A CONSTRUCTION OF REPRESENTATIONS OF AFFINE WEYL GROUPS

DANIEL S. SAGE DEPARTMENT OF MATHEMATICS UNIVERSITY OF UTAH SALT LAKE CITY, UTAH 84112

Published in Compositio Mathematica 108, 1997, p. 241-245.

In [KL1], Kazhdan and Lusztig have given an elementary procedure for constructing Springer representations of the Weyl group W of a complex semisimple group G in the homology of the varieties ℬ_{u}, where ℬ_{u} is the variety of Borel subgroups of G containing a fixed unipotent element u. Their method is a special case of a construction of W -actions on the homology of a larger class of varieties. They start with a complex algebraic variety X and consider a locally trivial principal Gbundle E → X. They form the associate ℬ-bundle Ê = E ×_{G}ℬ → X and assume given an algebraic fiberwise C-action φ on Ê. They denote the fixed point set of the action by Ê_{φ}; it is a closed subvariety of Ê. They fix a set S of simple reflections of the Coxeter group W, and for each s ∈ S, construct a map α_{s} : Ê_{φ} → Ê_{φ}, defined up to proper homotopy. The map s ↦ (α_{s})_{∗} then defines a group homomorphism W → Aut(H_{∗}(Ê_{φ})), where H_{∗}(Ê_{φ}) denotes the Borel-Moore homology groups with rational coefficients. Springer representations may be obtained using this procedure by taking X to be a point and E to be G, so that Ê = ℬ. Any unipotent element u ∈ G induces a C-action on ℬ via (t, B) ↦ u^{t}Bu^{−t}, where u^{t} is the image of t in the one parameter subgroup of G defined by u. Since the fixed point variety is ℬ_{u}, we get representations of W in H_{∗}(ℬ_{u}).

Because Kazhdan and Lusztig’s construction of W -actions relies heavily on the existence of simple reflections in Weyl groups, it is natural to conjecture that their procedure generalizes to other Coxeter groups. It is indeed the case that representations of affine Weyl group may be defined in this way.

Let G be a semisimple, simply connected, complex algebraic group with Lie algebra 𝔤 and affine Weyl group Ŵ. Let F = C((π)) be the field of formal power series in one variable with A = C[[π]] its ring of integers. We extend scalars to form G(F), G(A), 𝔤_{F} = 𝔤 ⊗_{C}F, and 𝔤_{A} = 𝔤 ⊗_{C}A. Fix a Borel subalgebra 𝔟 ∈ ℬ. An Iwahori subalgebra of 𝔤_{F} is any G(F)-conjugate of the pullback 𝔟̂ of 𝔟 under the natural projection 𝔤_{A} → 𝔤; we denote the space of Iwahori subalgebras by ℬ̂. If B ⊂ G corresponds to 𝔟 with preimage B̂ ⊂ G(A) under the map G(A) → G, then ℬ̂ may be identified with the quotient space G(F)/B̂. The set ℬ̂ has the structure of an infinite-dimensional projective variety over C, i.e. it is an increasing union of finite-dimensional projective varieties [KL2].

Given N ∈ 𝔤_{F}, we define ℬ̂_{N} to be the space of Iwahori subalgebras containing N. This is a closed subvariety of ℬ̂. We shall restrict our attention to the topologically nilpotent (or nil) elements N, those elements such that (ad N)^{m} → 0 in the power series topology on End 𝔤_{F} as m → ∞. More concretely, these are the elements of 𝔤_{F} which may be conjugated into some N′ ∈ 𝔤_{A} whose image under projection to 𝔤 is nilpotent. Kazhdan and Lusztig have shown in [KL2] that for N nil, the closed subvariety ℬ̂_{N} is finite-dimensional (although possibly having infinitely many irreducible components) precisely when N is regular semisimple. If we further assume that N is elliptic, i.e. the connected centralizer of N in G(F) is an anisotropic maximal torus, then 𝒫̂_{N} is a projective variety in the usual sense.

Let N be a fixed nil-elliptic element of 𝔤_{F}. We define an algebraic C-action on ℬ̂ via φ(t, 𝔟) = exp(tN)(𝔟); the exponential makes sense because tN is nil for every t ∈ C. The fixed point set of this action is simply ℬ̂_{N}. We then have the following result on the existence of affine Springer representations:

Theorem . Let G be a complex, semisimple, simply connected algebraic group, 𝔤 its Lie algebra, and F = C((π)) the field of formal power series. Then for any nilelliptic element N ∈ 𝔤_{F}, the affine Weyl group Ŵ acts naturally on the homology of B̂_{N}.

Remarks. 1. Kazhdan and Lusztig have shown that Ŵ acts on the top homology of B̂_{N} by a different method in [KL2]. Lusztig also has another construction of affine Springer representations [L].

