Let $H^*(\calB_e)$ be the cohomology of the Springer fibre for the nilpotent
element $e$ in a simple Lie algebra $\g$, on which the Weyl group $W$ acts by
the Springer representation. Let $\Lambda^i V$ denote the $i$th exterior power
of the reflection representation of $W$. We determine the degrees in which
$\Lambda^i V$ occurs in the graded representation $H^*(\calB_e)$, under the
assumption that $e$ is regular in a Levi subalgebra and satisfies a certain
extra condition which holds automatically if $\g$ is of type A, B, or C. This
partially verifies a conjecture of Lehrer--Shoji, and extends the results of
Solomon in the $e=0$ case and Lehrer--Shoji in the $i=1$ case. The proof
proceeds by showing that $(H^*(\calB_e) \otimes \Lambda^* V)^W$ is a free
exterior algebra on its subspace $(H^*(\calB_e)\otimes V)^W$.Comment: 8 page

Henderson, Anthony

English

arXiv

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1055–1058Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comGroup Theory/Lie AlgebrasExterior powers of the reflection representation in the cohomology ofSpringer fibresLes puissances extérieures de la représentation géométrique dans la cohomologie desfibres de SpringerAnthony Henderson 1School of Mathematics and Statistics, University of Sydney, NSW 2006, Australiaa r t i c l e i n f o a b s t r a c tArticle history:Received 3 February 2010Accepted after revision 15 September 2010Available online 25 September 2010Presented by Gérard LaumonLet H∗(Be) be the cohomology of the Springer fibre for the nilpotent element e in a simpleLie algebra g. Let Λi V denote the ith exterior power of the reflection representation of W .We determine the degrees in which Λi V occurs in the graded representation H∗(Be),under the assumption that e is regular in a Levi subalgebra and satisfies a certain extracondition which holds automatically if g is of type A, B, or C. This partially verifies aconjecture of Lehrer and Shoji.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éSoit H∗(Be) la cohomologie de la fibre de Springer pour l’élément nilpotent e de l’algèbrede Lie simple g. Soit Λi V la i-ème puissance extérieure de la représentation géométriquede W . Nous trouvons les degrés des contributions de Λi V à la représentation graduéeH∗(Be), si e est régulier dans une sous-algèbre de Levi et satisfait à une autre conditionqui est vraie si g est de type A, B, ou C. Ce résultat démontre partiellement une conjecturede Lehrer et Shoji.© 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.1. IntroductionLet g be a simple complex Lie algebra of rank . Let W denote the Weyl group of g, and let V be the reflection represen-tation of W . It is well known that the exterior powers Λi V , for i = 0,1, . . . , , are inequivalent irreducible representationsof W , each of which is self-dual.Let e be a nilpotent element of g. The Springer fibre Be is the variety of Borel subalgebras of g containing e. Let H∗(Be)denote the graded cohomology ring of Be with complex coefficients; the cohomology lives solely in even degrees, so H∗(Be)is commutative. We have the Springer representation of W on each H2 j(Be) (see [5], [2, Chapter 9]). Let s (depending on e)denote the multiplicity of the irreducible representation V in the total representation H∗(Be). Let m1,m2, . . . ,ms be themultiset of nonnegative integers, listed in increasing order, which are the halved degrees of the occurrences of V in thegraded representation H∗(Be). That is, we have by definition∑j dim(H2 j(Be) ⊗ V )W q j = qm1 + qm2 + · · · + qms .E-mail address: anthony.henderson@sydney.edu.au.1 The author’s research is supported by ARC grant DP0985184.1631-073X/$ – see front matter © 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.doi:10.1016/j.crma.2010.09.0151056 