aa r X i v : . [ m a t h . R T ] O c t GENERALIZED SPRINGER THEORYAND WEIGHT FUNCTIONS
G. LusztigIntroduction
The generalized Springer correspondence [L1] is a bijection beween, on theone hand, the set of pairs consisting of a unipotent class in a connected reductivegroup G and an irreducible G -equivariant local system on it and, on the other hand,the union of the sets of irreducible representations of a collection of Weyl groupsassociated to G . (The classical case involves only some irreducible local systemsand only one Weyl group.) In this paper we show that each Weyl group appearingin the collection has a natural weight function (see 0.2). We also show how toextend each of these weight functions to an affine Weyl group; in fact, we describetwo such extensions, one in terms of G and one in terms of the dual group G ∗ .The one in terms of G ∗ has a surprising representation theoretic interpretation,see 3.3. Let G be a connected reductive group over C . We fix a primenumber l . By local system we mean a ¯ Q l -local system. The centralizer of anelement x of a group Γ is denoted by Z Γ ( x ). The identity component of analgebraic group H is denoted by H . For an algebraic group H let Z H be thecentre of H . For a connected affine algebraic group H let U H be the unipotentradical of H . If ( W, S ) is a Coxeter group with length function l we say that L : W −→ N is a weight function if L ( ww ′ ) = L ( w ) + L ( w ′ ) whenever w, w ′ in W satisfy l ( ww ′ ) = l ( w ) + l ( w ′ ).
1. A weighted Weyl group An induction datum for G is a triple ( L, O , E ) where L is aLevi subgroup of a parabolic subgroup of G , O is a unipotent conjugacy class of L and E is an irreducible L -equivariant local system on O (up to isomorphism) whichis cuspidal (in a sense that will be made precise in 1.3). To an induction datum( L, O , E ) we will associate a complex of sheaves K on G as follows. We choose aparabolic subgroup P for which L is a Levi subgroup; let pr : Z L O U P −→ O be Supported in part by National Science Foundation grant 1303060. Typeset by
AMS -TEX G. LUSZTIG the projection (we identify Z L O U P , a subvariety of P , with Z L × O × U P ). Wehave a diagram Z L × O a ←− ˜ P b −→ P c −→ G where˜ P = { ( h, g ) ∈ G × G ; h − gh ∈ Z L O U P } , P = { ( hP, g ) ∈ G/P × G ; h − gh ∈ Z L O U P } , a ( h, g ) = pr ( h − gh ), b ( h, g ) = ( hP, g ), c ( hP, g ) = g .We have a ∗ ( ¯ Q l ⊠ E ) = b ∗ ˜ E where ˜ E is a well defined local system on P . Thus, K = c ! ˜ E is well defined. According to [L1], K is an intersection cohomologycomplex on G whose support is ∪ h ∈ G h Z L ¯ O U P h − ; ¯ O is the closure of O ).Let X G be the (finite) set consisting of all pairs ( C , S ) where S is a unipotentconjugacy class in G and S is an irreducible G -equivariant local system on C (upto isomorphism). Let [ L, O , E ] be the set of all ( C , S ) ∈ X G such that S is adirect summand of the local system on C obtained by restricting some cohomologysheaf of K | C . Note that subset [ L, O , E ] depends only on the G -conjugacy class of( L, O , E ). For example, if L is a maximal torus of G (so that P is a Borel subgroup, O = { } and E = ¯ Q l ), we have P = { ( hP, g ) ∈ G/P × G ; h − gh ∈ P } and c : P −→ G is the Springer resolution; in this case, K = c ! ¯ Q l . X G . Following [L1] we define a partition of X G into subsets called blocks . If ( C , S ) ∈ X G we say that S is cuspidal if { ( C , S ) } is a block by itself