2. Kato has shown that a modified version of the affine Weyl group acts on the top homologies of the classical fixed point varieties ℬ_{u} [K].

Choose a set S of simple reflections for Ŵ. For each s ∈ S, we let 𝒫̂_{s} be the set of minimal parahoric subalgebras of type s. More concretely, if we fix an Iwahori subalgebra 𝔟̂ with corresponding Iwahori subgroup B̂, then 𝔭̂_{s} = 𝔟̂ + s · 𝔟 is a subalgebra of 𝔤_{F} with stabilizer P̂_{s}. A minimal parahoric subalgebra of type s is just a conjugate of 𝔭̂_{s}, so 𝒫̂_{s} may be identified with G(F)/P̂_{s}. This quotient space may be endowed with the structure of an infinite-dimensional variety compatible with the variety structure on ℬ̂ and the natural map ℬ̂ →^{ρ} 𝒫̂_{s}, which becomes a P^{1}-fibration.

The C-action on ℬ̂ induces a unique algebraic C-action (also denoted φ) on 𝒫̂_{s} which makes the diagram

C × ℬ̂ ⟶^{φ} ℬ̂
↓  ↓
C × 𝒫̂_{s} ⟶^{φ} 𝒫̂_{s}

commute. We consider the restriction ℰ^{s} → 𝒫̂_{s,N} of the P^{1}-bundle ℬ̂ → 𝒫̂_{s} to the fixed point variety 𝒫̂_{s,φ} = 𝒫̂_{s,N}. The map φ is a fibrewise C-action on ℰ^{s} with fixed point set ℬ̂_{N}. Since a C-action on P^{1} is just a conjugate of a translation (viewing P^{1}(C) as the two-sphere), there is a fixed point in each fiber. (The translation may be the identity transformation, in which case the entire fiber is fixed.) The variety 𝒫̂_{s,N} is thus the image of ℬ̂_{N} under the natural projection map and is itself a finite-dimensional algebraic variety.

We now construct maps ℬ̂_{N} →^{α_{s}} ℬ̂_{N} (defined up to homotopy) following the procedure of Kazhdan and Lusztig. We choose Riemannian metrics on the fibers, depending continously on the base space, so that each fiber is isometric to S^{2}. We also take a closed neighborhood M of the fixed point variety such that the inclusion ℬ̂_{N} ↪^{i} M is a proper homotopy equivalence. The map α_{s} is then the composition of the antipodal map on the fibers and a flow via the action into M, followed by a proper homotopy inverse to i.

In order to show that the (α_{s})_{∗}’s define a representation of Ŵ in H_{∗}(ℬ̂_{N}), it suffices to show that they satisfy the relations of Ŵ viewed as an abstract Coxeter group with simple reflections S. Moreover, since Coxeter group relations involve at most two simple reflections at a time, it will be enough to show that the above maps induce representations of W_{s} in homology for each s, where W_{s} is the finite Weyl group obtained by omitting the generator s and ignoring relations involving it. This is possible even in the case of SL_{2}(C) where the affine Weyl group (the infinite dihedral group) has only two simple reflections because there is no relation linking the two.

For each s, consider the space 𝒦_{s} of maximal parahoric subalgebras of type s, i.e. those parahoric subalgebras corresponding to the subset S \ {s} of S. If we identify ℬ̂ with G(F)/B̂, where B̂ is the stabilizer of a fixed Iwahori subalgebra 𝔟̂, and let K_{s} and 𝔨_{s} be the maximal parahorics of type s containing B̂ and 𝔟̂ respectively, then 𝒦_{s} ≃ G(F)/K_{s}. Just as for 𝒫̂_{s}, the set 𝒦_{s} has the structure of an infinite-dimensional variety compatible with the variety structure on ℬ̂ and the natural projection (and with the structures on 𝒫̂_{t} for t ≠ s). It is known that K_{s} is a projective limit of algebraic groups (\overline{K}_{s}^{n})_{n≥1} defined over the residue field C where \overline{K}_{s}^{1} is a complex semisimple group (denoted for convenience G_{s}) with Weyl group W_{s} [BT]. There is a natural surjective homomorphism K_{s} → G_{s}; let K_{s}^{1} be its kernel and 𝒦_{s}^{1} = G(F)/K_{s}^{1}.

The inclusions K_{s}^{1} ⊂ B̂ ⊂ K_{s} give us a principal G_{s}-bundle 𝒦_{s}^{1} → 𝒦_{s} with associate ℬ_{s}-bundle ℬ̂ → 𝒦_{s}, where ℬ_{s} is the flag variety of G_{s}. The C-action on ℬ̂ induces a unique algebraic C-action on 𝒦_{s} making a diagram analogous to the above diagram commute. Let 𝒦_{s,φ} = 𝒦_{s,N} be the fixed point variety.

