A topological approach to induction theorems in Springer theory

We give a self-contained account of a construction due to Rossmann which lifts Springer's action of a Weyl group on the cohomology of a Springer fiber to an action on its homotopy type. We use this construction to produce a generalization of an"induction theorem"of Alvis and Lusztig, which relates the Springer representations attached to a reductive group to those attached to a Levi subgroup. Our generalization applies to more general centralizers and to representations of Weyl groups on mod p cohomology.


Introduction
The Springer correspondence is a connection between the geometry of the space of unipotent elements N in an algebraic group G and the representation theory of the Weyl group W associated to G. Representations of W are constructed on the cohomology of the fibers of Springer's resolution of singularitiesÑ → N . It has been clear for a long time that representations can be constructed this way over any ground ring, but the focus of the theory has overwhelmingly been on representations in characteristic zero. Recently Juteau [J] has begun a systematic study of the modular version of this theory, extending the intersection homology techniques used by Lusztig,Borho,and MacPherson [L2,BM] and studying the extent to which the Brauer-Cartan (or "cde") triangle can be seen in the geometry of N .
This paper is another contribution to the modular gap in Springer theory. Our perspective is topological: where Juteau approaches modular representations via the techniques of Lusztig-Borho-MacPherson, we follow the approach of Grothendieck-Brieskorn-Slodowy. We use this construction to produce "induction theorems" which relate the Springer correspondences for G and for a subgroup of G.
1.1. Modular representations via Rossmann's homotopy action. A major part of this paper is a review of and advocacy for an approach to Springer theory due to Rossman [R]. Write F ν ⊂Ñ for the fiber of Springer's resolution over ν ∈ N . The existence of the W -action on H * (F ν ), constructed first by Springer in 1976, came as a surprise especially because in simple examples it is easy to see that it cannot possibly be induced by an action of W on the space F ν . For a long time it was unknown whether the action on H * (F ν ) could be lifted to an action on the homotopy type of F ν (though early progress was made in [KL]). Rossmann gave an affirmative solution to this problem in 1993.
In the first five sections of this paper we give a self-contained account of Rossmann's construction. We produce an action of the braid group B W associated to W on a dim(G)dimensional manifold-with-partial-boundary U × C ν which is homotopy equivalent to F ν , and show that the kernel of the projection B W → W acts by transformations homotopic to the identity. This is a little weaker than producing a space homotopy equivalent to F ν on which W acts-this is still an open problem as far as I know. Nevertheless the construction does give an action of W on every functorial homotopy invariant of F ν , in particular on mod p cohomology.
The spaces U and C ν are easy to describe: U is the contractible universal cover of the set of regular elements in a small ball around 1 ∈ W \\T , and C ν is a regular neighborhood with boundary of F ν inÑ . Rossmann's construction follows a standard construction (due originally to Grothendieck, Brieskorn, and Slodowy [B, S]) of the Springer representations via nearby cycles. In that approach, the essential point is that a Springer fiber has the same cohomology as the appropriate Milnor fibers of the projection f : G → W \\T . (This is expressed sheaf-theoretically as an isomorphism between the nearby cycles sheaf of f and the direct image of the constant sheaf under the Springer mapÑ → N . See [G] for a nice explanation.) The essential point of Rossmann's refined construction is that these Milnor fibers are actually homeomorphic to C ν .