said tobe a cuspidal block. The definition of blocks is by induction on dim G . If G = { } ,then X G has a single element; it forms a block. For general G , the non-cuspidalblocks of X G are exactly the subsets of X G of the form [ L, O , E ], where ( L, O , E )is an induction datum for G with L = G . (Note that the notion of cuspidalityof E is known from the induction hypothesis since dim L < dim G .) The cuspidalblocks of X G are the one element subsets of X G which are not contained in anynon-cuspidal block. The correspondence ( L, O , E ) [ L, O , E ] defines a bijectionbetween the set of induction data of G (up to conjugation) and the set of blocksof X G , see [L1]. Let L, O , E , P, c : P −→ G be as in 1.1 and let x ∈ O . Let P x = c − ( x ). Thus, P x = { hP ∈ G/P ; h − xh ∈ O U P } . In [L3, § P x is called a generalized flagmanifold . This is justified by the following result in [L3, 11.2] in which U = U Z G ( x ) .(a) The conjugation action of Z G ( x ) on P x is transitive. If hP ∈ P x then β P := ( hP h − ∩ Z G ( x )) U is a Borel subgroup of Z G ( x ) . The map hP −→ β P from P x to the variety of Borel subgroups of Z G ( x ) is a fibration. The fibres are exactlythe orbits of the conjugation action of U on P x hence are affine spaces. We have the following result.(b) dim P x = (dim Z G ( x ) − dim Z L ( x )) / ENERALIZED SPRINGER THEORY AND WEIGHT FUNCTIONS 3 the Z G ( x )-orbit of P in G/P and that this orbit is connected so that, by (a), itequals P x . This proves (b).Let W be the Weyl group of G , a finite Coxeter group, and let S be the set ofsimple reflections of W . For any J ⊂ S let W J be the subgroup of W generatedby J and let w J be the longest element of W J .Now P is a parabolic subgroup of type I for a well defined subset I of S .Let W be the set of all w ∈ W such that wW I = W I w and w has minimallength in wW I = W I w . This is a subgroup of W . For any s ∈ S − I we have w I ∪ s w I = w I w I ∪ s hence σ s = w I ∪ s w I = w I w I ∪ s satisfies σ s = 1. Moreover wehave σ s ∈ W .Let x ∈ O . Let b be the dimension of the variety of Borel subgroups of P thatcontain x . For any s ∈ S − I let P s be the unique parabolic subgroup of type I ∪ s that contains P and let P s,x = { hP ∈ P s /P ; h − xh ∈ O U P } . This is the analogue of P x when G is replaced by P s /U P s hence is again a gener-alized flag manifold. We set L ( s ) = dim P s,x . One can verify that(c) W is a Weyl group with Coxeter generators { σ s ; s ∈ S − I } (see [L1]) and(d) σ s
7→ L ( s ) is the restriction to { σ s ; s ∈ S − I } of a weight function ˜ L on W .To verify (d), we note that L ( s ) can be computed explicitly in each case using(b) for P s /U P s instead of G . (See the next section.) We now assume that G is almost simple, simply connected. We describe ineach case where L is not a maximal torus, the assignment ( G, L, O , E )
7→ W andthe values of the function L ; we will write ( G, L ) instead of (
G, L, O , E ) and willspecify G, L by the type of
