Journal of Algebra 368 (2012) Contents lists available at SciVerse ScienceDirect. Journal of Algebra.

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1 Journal of Algebra 368 (2012) Contents lsts avalable at ScVerse ScenceDrect Journal of Algebra An algebro-geometrc realzaton of equvarant cohomology of some Sprnger fbers Shrawan Kumar a,1, Claudo Proces b, a Department of Mathematcs, Unversty of North Carolna, Chapel Hll, NC , USA b Dpartmento d Matematca, Sapenza Unverstà d Roma, Pazzale Aldo Moro 5, Roma, Italy artcle nfo abstract Artcle hstory: Receved 30 March 2012 Avalable onlne xxxx Communcated by Dhua Jang Keywords: Equvarant cohomology Sprnger fbers We gve an explct affne algebrac varety whose coordnate rng s somorphc (as a W -algebra) wth the equvarant cohomology of some Sprnger fbers Elsever Inc. All rghts reserved. 1. Introducton Let G be a connected smply-connected semsmple complex algebrac group wth a Borel subgroup B and a maxmal torus T B. LetP B be a (standard) parabolc subgroup of G. LetL T be the Lev subgroup of P and let S be the connected center of L (.e., S s the dentty component of the center of L). Then, S T. We denote the Le algebras of G, T, B, P, L, S by the correspondng Gothc characters: g, t, b, p, l, s respectvely. Let W be the Weyl group of G and W L W the Weyl group of L. Letσ = σ l be a prncpal nlpotent element of l. LetX = G/B be the full flag varety of G and let X σ X be the Sprnger fber correspondng to the nlpotent element σ (.e., X σ s the subvarety of X fxed under the left multplcaton by Exp σ, endowed wth the reduced subscheme structure). Observe that S keeps the varety X σ stable under the left multplcaton of S on X. Defnton 1.1. Let Z l be the reduced closed subvarety of t t defned by Z l := { (x, wx): w W, x s }. * Correspondng author. E-mal addresses: shrawan@emal.unc.edu (S. Kumar), proces@mat.unroma1.t (C. Proces). 1 Supported by NSF grants /$ see front matter 2012 Elsever Inc. All rghts reserved.

2 S. Kumar, C. Proces / Journal of Algebra 368 (2012) Snce Z l s a cone nsde t t, the affne coordnate rng C[Z l ] s a non-negatvely graded algebra. Moreover, the projecton π 1 : Z l s on the frst factor gves rse to an S(s )-algebra structure on C[Z l ]. Also, defne an acton of W on Z l by v (x, wx) = (x, vwx), for x s, v, w W. Ths acton gves rse to a W -acton on C[Z l ], commutng wth the S(s ) acton on C[Z l ]. Infact,eventhoughwedonotneedt,W s precsely the automorphsm group of C[Z l ] as S(s )- algebra. For p = b, the Lev subalgebra l concdes wth t, σ t = 0 and X σ = X. In ths case, s = t and we abbrevate Z t by Z.Clearly,Z l (for any Lev subalgebra l) s a closed subvarety of Z. The followng theorem s our man result. Theorem 1.2. Wth the notaton as above, assume that the canoncal restrcton map H (X) H (X σ ) s surjectve, where H denotes the sngular cohomology wth complex coeffcents. Then, there s a graded S(s )- algebra somorphsm φ l : C[Z l ] H S (X σ ), where H S denotes the S-equvarant cohomology wth complex coeffcents. Moreover, the followng dagram s commutatve: C[Z] φ t H T (X) C[Z l ] φ l H S (X σ ), (1) where the vertcal maps are the canoncal restrcton maps. In partcular, we get an somorphsm of graded algebras φ o l : C S(s ) C[Z l ] H (X σ ), makng the followng dagram commutatve: C S(t ) C[Z] φ o t H (X) C S(s ) C[Z l ] φ o l H (X σ ), (2) where the vertcal maps are the canoncal restrcton maps and C s consdered as an S(s )-module under the evaluaton at 0. Moreover, the somorphsm φ o l s W -equvarant under the Sprnger s W -acton on H (X σ ) and the W - acton on C S(s ) C[Z l ] nduced from the W -acton on C[Z l ] defned above.