1.2. Induction theorems. Let H ⊂ G be a connected reductive subgroup of G. Suppose that ν lies in H. Then we may study the Springer representations for the pair ν ∈ H and for the pair ν ∈ G. A basic question is how the representations of W H on H * (F ν,H ) and of W G on H * (F ν,G ) are related. In case H is a Levi subgroup, Alvis and Lusztig [AL,L1] showed that the characters of the W G -modules Ind W G W H H * (F ν,H ) and H * (F ν,G ) are the same. Theorem 1.1 (Alvis-Lusztig, [AL,L1]). Let G be a reductive group, let L be a Levi subgroup, and let ν be a unipotent element of L. Let W L ⊂ W G denote the Weyl groups of L and G, and let F ν,L and F ν,G denote the Springer fibers associated to ν in L and G. There is a filtration by W G -submodules of so that the associated graded satisfies In this paper we use Rossmann's construction to give a topological explanation for this. Our proof is reminiscent of topological work on the "fundamental lemma" in the Langlands program [GKM, N]: we use localization in circle-equivariant cohomology to make comparisons. This technique, using Z/p-equivariant localization instead, applies in the modular setting as well. We will show Theorem 1.2. Let G be a reductive group, let ν ∈ G be a unipotent element, and let g ∈ G be an element of order p (p a prime) which is contained in the connected component of the centralizer of ν. Let Z ⊂ G be the centralizer of g in G; note that ν is contained in Z. Suppose furthermore that Z is connected (this is true, for instance, if G is simply connected.) Let W Z and W G denote the Weyl groups of Z and G, and let F ν,Z and F ν,G be the Springer fibers associated to ν in Z and G. There is a filtration by W G -modules of whose associated graded satisfies

Springer glossary
For our purposes, Springer theory is the geometry of the spaces and maps in this diagram: Let us briefly indicate what these spaces are; for more details we refer to [G].
subgroup and x is an element of [B, B].
The maps j andj are just the inclusions. The maps p and q are given by (x, B) → x and π is the quotient map. The mapf carries (x, B) to its image in B/ [B, B] ∼ = T , and f is the unique map making the right-hand square commute. Alternatively, letting G act on itself by conjugation, one may define f to be the GIT quotient map G → G\\G composed with a canonical isomorphism W \\T ∼ = G\\G. Each of the spacesG, G, T and W \\T have dense open subsets of regular semisimple elements; we write them asG rss ⊂G, G rss ⊂ G, etc. Explicitly, we have • G rss is the set of regular semisimple elements in G. •G rss is the set of pairs (x, B) ∈G where x is a regular semisimple element.
• T rss is the set of regular semisimple elements in T .
• (W \\T ) rss is the set of W -orbits of regular semisimple elements in T .
Each of the mapsf , f, q, π restricts to a submersion on the set of regular elements, and the squareG is cartesian. In particular, the vertical arrows are covering spaces with deck group W .
Remark 2.1. If X is a subset of one ofG, G, T or W \\T we will write X rss for the intersection of X withG rss , G rss , T rss or (W \\T ) rss respectively.

Metric structures and a trivialization off
Fix a maximal compact subgroup K of G, and endow G with a Hermitian metric which is left and right invariant under K. This structure provides, for each Borel subgroup, a canonical diffeomorphism B ∼ = T × [B, B]. Indeed, B ∩ K is a maximal compact torus of B, and its Zariski closure in B gives a section of the principal [B, B] B] can be obtained by exponentiating the orthogonal decomposition of the Lie algebra Denote the projection B → [B, B] by u B and the projection B → [B, B] ⊥ by s B (the u and s stand for "unipotent" and "semisimple"). The u B assemble to a trivialization off . That is, the mapG is a diffeomorphism and commutes with the projections to T .
Remark 3.1. We will actually only make use of the fact thatf :G → T is a submersionthis is a weaker statement becausef is not proper.
The Hermitian structure on G induces one on the universal torus T , via the isomorphism T ∼ = [B, B] ⊥ ; the metric is independent of B. Let us write d(x, y) for the induced distance function on these spaces. Let us also define a metric on the space W \\T , by setting for the distance between the W -orbit of x and the W -orbit of y.
For each δ > 0, write B(δ, T ) for the open δ-ball centered at the identity in T , and write B(δ, W \\T ) for the open δ-ball centered at the identity orbit in W \\T .