G, L/ Z L . The notation for Weyl groups is the usualone, with the convention that a Weyl group of type A is { } .(a) ( A kn − , A kn − ) A k − , n ≥ , k ≥ L = n, n, . . . , n ;(b) ( C t + t + k , C t + t ) C k , t ≥ , k ≥ L = 1 , , . . . , , t + 1;(c) ( C t +3 t + k +1 , C t +3 t +1 ) C k , t ≥ , k ≥ L = 1 , , . . . , , t + 2;(d) ( B t +2 t + k , B t +2 t ) B k , t ≥ , k ≥ L = 1 , , . . . , , t + 1; G. LUSZTIG (e) ( B t +3 t +2 k , B t +3 t × A k ) C k , t ≥ , k ≥ L = 2 , , . . . , , t + 2;(f) ( B t +5 t +2 k +1 , B t +5 t +1 × A k ) C k , t ≥ , k ≥ L = 2 , , . . . , , t + 1;(g) ( D t + k , D t ) B k , t ≥ , k ≥ L = 1 , , . . . , , t ;(h) ( D t + t +2 k , D t + t × A k ) C k , t ≥ , k ≥ L = 2 , , . . . , , t − D t − t +2 k , D t − t × A k ) C k , t ≥ , k ≥ L = 2 , , . . . , , t ;(j) ( E , A ) G ; L = 1 , E , A ) F ; L = 1 , , , E , E ) A ;(m) ( F , F ) A ;(n) ( G , G ) A . (In the case where W is of type B k = C k the name we have chosen is such that itagrees with the type of the affine Weyl group ˆ W in 1.5.)In the case where L is a maximal torus that is, ( L, O , E ) is as in 1.2, we have W = W ; the function L is constant equal to 1. Let L, O , E , P be as in 1.1 and let x ∈ O . Let Ω be the set of P -orbitson G/P (under the action by left translation). For ω ∈ Ω we set P ωx = P x ∩ ω so that we have a partition P x = ⊔ ω P ωx where each P ωx is locally closed in P x .Let N L be the normalizer of L in G . We can identify N L/L with a subset ofΩ by nL P − orbit of nP where n ∈ N L . We can also identify
N L/L = W canonically so that we can identify W with a subset of Ω. One can show that(a) If w ∈ W then P wx is an affine space of dimension ˜ L ( w ) . Let w be the longest element of W . Since P w x is open in P x we deduce that(b) dim P x = ˜ L ( w ) . ENERALIZED SPRINGER THEORY AND WEIGHT FUNCTIONS 5
2. A weighted affine Weyl group
In this subsection we describe an affine analogue of the generalized Springertheory. We assume that G is almost simple, simply connected and that ( L, O , E )are as in 1.1. Let ˆ G = G ( C (( ǫ ))) where ǫ is an indeterminate. We can find aparahoric subgroup ˆ P of ˆ G whose prounipotent radical U ˆ P satisfies ˆ P = U ˆ P L , U ˆ P ∩ L = { } . Let ˆ W be the affine Weyl group defined by ˆ G . It is a Coxeter groupwith set of simple reflections ˆ S . We have S ⊂ ˆ S naturally and the subgroupof ˆ W generated by S can be identified with W . In particular the subset I ⊂ S can be viewed as a subset of ˆ S . Let ˆ S ′ be the set of s ∈ ˆ S − I such that I ∪ s generate a finite subgroup of ˆ W ; this set contains S − I . Let ˆ W be the subgroupof ˆ W defined in terms of ˆ W , W, u = 1 as in [L4, 25.1]. This is a Coxeter group(in fact an affine Weyl group) with generators { σ s ; s ∈ ˆ S ′ } . It contains W as thesubgroup generated by S − I .For any g ∈ ˆ G let ˆ P g = { h ˆ P ∈ ˆ G/ ˆ P ; h − gh ∈ Z L O U ˆ P } . If g ∈ ˆ G is regularsemisimple, then ˆ P g can be viewed as an increasing union of algebraic varieties ofbounded dimension. Moreover, E gives rise to a local system ˆ E on ˆ P g in the sameway as E gives rise to a a local system ˜ E on P in 1.1. Then the homology groups H i ( ˆ P g , ˆ E ) are defined; they are (possibly infinite dimensional) ¯ Q l -vector spaces.Using the method in [L5] (patching together various generalized Springer repre-sentations for groups of rank 2 considered in [L1]) we see that ˆ W acts naturallyon H i ( ˆ P g , ˆ E ).We now describe the type of the affine Weyl group ˆ W .In 1.5(a), ˆ W has type ˜ A k − .In 1.5(b), ˆ W has type ˜ C k .In 1.5(c), ˆ W has type ˜ C k .In 1.5(d), ˆ W has type ˜ B k .In 1.5(e), ˆ W has type ˜ C k .In 1.5(f), ˆ W has type ˜ C k .In 1.5(g), ˆ W has type ˜ B k .In 1.5(h), ˆ W has type ˜ C k .In 1.5(i), ˆ W has type ˜ C k .In 1.5(j), ˆ W has type ˜ G .In 1.5(k), ˆ W has type ˜ F .In 1.5(l),(m),(n), ˆ W has type ˜ A .In [L2, 2.6] it is shown that the Weyl group W