3 72 S. Kumar, C. Proces / Journal of Algebra 368 (2012) Proof of the theorem Before we come to the proof of the theorem, we need the followng lemma. (See, e.g., [C, Theorem 2].) Lemma 2.1. For any w W, there exsts a unque w W L such that w wb X S σ X. Moreover, ths nduces a bjecton W L \W X S σ. We also need the followng smple (and well-known) result. Lemma 2.2. Let S = S(V ) be the symmetrc algebra for a fnte dmensonal vector space V and let M, N, R be three S-modules. Assume that N and R are S-free of the same fnte rank and M s an S-submodule of R. Then, any surjectve S-module morphsm φ : M N s an somorphsm. We now come to the proof of the theorem. Proof of the theorem. Consder the equvarant Borel homomorphsm β : S ( t ) H T (X) obtaned by λ c 1 (L λ ), where λ t and c 1 (L λ ) s the T -equvarant frst Chern class of the lne bundle L(λ) on X correspondng to the character e λ, and extended as a graded algebra homomorphsm. Ths gves rse to an algebra homomorphsm χ : C[t t] S ( t ) S ( t ) H T (X), p q p β(q), where p denotes the multplcaton n the T -equvarant cohomology by p S(t ) H T (pt). Its well known that χ s surjectve. Moreover, both the restrcton maps H T (X) H S (X) H S (X σ ) are surjectve; ths follows snce both the spaces X and X σ have cohomologes concentrated n even degrees (cf. [DLP]). (Use the degenerate Leray Serre spectral sequence and the assumpton that the restrcton map H (X) H (X σ ) s surjectve.) Consder the canoncal surjectve map θ : C[t t] C[Z l ]. Then, of course, { Ker θ = p q : p, q ( S t ) } and p (x) q (wx) = 0, for all x s and w W. (3) We clam that Ker θ Ker γ, (4)

4 S. Kumar, C. Proces / Journal of Algebra 368 (2012) where γ s the composte map C[t t] χ H T (X) H S (X σ ). Snce X σ has cohomologes only n even degrees, by the degenerate Leray Serre spectral sequence, H S (X σ ) s a free S(s )-module. In partcular, by the Borel Atyah Segal Localzaton Theorem (cf. [AP, Theorem 3.2.6]), ( ) H S (X σ ) H S X S σ. Thus, to prove the clam (4), t suffces to prove that for any p q Ker θ, ( γ p q 0. ) X Sσ It s easy to see that the Borel homomorphsm β restrcted to the T -fxed ponts X T satsfes: Thus, for any w W, β(q)(wb) = wq, for any q S ( t ) and w W. ( ) ( (w ) γ p q wb = (p ) ( w wq ) ) s, where w s as n Lemma 2.1. From the descrpton of Ker θ gven n (3), we thus get that the clam (4) s true. Hence, the map θ descends to a surjectve S(s )-algebra homomorphsm φ l : C[Z l ] H S (X σ ). Agan usng the Localzaton Theorem, the free S(s )-module H S (X σ ) s of rank = # W L \W,snce # X S σ = # W L\W by Lemma 2.1. Also, the projecton on the frst factor π 1 : Z l s s a fnte morphsm wth all ts fbers of cardnalty # W L \W. To see ths, consder the surjectve morphsm α : s W /W L Z l, (x, ww L ) (x, wx). Then, π 1 α : s W /W L s s agan the projecton on the frst factor, whch s clearly a fnte morphsm and hence so s π 1. Now, takng M = C[Z l ], N = H S (X σ ), R = C[s W /W L ] and V = s n Lemma 2.2, we get that φ l s an somorphsm, where the ncluson M R s nduced from the surjectve morphsm α : s W /W L Z l. The commutatvty of the dagram (1) clearly follows from the above proof. Snce H (X σ ) s concentrated n even degrees, by the degenerate Leray Serre spectral sequence, we get that H (X σ ) C S(s ) H S (X σ ). From ths the In partcular part of the theorem follows. From the defnton of the map φ t, t s clear that φt o s W -equvarant wth respect to the acton of W on C S(t ) C[Z] nduced from the acton of W on C[Z] as defned n Defnton 1.1 and the standard acton of W on H (X). Moreover, the restrcton map H (X) H (X σ ) s W -equvarant wth respect to the Sprnger s W acton on H (X σ ) (cf. [HS, Theorem 1.1]). Thus, the W -equvarance of φl o follows from the commutatvty of the dagram (2). Ths completes the proof of the theorem.