Some spaces near a unipotent element
Fix a unipotent element ν ∈ N , and let ǫ and δ be positive real numbers with δ ≪ ǫ ≪ 1. Define the following spaces: • Let F ν ⊂Ñ be the Springer fiber over ν: • Let E ν (ǫ, δ) ⊂ G be the intersection of a closed ǫ-ball around ν with the inverse image of an open δ-ball around 1 ∈ W \\T . If B is any Borel subgroup containing x then we may write this as ⊂Ñ be the intersection of D ν withÑ : (SinceÑ → N is a resolution of singularities, a standard argument shows that C ν deformation retracts onto F ν . See proposition 4.1.) E ν and D ν fit into the following commutative square: This square becomes cartesian when the maps are restricted to the regular parts of each space: The action of the compact centralizer. Write K(ν) = K ∩ Z G (ν) for the maximal compact subgroup of the centralizer of ν which is contained in K. The distance from ν with respect to the hermitian metric on G is invariant under K(ν), and therefore K(ν) acts on the spaces F ν , E ν (ǫ, δ), D ν (ǫ, δ), and C ν (ǫ). Moreover, the fibers of D ν → B(δ, T ) and E ν → B(δ, W \\T ) are preserved by this action. We also have Proposition 4.1. For ǫ sufficiently small, the inclusion F ν → C ν (ǫ) admits a K(ν)-equivariant deformation retraction.
We will use the following very special cases of Thom's isotopy lemmas: , there are homeomorphisms making the following diagram commute Furthermore, all maps in this square commute with the projections to B.
Proofs of the full isotopy lemmas may be found in [M], however we remark that the special cases we consider here may be proved by much more elementary means. Since ρ and ρ|Ñ are real algebraic and smooth, their critical values are isolated by Sard's theorem. It follows that there exists a positive real number ǫ 0 such that neither ρ nor ρ|Ñ have critical values in the open interval (0, ǫ 0 ). In particular, ∂D ν (ǫ, δ) (and ∂C ν (ǫ)) is a manifold for every ǫ in (0, ǫ 0 ). By the first isotopy lemma, the fact that D ν (ǫ, δ) → B(δ, T ) is a trivial fiber bundle is a consequence the following claim: there exists δ 0 such that the projections D • ν (ǫ, δ) → B δ (T ) and ∂D ν (ǫ, δ) → B δ (T ) are submersions for all ǫ < ǫ 0 and δ < δ 0 . Furthermore, applying Thom's second isotopy lemma to the case where M = D ν (ǫ, δ), N = K(ν) × D ν (ǫ, δ), and v : N → M is the action map gives a K(ν)-equivariant trivialization. Thus, let us prove the claim.
Since q :G → T is a submersion (by section 3) and is also a submersion for any positive ǫ and δ. We are left with showing that q| ∂Dν (ǫ,δ) is a submersion for ǫ < ǫ 0 and δ sufficiently small. To show that q| ∂Dν (ǫ,δ) does not have critical values in a sufficiently small neighborhood of 1 ∈ T , it suffices to show (since q| ∂Dν (ǫ,δ) is proper and the set of critical values is closed) that 0 itself is not criticali.e. that at any point (x, B) ∈Ñ = q −1 (0), the vertical tangent space to q at (x, B) and the tangent space to ∂D ν (ǫ, δ) at (x, B) are transverse.
Let us call these tangent spaces T 1 and T 2 , respectively. Since ∂D ν (ǫ, δ) is of real codimension one inG, it is equivalent to showing that T 2 does not contain T 1 . But T 1 is the tangent space toÑ and T 2 is the kernel of dρ x,B , so if T 1 ⊂ T 2 then (x, B) is a critical point of ρ|Ñ , which contradicts the assumption that ǫ < ǫ 0 . This completes the proof.
In particular, the Borel construction of E ν (ǫ, δ) with respect to any subgroup K ′ ⊂ K(ν) may be exhibited as a fiber bundle over B(δ, W \\T ) rss , whose fibers are Borel constructions of the K ′ -space C ν (ǫ). Indeed, letting EK ′ denote a contractible space on which K ′ acts freely, the map K ′ \(E ν (ǫ, δ) × EK ′ ) → B(δ, W \\T ) rss that takes the K ′ -orbit of (x, y) to f (x) is well-defined and provides such a fibration. We will use this fact in the next section.