can be identified with the Weylgroup of Z G ( x ) /U Z G ( x ) where x ∈ O . The results above show that ˆ W can beidentified with the affine Weyl group associated with Z G ( x ) /U Z G ( x ) . For any s ∈ ˆ S ′ let ˆ P s be a parahoric subgroup of type I ∪ { s } containingˆ P and let U ˆ P s the prounipotent radical of ˆ P s . Then ( L, O , E ) can be viewed G. LUSZTIG as an induction datum for the connected reductive group ˆ P s /U ˆ P s . Let L ( s ) bethe dimension of the generalized flag manifold associated to the induction datum( L, O , E ) of ˆ P s /U ˆ P s . (When s ∈ S − I this agrees with the definition of L ( s ) in1.4.) One can verify that(a) σ s
7→ L ( s ) is the restriction to { σ s ; s ∈ ˆ S ′ } of a weight function ˜ L on theCoxeter group ˆ W . Let x ∈ O ⊂ L ⊂ ˆ G . We say that ˆ P x = { h ˆ P ∈ ˆ G/ ˆ P ; h − xh ∈ O U ˆ P } is ageneralized affine flag manifold. Let ˆΩ be the set of ˆ P -orbits on ˆ G/ ˆ P (under theaction by left translation). For ω ∈ ˆΩ we set ˆ P ωx = ˆ P x ∩ ω so that we have apartition ˆ P x = ⊔ ω ˆ P ωx where each ˆ P ωx is an algebraic variety. In analogy with 1.6,we can identify ˆ W with a subset of ˆΩ. It is likely that the following affine analogueof 1.6(a) holds.(a) If w ∈ ˆ W then ˆ P wx is an affine space of dimension ˜ L ( w ) .
3. Another weighted affine Weyl group
We again assume that G is almost simple, simply connected. We denote by G ∗ a simple adjoint group over C of type dual to that of G . Let ( L, O , E ) be aninduction datum for G . Let G ∗ (resp. L ∗ ) be a connected reductive group over C of type dual to that of G (resp. L ); we can regard L ∗ as the Levi subgroupof a parabolic subgroup of G ∗ . Let E = j ! ( ¯ Q l ⊠ E ) where j : Z L × O = Z L O −→ L is the obvious imbedding. Then E [ d ] (where d = dim( Z L O )) is a charactersheaf on L . The classification of character sheaves of L associates to E [ d ] a triple( s, C, c ) where s is a semisimple element of finite order of L ∗ , C is a connectedcomponent of H = Z L ∗ ( s ) and c is a two-sided cell of the Weyl group W ′ of H which is stable under the conjugation by any element of C . (The triple ( s, C, c )is defined up to L ∗ -conjugacy.) Let W a be the affine Weyl group associated to( Z G ∗ ( s ) / centre)( C (( ǫ ))). Then W ′ can be viewed as a finite (standard) parabolicsubgroup of W a . Note that conjugation by an element of C induces a Coxetergroup automorphism γ : W a −→ W a which leaves W ′ stable.We describe in each case where L is not a maximal torus, th assignment( G, L, O , E ) ( W a , W ′ ); we will write ( G, L ) instead of (
G, L, O , E ) and willspecify G, L by the type of
G, L/ Z L . The notation for Weyl groups and affineWeyl groups is the usual one, with the convention that a Weyl group or affineWeyl group of type A , B , C , D , D is { } . The cases (a)-(n) below correspondto the cases (a)-(n) in 1.5.(a) ( A kn − , A kn − ) ( ˜ A nk − , A ) , n ≥ , k ≥ C t + t + k , C t + t ) ( ˜ B t + t + k × ˜ D t , B t + t × D t ) , t ≥ , k ≥ C t +3 t + k +1 , C t +3 t +1 ) ( ˜ D t +2 t + k +1 × ˜ B t + t , D t +2 t +1 × B t + t ) , t ≥ , k ≥ ENERALIZED SPRINGER THEORY AND WEIGHT FUNCTIONS 7 (d) ( B t +2 t + k , B t +2 t ) ( ˜ C t + t × ˜ C t + t + k , C t + t × C t + t ) , t ≥ , k ≥ B t +3 t +2 k , B t +3 t × A k ) ( ˜ C t + t + k × ˜ A t + t − × ˜ C t + t + k , C t + t × A t + t − × C t + t ) , t ≥ , k ≥ B t +5 t +2 k +1 , B t +5 t +1 × A k ) ( ˜ C t + t × ˜ A t +3 t +2 k × ˜ C t + t , C t + t × A t +3 t × C t + t ) , t ≥ , k ≥ D t + k , D t ) ( ˜ D t × ˜ D t + k , D t × D t ) , t ≥ , k ≥ D t + t +2 k , D t + t × A k ) ( ˜ D t × ˜ A t + t +2 k − × ˜ D t , D t × A t + t − × D t ) , t ≥ , k ≥ D t − t +2 k , D t − t × A k ) ( ˜ D t + k × ˜ A t − t − × ˜ D t + k , D t × A t − t − × D t ) , t ≥ , k ≥ E , A ) ( ˜ D , A );(k) ( E , A ) ( ˜ E , A );(l) ( E , E ) ( ˜ E , E );(m) ( F , F ) ( ˜ F , F );(n) ( G , G ) ( ˜ G , G ) . We set n t = 1 if t is even, n t = 2 if t is odd. In (a) with k ≥ γ has order n ; itpermutes cyclically the n copies of A k − ; in (a) with k = 1, we have γ = 1. In (b)with t ≥ γ has order n t ; it acts only on the ˜ D -factor; in (b) with t = 1, we have γ = 1. In (c) with ( t, k ) = (0 , γ has order n t +1 ; it acts only on the ˜ D -factor;in (c) with ( t, k ) = (0 , γ = 1. In (d) we have γ = 1. In (e), γ has G. LUSZTIG order 2; it interchanges the two ˜ C -factors and acts nontrivially on the ˜ A -factor.In (f) with ( t, k ) = (0 , γ has order 2; it interchanges the two ˜ C -factors and actsnontrivially on the ˜ A -factor; in (f) with ( t, k ) = (0 , γ = 1. In (g) with( t, k ) = (1 , γ has order n t ; it acts on the ˜ D t + k -factor. In (g) with ( t, k ) = (1 , γ = 1. In (h) with ( t, k ) = (1 , γ has order 2 n t ; it interchanges thetwo ˜ D factors and acts nontrivially on the ˜ A -factor. In (h) with ( t, k ) = (1 , γ has order 2. In (i) with ( t, k ) = (1 , γ has order 2 n t ; it interchanges the two ˜ D factors. In (i) with ( t, k ) = (1 , γ = 1. In (j), γ has order 3; in (k), γ has order 2. In (l),(m),(n), we have γ = 1.We now describe in each case the two-sided cell c of W ′ . If W ′ = { } then c = { } . If W ′ = { } , we write W ′ = W ′ × . . . × W ′ m where W ′ i are irreducibleWeyl groups and c = c × . . . × c m where c i is a two-sided cell in W ′ i . For any i such that W ′ i is of type A r , r ≥
1, we have r + 1 = ( h + h ) / h and c i isthe two-sided cell associated to a unipotent cuspidal representation of a nonsplitgroup of type A r over F q . For any i such that W ′ i is of type B r or C r with r ≥ r = h + h for some h and c i is the two-sided cell associated to a unipotentcuspidal representation of a group of type B r or C r over F q . For any i such that W ′ i is of type D r with r ≥
4, we have r = h for some h and c i is the two-sidedcell associated to a unipotent cuspidal representation of a group of type D r over F q (which is split if h is even, nonsplit if h is odd). If W ′ is of type E , F or G , c is the two-sided cell associated to a unipotent cuspidal representation of a groupof type E , F or G over F q . We associate to an induction datum ( L, O , E ) of G an affine Weyl group W a .We define W a in terms of ( W a , W ′ , γ ) as in [L4, 25.1]. In more detail, let S be theset of simple relections of W a . For any subset J of S let W aJ be the subgroup of W a generated by J ; when W aJ is finite let w J be the longest element of W aJ . Let J ′ be the set of simple reflections of W ′ . Let ˜ W a be the set of all w ∈ W a suchthat wW aJ ′ = W aJ ′ w and w has minimal length in wW aJ ′ = W aJ ′ w and let W a bethe fixed point set of γ : ˜ W a −→ ˜ W a . Note that ˜ W a , W a are subgroup of W a .Let K be the set of all γ -orbits k on S − J ′ such that W aJ ′ ∪ k is finite. In eachcase (a)-(n), for any k ∈ K we have w J ′ ∪ k w J ′ = w J ′ w J ′ ∪ k hence τ k = w J ′ ∪ k w J ′ = w J ′ w J ′ ∪ k satisfies τ k = 1. Moreover we have τ k ∈ W a . Let a : W a −→ N bethe a -function of the Coxeter group W a (with standard length function), see [L4, § L : K −→ N by L ( k ) = a ( c τ k ) − a ( c ) where a ( c τ k ), a ( c ) denotesthe (constant) value of the a -function on c τ k , c (see [L4, 9.13]). One can verifythat W a is an affine Weyl group with Coxeter generators { τ k ; k ∈ K } and that τ k