A. Henderson / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1055–1058In the special case e = 0, it is well known (see [5, §5]) that H∗(B) is isomorphic to the coinvariant algebra C∗(W ) of W ,that s = , and that m1, . . . ,m are the exponents of W . More generally, if e is a regular nilpotent in a Levi subalgebraof semisimple rank r, then it was proved by Lehrer and Shoji in [3, Theorem 2.4] (see also [9]) that s =  − r and thatm1, . . . ,ms are the coexponents of the corresponding parabolic hyperplane arrangement, in the sense of Orlik and Solomon.See [8] for some other related interpretations of these coexponents. For g of classical type and general e, the numbersm1,m2, . . . ,ms were calculated by Spaltenstein in [9, Propositions 1.6–1.9].Lehrer and Shoji conjectured that, at least in the parabolic case which they considered, the occurrences of each exteriorpower Λi V in H∗(Be) were also controlled in a natural way by m1,m2, . . . ,ms .Conjecture 1.1. (See [3, Conjecture 8.3].) Suppose that e is a regular nilpotent in a Levi subalgebra. Then for any i = 0,1, . . . , , wehave∑j dim(H2 j(Be) ⊗ Λi V )W q j = ei(qm1 ,qm2 , . . . ,qms ) (the ith elementary symmetric polynomial in qm1 ,qm2 , . . . ,qms , whichis defined to be zero if i > s).The e = 0 case of this conjecture had already been proved by Solomon in [6]; indeed, he proved the stronger statementthat the algebra (C∗(W ) ⊗ Λ∗V )W is a free exterior algebra on (C∗(W ) ⊗ V )W .The main result of this Note is the following generalization of Solomon’s result, which implies various cases of Conjec-ture 1.1:Theorem 1.2. Suppose that e is regular in a Levi subalgebra of g, and define s and m1, . . . ,ms as above. Also suppose that there is aparabolic subgroup W K of W such that the following two conditions hold:(1) There exist invariant polynomials f1, f2, . . . , f s ∈ (S∗V )W , homogeneous of degrees m1 + 1,m2 + 1, . . . ,ms + 1, whose restric-tions to the reflection representation V K of W K form a set of fundamental invariants for W K .(2) The nilpotent orbit of e intersects the nilradical of the parabolic subalgebra pK associated to W K .(See Section 2 for the definitions of V K and pK .) Then the algebra (H∗(Be) ⊗ Λ∗V )W is a free exterior algebra on (H∗(Be) ⊗ V )W .More precisely, the natural homomorphism ψ : Λ∗((H∗(Be) ⊗ V )W ) → (H∗(Be) ⊗ Λ∗V )W is an isomorphism.Here the domain and codomain of ψ are (N×N)-graded algebras over C, where the (i, j)-components are Λi((H2 j(Be)⊗V )W ) and (H2 j(Be)⊗Λi V )W respectively, and in both cases the algebra multiplication is graded-commutative with respectto the N-grading labelled by i; the homomorphism ψ is induced by the inclusion of the subspace (H∗(Be) ⊗ V )W in(H∗(Be) ⊗ Λ∗V )W . Since the graded degrees of this subspace are (1,m1), (1,m2), . . . , (1,ms), the statement that ψ is anisomorphism implies Conjecture 1.1.Simple calculations2 verify the following results:Proposition 1.3. If g is of type A, then the assumptions of Theorem 1.2 hold for any e.Proposition 1.4. If g is of type B or C, then the assumptions of Theorem 1.2 hold for any e which is regular in a Levi subalgebra.Hence Conjecture 1.1 is proved in types A–C. By contrast, suppose that g is of type D4 and e has Jordan type (3212) inthe natural representation on C8. Then e is regular in a Levi subalgebra of type A2, but we have m2 = 2 and there are noW -invariant polynomials of degree 3, so condition (1) of Theorem 1.2 cannot be satisfied.2. Proof of Theorem 1.2Continue the notation of the introduction. Let h ⊂ b be a Cartan subalgebra and Borel subalgebra of g, and let Π ⊂Φ+ ⊂ Φ be the corresponding set of simple roots, positive roots, and roots. We identify W with the subgroup of GL(h)generated by the simple reflections