5 74 S. Kumar, C. Proces / Journal of Algebra 368 (2012) Remark 2.3. (1) By the Jordan block decomposton, any nlpotent element σ sl(n) (up to conjugacy) s a regular nlpotent element n a standard Lev subalgebra l of sl(n). Moreover, the canoncal restrcton map H (X) H (X σ ) s surjectve n ths case. In fact, as proved by Spaltensten [S], n ths case there s a pavng of X by affne spaces as cells such that X σ s a closed unon of cells (cf. also [DLP]). Thus, the above theorem, n partcular, apples to any nlpotent element σ n any specal lnear Le algebra sl(n). (2) A certan varant (though a less precse verson) of our Theorem 1.2 for g = sl(n) s obtaned by Goresky and MacPherson [GM, Theorem 7.2]. (3) For a general semsmple Le algebra g, t s not true that the restrcton map H (X) H (X σ ) s surjectve for any regular nlpotent element n a Lev subalgebra l. Take, e.g., g of type C 3 and σ correspondng to the Jordan blocks of sze (3, 3). In ths case, the centralzer of σ n the symplectc group Sp(6) s connected and X σ s two-dmensonal. The cohomology of X σ as a W -module s gven as follows: Of course, H 0 (X σ ) s the one-dmensonal trval W -module; H 2 (X σ ) s the sum of the threedmensonal reflecton representaton wth a one-dmensonal representaton; and H 4 (X σ ) s a threedmensonal rreducble representaton. (4) It wll be nterestng to deduce the result by De Concn and Proces on the dentfcaton of the cohomology of Sprnger fbers for sl(n) as the coordnate rng of a certan scheme (cf. [DP, 4]) from our Theorem 1.2. (5) Stroppel has dentfed the cohomology of Sprnger fbers for sl(n) wth the center of the proncpal block O p 0 of the correspondng parabolc category O (cf. [St]). Acknowledgments We thank Erc Sommers for the example gven n Remark 2.3(3). Ths work was started when both the authors were vstng the Abdus Salam Internatonal Centre for Theoretcal Physcs (ICTP) durng December, 2002, hosptalty of whch s gratefully acknowledged. The work was completed durng the followng Sprng, References [AP] C. Allday, V. Puppe, Cohomologcal Methods n Transformaton Groups, Cambrdge Stud. Adv. Math., vol. 32, Cambrdge Unversty Press, [C] J. Carrell, Orbts of the Weyl group and a theorem of De Concn and Proces, Compos. Math. 60 (1986) [DLP] C. De Concn, G. Lusztg, C. Proces, Homology of the zero-set of a nlpotent vector feld on a flag manfold, J. Amer. Math. Soc. 1 (1988) [DP] C. De Concn, C. Proces, Symmetrc functons, conjugacy classes and the flag varety, Invent. Math. 64 (1981) [GM] M. Goresky, R. MacPherson, On the spectrum of the equvarant cohomology rng, Canad. J. Math. 62 (2010) [HS] R. Hotta, T.A. Sprnger, A specalzaton theorem for certan Weyl group representatons and an applcaton to the Green polynomals of untary groups, Invent. Math. 41 (1977) [S] N. Spaltensten, The fxed pont set of a unpotent transformaton on the flag manfold, Nederl. Akad. Wetensch. Proc. Ser. A 79 (1976) [St] C. Stroppel, Parabolc category O, perverse sheaves on Grassmannans, Sprnger fbers and Khovanov homology, Compos. Math. 145 (2009)