7→ L ( k ) is the restriction to { τ k ; k ∈ K } of a weight function on W a .We describe below the type of the affine Weyl group W a and the values of theweight function L on K .In 3.1(a), W a has type ˜ A k − , L = n, n, . . . , n .In 3.1(b), W a has type ˜ B k , L = 1 , , . . . , , t + 1.In 3.1(c), W a has type ˜ B k , L = 1 , , . . . , , t + 2. ENERALIZED SPRINGER THEORY AND WEIGHT FUNCTIONS 9
In 3.1(d), W a has type ˜ C k , L = 1 , , . . . , , t + 1.In 3.1(e), W a has type ˜ C k , L = 2 , , . . . , , t + 2.In 3.1(f), W a has type ˜ C k , L = 1 , , , . . . , , t + 1.In 3.1(g), W a has type ˜ B k , L = 1 , , . . . , , t .In 3.1(h), W a has type ˜ C k , L = 1 , , , . . . , , t − W a has type ˜ B k , L = 2 , , . . . , , t .In 3.1(j), W a has type ˜ G , L = 1 , , W a has type ˜ F , L = 1 , , , , W a has type ˜ A .In the case where L is a maximal torus that is, ( L, O , E ) is as in 1.2, we have s = 1, W a is an affine Weyl group of type dual to that of G , W ′ = { } , c = 1, and γ = 1; W a = W a ; the function L is constant equal to 1.We see that W in 1.4 is naturally imbedded (as a Coxeter group) in W a so that W a is an affine Weyl group associated to W and that L in 1.4 is the restrictionof L . Let ¯ F q be an algebraic closure of the finite field F q . The pair Z G ∗ ( s ) ⊃ Z L ∗ ( s )has a version G ′ ⊃ G ′ with G ′ , G ′ being connected reductive groups over ¯ F q of thesame type as ( Z G ∗ ( s ) , Z L ∗ ( s )). Let G ⊃ G be obtained from G ′ ⊃ G ′ by dividingby the centre of G ′ . Let F : G −→ G be the Frobenius map for an F q -rationalstructure on G which induces on the Weyl group of G the same automorphismas γ in 3.1. We can then form the corresponding group G ( F q (( ǫ ))) where ǫ isan indeterminate and its subgroup G ( F q ). This subgroup can be regarded asthe reductive quotient of a parahoric subgroup P of G ( F q (( ǫ ))); moreover thissubgroup carries a unipotent cuspidal representation as in the last paragraph of3.1. We can induce this representation from P to G ( F q (( ǫ ))). The endomorphismalgebra of this induced representation is known to be an extended affine Heckealgebra with explicitly known (possibly unequal) parameters. An examination ofthe cases (a)-(n) in 3.2 shows that these parameters are exactly those describedby the function L in 3.2. References [L1] G.Lusztig,
Intersection cohomology complexes on a reductive group , Inv.Math. (1984),205-272.[L2] G.Lusztig, Cuspidal local systems and graded Hecke algebras I , Publications Math. IHES (1988), 145-202.[L3] G.Lusztig, Cuspidal local systems and graded Hecke algebras II , Representations of groups,ed. B.Allison et al., Canad. Math. Soc. Conf. Proc., vol. 16, Amer. Math. Soc., 1995,pp. 217-275.[L4] G.Lusztig,
Hecke algebras with unequal parameters , CRM Monograph Ser., vol. 18, Amer.Math. Soc., 2003.[L5] G.Lusztig,
Unipotent almost characters of simple p -adic groups , De la G´eometrie Alg´ebriqueaux Formes Automorphes, Ast´erisque, vol. 369-370, Soc. Math. France, 2015., De la G´eometrie Alg´ebriqueaux Formes Automorphes, Ast´erisque, vol. 369-370, Soc. Math. France, 2015.