sα for α ∈ Π ; the reflection representation V of W is merely h itself.Let J ⊆ Π be a subset of size r, and set s = − r. We have a Levi subalgebra l J and parabolic subalgebra p J containing hand b respectively, a parabolic subsystem Φ J of Φ , and a parabolic subgroup W J of W . Define V J = ⋂α∈ J ker(α) = V W J .We write V J for the unique W J -invariant complement to V J in V , which is the reflection representation of W J . Note thatdim V J = s and dim V J = r. Let A J and A J be the hyperplane arrangements in V J and V J respectively induced by the roothyperplanes in V .We assume for the remainder of the section that e is parabolic of type J , meaning that the orbit of e contains the regularnilpotent elements of l J . As mentioned in the introduction, Lehrer and Shoji proved in this case that∑j dim(H2 j(Be) ⊗V )W q j = qm1 + qm2 + · · · + qms , where m1, . . . ,ms are the coexponents of the arrangement A J . (See [3, Theorem 2.4]; themissing case in type D is covered by the results of Spaltenstein [9].)An important special feature of the parabolic case is Lusztig’s Induction Theorem for Springer representations.2 The details may be found in the preprint version of this Note, arXiv:1001.3164.A. Henderson / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1055–1058 1057Theorem 2.1. (See [4].) The representation of W on H∗(Be), neglecting the grading, is isomorphic to the induction IndWW J (C) of thetrivial representation of W J .Corollary 2.2. We have dim(H∗(Be) ⊗ Λ∗V )W = 2s , so the domain and codomain of ψ have the same dimension.Proof. By Frobenius Reciprocity, we know that dim(IndWW J (C) ⊗ Λ∗V )W = dim(Λ∗V )W J . From the fact that (Λ∗V )sα =Λ∗(ker(α)) for all α ∈ J one deduces (Λ∗V )W J = Λ∗(V J ), and the corollary follows. So to prove Theorem 1.2, it suffices to show that the homomorphism ψ is injective. Since the domain is an exterioralgebra, this will follow if we can show that ψ(Λs((H∗(Be) ⊗ V )W )) = 0.Using the W -equivariant isomorphism of V with its dual, we will identify the symmetric algebra S∗V with the ringof polynomial functions on V . It is well known that the invariant subring (S∗V )W is freely generated by  homogeneouspolynomials, called fundamental invariants for W . The coinvariant algebra C∗(W ) of W is the quotient S∗V /I , where I isthe ideal generated by these fundamental invariants.Now there is a canonical (degree-doubling) W -equivariant algebra homomorphism S∗V → H∗(B) which identifies H∗(B)with C∗(W ) (see [5, §5]). Composing this with the natural homomorphism H∗(B) → H∗(Be), which is W -equivariantby [9, Lemma 1.4] (this fact in characteristic p was [1, Theorem 1.1]), we obtain a W -equivariant homomorphismϕ : S∗V → H∗(Be). Note that the image of ϕ is contained in the subspace H∗(Be)A(e) of invariants for the componentgroup of the centralizer of e in the adjoint group of g; in particular, ϕ is not surjective in general. However, it may happenthat the induced map (S∗V ⊗ V )W → (H∗(Be) ⊗ V )W is surjective even if ϕ is not. We will see that this occurs under theassumptions of Theorem 1.2, which means that a calculation with polynomials on V suffices to prove what we want.Henceforth we let K ⊆ Π be a subset satisfying conditions (1) and (2) of Theorem 1.2. Note that condition (1) entails|K | = s. Choose a basis v1, v2, . . . , v of V such that v1, . . . , vs is a basis of V K . Since the exterior derivative S∗V →S∗V ⊗ V : f 	→ ∑ ∂ f∂v j⊗ v j is W -equivariant, we have the following s elements of (H∗(Be) ⊗ V )W :n∑j=1ϕ(∂ f1∂v j)⊗ v j,n∑j=1ϕ(∂ f2∂v j)⊗ v j, . . . ,n∑j=1ϕ(∂ f s∂v j)⊗ v j.We can prove both that these form a basis of (H∗(Be) ⊗ V )W , and that ψ(Λs((H∗(Be) ⊗ V )W )) = 0 as required, by provingthe single fact that ϕ() = 0, where  is the determinant of