We fix n ≥
2. Let A be the set of all ( x ≥ x ≥ . . . ) where x i ∈ N are zero forlarge i such that x + x + · · · = 2 n ,for any i ≥ x i − = x i mod 2,for any i ≥ x i − , x i are odd we have x i − = x i .Let A ′ be the set of all (( y ≥ y ≥ . . . ) , ( z ≥ z ≥ . . . )) where y i , z i ∈ N are zerofor large i such that( y + y + . . . ) + ( z + z + . . . ) = n , y i +1 ≤ z i ≤ y i + 1 for i ≥ h : A −→ A ′ by( x ≥ x ≥ . . . ) (( y ≥ y ≥ ) , ( z ≥ z ≥ ))where ONJUGACY CLASSES IN THE WEYL GROUP AND UNIPOTENT CLASSES, II 21 y i = x i / , z i = x i − / x i − , x i are even, y i = ( x i − / , z i = ( x i − + 1) / x i − = x i are odd.Define h ′ : A ′ −→ A by(( y ≥ y ≥ . . . ) , ( z ≥ z ≥ . . . )) ( x ≥ x ≥ . . . )where x i = 2 y i , x i − = 2 z i if z i ≤ y i , x i = 2 y i + 1 , x i − = 2 z i − z i = y i + 1.Note that h, h ′ are inverse bijections. We preserve the setup of 5.1 and we assume that G is simple of type C n .Let W be the group of permutations of [1 , n ] which commute with the involution χ : i n − i + 1. We identify W with W as in [L8, 1.4]. To ( x ≥ x ≥ . . . ) ∈ A we associate an element of W = W which is a product of disjoint cycles withsizes given by the nonzero numbers in x , x , . . . where each cycle of even size is χ -stable and each cycle of odd size is (necessarily) not χ -stable. This identifies A with the subset W ♠ of W . We identify in the standard way ˆ W with the set ofall (( y ≥ y ≥ . . . ) , ( z ≥ z ≥ . . . )) where y i , z i ∈ N are zero for large i suchthat ( y + y + . . . ) + ( z + z + . . . ) = n . Under this identification A ′ becomes S W (the special representations of W ). The bijection τ : W ♠ ∼ −→ S W becomesthe bijection h : A ∼ −→ A ′ in 5.1. Now assume that G is simple of type B n ( n ≥ G be a simple groupof type C n over C . Then W can be viewed both as the Weyl group of G andthat of G ′ . The bijection sets W ♠ , S W and the bijection τ : W ♠ ∼ −→ S W is thesame from the point of view of G ′ as from that of G (see 0.6) hence it is describedcombinatorially as in 5.2.In the case where G is simple of type A n ( n ≥
1) then W ♠ = W , S W = ˆ W are all naturally parametrized by partitions of n and the bijection τ is the identitymap in these parametrizations. We fix n ≥
2. Let C be the set of all (( x ≥ x ≥ . . . ) , ( e , e , . . . )) where x i ∈ N are zero for large i and e i ∈ { , } are such that x + x + · · · = 2 n ,for any i ≥ x i − = x i mod 2, e i − = e i ,for any i ≥ x i − , x i are odd we have x i − = x i , e i − = e i = 0,for any i ≥ x i − , x i are even and e i − = e i = 0 we have x i − = x i ,for any i ≥ x i = 0 we have x i − = 0 and e i − = e i = 0,for any i ≥ x i = x i +1 are even we have e i = e i +1 = 0.Let C ′ be the set of all (( y ≥ y ≥ . . . ) , ( z ≥ z ≥ . . . )) where y i , z i ∈ N arezero for large i such that( y + y + . . . ) + ( z + z + . . . ) = n , y i +1 − ≤ z i ≤ y i for i ≥ k : C −→ C ′ by (( x ≥ x ≥ . . . ) , ( e , e , . . . )) (( y ≥ y ≥ . . . ) , ( z ≥ z ≥ . . . ))where y i = ( x i + 1) / , z i = ( x i − − / x i − = x i are odd, y i = x i / , z i = x i − / x i − = x i are even and e i − = e i = 0, y i = ( x i + 2) / , z i = ( x i − − / x i − , x i are even and e i − = e i = 1.Define k ′ : C ′ −→ C by(( y ≥ y ≥ . . . ) , ( z ≥ z ≥ . . . )) −→ (( x ≥ x ≥ . . . ) , ( e , e , . . . ))where x i = 2 y i , x i − = 2 z i , e i = e i − = 0 if y i = z i , x i = 2 y i − , x i − = 2 z i + 1 , e i = e i − = 0 if y i = z i + 1, x i = 2 y i − , x i − = 2 z i + 2 , e i = e i − = 1, if y i ≥ z i + 2.Note that k, k ′ are inverse bijections.Let C be the set of all (( x ≥ x ≥ . . . ) , ( e , e , . . . )) where x i ∈ N are zerofor large i and e i ∈ { , } are such that x + x + · · · = 2 n , x i − = x i are even, e i − = e i = 0 for i ≥ C ⊂ C .Let C ′ be the set of all (( y ≥ y ≥ . . . ) , ( z ≥ z ≥ . . . )) where y i , z i ∈ N arezero for large i such that( y + y + . . . ) + ( z + z + . . . ) = n , y i = z i for i ≥ C ′ ⊂ C ′ . Now k : C ∼ −→ C ′ restricts to a bijection k : C ∼ −→ C ′ . Itmaps (( y ≥ y ≥ . . . ) , ( y ≥ y ≥ . . . )) to ((2 y , y , y , y , . . . ) , (0 , , , . . . )). We preserve the setup of 5.4 and we assume that n ≥ G is simpleof type D n . Let W be as in 5.2 and let W ′ be the subgroup of W consistingof even permutations of [1 , n ]. We identify W = W ′ as in [L8, 1.4, 1.5]. Let χ : [1 , n ] −→ [1 , n ] be as in 5.2. Let W ♠ , (resp. W ♠ , ) be the set of specialconjugay classes in W which are not conjugacy classes of W (resp. form a singleconjugacy class in W ). Let S W , (resp. S W , ) be the set of special representationsof W which do not extend (resp. extend) to W -modules.To ( x ≥ x ≥ . . . ) ∈ C we associate an element of W ′ = W which is a productof disjoint cycles with sizes given by the nonzero numbers in x , x , . . . where eachcycle of even size is χ -stable and each cycle of odd size is (necessarily) not χ -stable.This identifies C − C with W ♠ , and C with W ♠ , modulo the fixed point freeinvolution given by conjugation by an element in W − W ′ .An element (( y ≥ y ≥ . . . ) , ( z ≥ z ≥ . . . )) ∈ C ′ can be viewed as in 5.2 as anirreducible representations of W . This identifies C ′ − C ′ with the S W , and C ′ with S W , modulo the fixed point free involution given by E E ′ where E ⊕ E ′ extendsto a W -module. Under these identifications, the bijection τ : W ♠ ∼ −→ S W becomesthe bijection ( C − C ) ⊔ C ⊔ C ∼ −→ ( C ′ − C ′ ) ⊔ C ′ ⊔ C ′ , ( a, b, c ) ( k ( a ) , k ( b ) , k ( c )). In 5.7-5.11 we describe explicitly the bijection τ : W ♠ ∼ −→ S W (in the casewhere G simple of exceptional type) in the form of a list of data α β where α is ONJUGACY CLASSES IN THE WEYL GROUP AND UNIPOTENT CLASSES, II 23 a special conjugacy class and β is a special representation (we use notation of [Ca]for the conjugacy classes in W and the notation of [S1] for the objects of ˆ W ). G . We have: A ǫA θ ′ G F . We have: A χ , A χ , A χ , A χ , ˜ A χ , D ( a ) χ D χ , C + A χ , F ( a ) χ , B χ , F χ , . E . We have: A A A A A + A A A + 2 A A D ( a ) A D A + A D ( a ) E ( a ) D E ( a ) E . E . We have: A A A (3 A ) ′ A A + A A + 