Corollary 5.5. Let A be a commutative ring. With ǫ and δ as above, let φ denote the map E rss ν → B(δ, W \\T ) rss . The higher direct image sheaves R i φ * A are locally constant on B(δ, W \\T ) rss . Moreover, the stalks are isomorphic to the cohomology sheaves H i (F ν ; A) with coefficients in A, and the monodromy action π 1 (B(δ, W \\T ) rss ) → Aut (H i It is shown in [H] that when A = Q ℓ , this action of W on H i (F ν ) coincides with the original action of Springer after tensoring with the sign representation of W . 5.1. Weyl group action on Springer fibers at the level of homotopy. Let us indicate how this result lifts Springer's action of W on the cohomology of F ν to an action of W on the homotopy type of F ν . Let U be the universal cover of B δ (W \\T ) rss (equivalently: of B δ (T ) rss ); it is well-known that U is contractible and that the deck group of U is B W , the braid group associated to W (this was finally proved for all reflection groups in [D]). It follows that B W acts on the pullback E rss ν × B δ (W \\T ) rss U of E rss ν to U. In fact, it acts freely. The lemma provides a homeomorphism between this pullback and C ν × U, and by proposition 4.1, this space is homotopy equivalent to the Springer fiber F ν . To complete the construction, we need to check that the subgroup B ′ W of "pure braids" (i.e. the kernel of the projection B W → W ) acts on U × C ν by transformations homotopic to the identity. This also follows from the lemma:

Induction theorems
In this section we will prove theorems 1.1 and 1.2. To do this we will need to consider the spaces F, E, D, C defined in section 4 for different reductive groups G. To keep them straight we need to complicate the notation introduced in sections 2 and 4 with subscripts.
• Write T G , W G , N G , andÑ G for the universal torus, unipotent cone, and Springer resolution associated to the complex connected reductive algebraic group G. • Write F ν,G , E ν,G (ǫ, δ), D ν,G (ǫ, δ), and C ν,G (ǫ) for the spaces associated to G defined in section 4.
• If G is endowed with a hermitian metric, write K G for the maximal compact which preserves it. Write K G (ν) for the intersection of K G with the centralizer of the unipotent element ν.
Proof of theorem 1.1. Since any maximal torus of L is also a maximal torus of G, we may identify the universal tori T L and T G , so we will omit the subscript. For appropriate ǫ and δ, δ ≪ ǫ ≪ 1, we have by corollary 5.4 a commutative square (Let us fix ǫ and δ, and for the rest of the proof omit them from the notation for E ν,L , E ν,G .) Let U(1) ⊂ G be a compact circle whose centralizer in G is L. Then U(1) ⊂ K G (ν) and the U(1)-fixed points E U (1) ν,G coincide with E ν,L . We will now deduce the theorem by standard "localization" techniques in U(1)-equivariant cohomology, and the degeneration of the Borel spectral sequence.
We may replace the top row of the diagram above by the Borel constructions with respect to U(1). That is, letting X hU (1) denote the Borel construction of a U(1)-space X, we have Let us denote the vertical maps by χ L and χ G , and the bottom horizontal map by ι. Define a locally constant sheaf F L on B(δ, W L \\T ) rss and F G on B(δ, W G \\T ) rss by The fibers of F L (resp. F G ) are isomorphic to the equivariant cohomology groups H * U (1) (F ν,L ; Q) (resp. H * U (1) (F ν,G ; Q)). Fix an isomorphism of graded rings H * U (1) (pt; Q) ∼ = Q[t], where t has degree 2; then F L and F G are sheaves of Q[t]-modules. Since E rss ν,L is the U(1)-fixed set in E rss ν,G , the localization theorem for U(1)-equivariant cohomology (see e.g. [Q,Theorem 4.4]) provides an isomorphism of graded Q[t, t −1 ]-modules Translating this into the language of W G -modules, we have Q[t, t −1 ] Since U(1) acts trivially on F ν,L , the group on the right is isomorphic to Ind W G W L H * (F ν,L ; Q)⊗ Q Q[t, t −1 ].
Consider the Leray spectral sequence for the composition of maps (E rss ν,G ) hU (1) → BU(1) × B(δ, W G \\T ) rss → B(δ, W G \\T ) rss Remark 6.1. In case g is not contained in the identity component of the centralizer of ν, we may still conclude from the existence of the Borel spectral sequence that is a subquotient of H k (F ν,G ; F p ), even if this spectral sequence does not degenerate.
Remark 6.2. Let us give an example (without proof) where the hypotheses of theorem 1.2 apply, and one where they do not. Let G = Sp(4), Z = SL(2) × SL(2) (note that this is not a Levi subgroup) and p = 2. Then Z is the centralizer of