the s × s matrix ( ∂ fa∂vb), with a and b ranging from 1 to s.Since v1, . . . , vs span the reflection representation of W K , we have w = ε(w) for all w ∈ W K . This forces  to bedivisible by the polynomial πK = ∏β∈Φ+K β . Condition (1) tells us that on restricting to V K ,  becomes the Jacobian matrixof the fundamental invariants of W K , which is well known to be a nonzero scalar multiple of the restriction to V K of πK(see [10]). So  is a nonzero scalar multiple of πK , and it suffices to prove that ϕ(πK ) = 0.By condition (2), we may suppose that e lies in the nilradical of pK . Then any Borel subalgebra contained in pK mustcontain e, so we have an inclusion B K ↪→ Be , where B K denotes the variety of Borel subalgebras contained in pK , whichcan be identified with the flag variety of lK . Hence it suffices to prove that πK is not in the kernel of the compositionS∗V → H∗(B) → H∗(B K ). But this composition is the canonical homomorphism identifying C∗(W K ) with H∗(B K ), whichmaps πK to a generator of the top-degree cohomology of B K (compare [1, Proposition 1.4], which uses exactly this argumentin the case when e lies in the Richardson orbit of pK ). This completes the proof of Theorem 1.2.AcknowledgementsThe main result of this Note, Theorem 1.2, dates from 1997, when I was a student at the University of Sydney, supervisedby Gus Lehrer. As the reader will observe, it is indebted to Lehrer’s ideas, and I thank him for his help and encouragement.I did not publish this result at the time, since it did not prove the motivating Conjecture 1.1 in general. Recently EricSommers [7] has completed the proof of Conjecture 1.1 by a different method, and also removed the assumption that e isregular in a Levi subalgebra. I thank him for his interest in my old result, and for the suggestion that it be published tosupply part of the general proof.References[1] R. Hotta, T.A. Springer, A specialization theorem for certain Weyl group representations and an application to the Green polynomials of unitary groups,Invent. Math. 41 (1977) 113–127.[2] J.E. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups, Mathematical Surveys and Monographs, vol. 43, American Mathematical Society,Providence, Rhode Island, 1995.[3] G.I. Lehrer, T. Shoji, On flag varieties, hyperplane complements and Springer representations of Weyl groups, J. Austral. Math. Soc. Ser. A 49 (3) (1990)449–485.[4] G. Lusztig, An induction theorem for Springer’s representations, in: Representation Theory of Algebraic Groups and Quantum Groups, in: Adv. Stud.Pure Math., vol. 40, Math. Soc. Japan, Kinokuniya, 2004, pp. 253–259.1058 A. Henderson / C. R. Acad. Sci. Paris, Ser. I 348 (2010) 1055–1058[5] T. Shoji, Geometry of orbits and Springer correspondence, in: Orbites unipotentes et représentations, I, in: Astérisque, vol. 168, Soc. Math. de France,Paris, 1988, pp. 61–140.[6] L. Solomon, Invariants of finite reflection groups, Nagoya Math. J. 22 (1963) 57–64.[7] E. Sommers, Exterior powers of the reflection representation in Springer theory, arXiv:1008.1180.[8] E. Sommers, P. Trapa, The adjoint representation in rings of functions, Represent. Theory 1 (1997) 182–189.[9] N. Spaltenstein, On the reflection representation in Springer’s theory, Comment. Math. Helv. 66 (4) (1991) 618–636.[10] R. Steinberg, Invariants of finite reflection groups, Canad. J. Math. 12 (1960) 616–618.

Exterior powers of the reflection representation in the cohomology of Springer fibres

Anthony Henderson

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Comptes Rendus Mathematique

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Exterior powers of the reflection representation in the cohomology of Springer fibres

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