Lemma . 𝒦_{s,N} is finite-dimensional.

Proof. It suffices to show that it is the image of ℬ̂_{N} under the natural projection. Note that φ is a fiberwise C-action on the restriction of the bundle ℬ̂ → 𝒦_{s} to the fixed point set 𝒦_{s,N}. Since the fibers are projective and C is a connected, solvable affine algebraic group, there is a fixed point, i.e. an element of ℬ̂_{N}, in each fiber.

Form a principal G_{s}-bundle by restricting the principal G_{s}-bundle 𝒦_{s}^{1} → 𝒦_{s} to the fixed point variety 𝒦_{s,N}. Taking the associate ℬ_{s}-bundle, we obtain the restriction of the natural fibration ℬ̂ → 𝒦_{s}. The map φ furnishes this bundle with an algebraic fiberwise C-action whose fixed point set is precisely ℬ̂_{N}. We are in the precise context of Kazhdan and Lusztig’s original construction and thus obtain a representation of W_{s} in H_{∗}(ℬ̂_{N}).

It only remains to show that the map ℬ̂_{N} →^{γ_{s}} ℬ̂_{N} is the same as α_{s} for each s ∈ W_{s}. However, in both procedures, the maps are constructed by passing to the P^{1}-bundle obtained by omitting every simple reflection besides s. In other words, the actual construction takes place in the bundle ℰ^{s} → P̂_{s,N} in both cases, and it does not matter that in the second method, we remove the other simple reflections in two steps instead of in one. Therefore, the (α_{s})_{∗}’s satisfy the Coxeter relations, so the map s ↦ (α_{s})_{∗} defines a group homomorphism Ŵ → Aut(H_{∗}(ℬ̂_{N})). This proves the theorem.

Remarks. 1. It is possible to prove that (α_{s})_{∗}^{2} = 1 (and in particular, to prove the theorem for the special case G = SL_{2}(C)) without considering the varieties of maximal parahoric subalgebras. Simply notice that for the algebraic P^{1}-bundle ℰ^{s} → P̂_{s,N}, the structure group is P GL_{2}(C) because these are the only algebraic automorphisms of P^{1}. It is then a general fact that this bundle must be the associate P^{1}-bundle to a principal P GL_{2}(C)-bundle. Therefore, Kazhdan and Lusztig’s result, applied to the group P GL_{2}(C), shows that s ↦ (α_{s})_{∗} gives a representation of the Weyl group Z/2Z.

2. It seems possible to extend this construction to the following more general situation. Let X be a complex algebraic variety and E →^{ρ} X a ‘principal G(F)- bundle’, i.e. let E be a set on which G(F) acts freely and transitively on the right with ρ defining a bijective correspondance between orbits in E and points of X. We can formally define the associated bundles E ×_{G(F)}𝒫̂ → X for any variety 𝒫̂ of parahoric subalgebras. Suppose that these varieties are endowed with the structures of infinite-dimensional complex varieties compatible with the natural projections. Now assume that we have an algebraic fiberwise C-action φ on Ê = E × _{G(F)}ℬ̂, whose fixed point variety Ê_{φ} is complex algebraic in the usual sense. The above argument, replacing 𝒫̂ by the associated 𝒫̂-bundle, then yields a representation of Ŵ in H_{∗}(Ê_{φ}). Affine Springer representations are the result of this construction for the special case where X is a point and E = G(F).

This paper is based on part of my doctoral thesis at the University of Chicago [S]. It is a great pleasure to thank my advisor, Robert Kottwitz, for many extremely helpful comments and observations. I would also like to thank George Lusztig for some helpful comments and Sam Evens for a very useful suggestion. Finally, I am happy to acknowledge the support of a National Science Foundation Graduate Fellowship and a University of Chicago McCormick Fellowship.

References

[BT] F. Bruhat and J. Tits, Groupes algébriques simples sur un corps local, Proceedings of a Conference on Local Fields, Springer-Verlag, Berlin, 1991, pp. 23–36.

[K] S. Kato, A realization of irreducible representations of affine Weyl groups, Indag. Math. 45 (1983), 193–201.

[KL1] D. Kazhdan and G. Lusztig, A topological approach to Springer’s representations, Adv. in Math. 38 (1980), 222–228.

[KL2] D. Kazhdan and G. Lusztig, Fixed point varieties on affine flag manifolds, Israel J. Math. 62 (1988), 129–168.

[L] G. Lusztig, personal communication, January 20, 1996.

[S] D. S. Sage, The geometry of fixed point varieties on affine flag manifolds, Ph.D. thesis, University of Chicago, 1995.

Department of Mathematics, University of Utah, Salt Lake City, Utah 84112

   E-mail address: sage@math.utah.edu