2 A A A A + 3 A ( A + A ) ′ D ( a ) D D ( a ) + A D ( a ) + 2 A A + A A A ′ A + A D ( a ) A + A D ( a ) + A E ( a ) E ( a ) D A D + A D ( a ) A E ( a ) E E ( a ) E ( a ) E ( a ) E . E . We have: A A A A A + A A + 2 A A A D ( a ) D ( a ) + A D ONJUGACY CLASSES IN THE WEYL GROUP AND UNIPOTENT CLASSES, II 25 (2 A ) ′ A D ( a ) + A A + A A + 2 A D ( a ) A + A A + A + A D ( a ) + A D + A E ( a ) E ( a ) D D ( a ) A A ′ A + A A + A E ( a ) D ( a ) E ( a ) + A E E ( a ) A D ( a ) E ( a ) E ( a ) E ( a ) E ( a ) D E ( a ) E ( a ) E ( a ) E ( a ) E . From the tables above we see that if W is of type G , F , E and < c k > isthe conjugacy class of the k -th power of a Coxeter element of W then:(type G ) < c >, < c > are special, < c > is not special;(type F ) < c >, < c >, < c > are special, < c > is not special;(type E ) < c >, < c >, < c >, < c >, < c > are special, < c > is notspecial.The numbers 2 , , W of type G , F , E is 2 × , × , × Theorem 0.6 suggests that the set W ♠ should also make sense in the casewhere W is replaced by a finite Coxeter group Γ (not necessarily a Weyl group).Assume for example that Γ is a dihedral group of order 2 m , m ≥
3, with standardgenerators s , s . We expect that if m is odd then Γ ♠ consists of 1, the conjugacyclass of s s and the conjugacy class containing s and s ; if m is even then Γ ♠ consists of 1, the conjugacy class of s s and the conjugacy class of s s s s . Thisagrees with the already known cases when m = 3 , , References [Ca] R.W.Carter,
Conjugacy classes in the Weyl group , Compositio Math. (1972), 1-59.[L2] G.Lusztig, Intersection cohomology complexes on a reductive group , Invent.Math. (1984), 205-272.[KL] D.Kazhdan and G.Lusztig, Fixed point varieties on affine flag manifolds , Isr.J.Math. (1988), 129-168.[L1] G.Lusztig, A class of irreducible representations of a Weyl group , Proc. Kon. Nederl.Akad. (A) (1979), 323-335.[L3] G.Lusztig, Notes on unipotent classes , Asian J.Math. (1997), 194-207.[L4] G.Lusztig, Unipotent elements in small characteristic , Transform.Groups (2005), 449-487.[L5] G.Lusztig, Unipotent elements in small characteristic II , Transform.Groups (2008),773-797.[L6] G.Lusztig, Unipotent elements in small characteristic III , J.Algebra (2011), 163-189.[L7] G.Lusztig,
On some partitions of a flag manifold , arxiv:0906.1505.[L8] G.Lusztig,
From conjugacy classes in the Weyl group to unipotent classes , arxiv:1003.0412.[L9] G.Lusztig,
On C-small conjugacy classes in a reductive group , arxiv:1005.4313.[S1] N.Spaltenstein,
Classes unipotentes et sous-groupes de Borel , Lecture Notes in Math.,vol. 946, Springer Verlag, 1982.[S2] N.Spaltenstein,
On the generalized Springer correspondence for exceptional groups , Alge-braic groups and related topics, Adv.Stud.Pure Math., vol. 6, North-Holland and Kinoku-niya, 1985, pp. 317-338.[S3] N.Spaltenstein,
Polynomials over local fields, nilpotent orbits and conjugacy classes inWeyl groups , Ast´erisque (1988), 191-217.[Xue] T.Xue,
On unipotent and nilpotent pieces , arxiv:0912.3820., arxiv:0912.3820.