Springer theory and the geometry of quiver ag varieties

Transcription

1 Springer heory and he geomery of quiver ag varieie Julia Anneliee Sauer Submied in accordance wih he requiremen for he degree of Docor of Philoophy The Univeriy of Leed School of Mahemaic Sepember 2013

2 The candidae conrm ha he work ubmied i her own and ha appropriae credi ha been given where reference ha been made o he work of oher. Thi copy ha been upplied on he underanding ha i i copyrigh maerial and ha no quoaion from he hei may be publihed wihou proper acknowledgemen. c The Univeriy of Leed and Julia Anneliee Sauer 2

3 Acknowledgemen I hank my upervior Dr. Andrew Hubery for hi conan uppor hrough regular meeing, he helped o anwer a grea many queion and o olve mahemaical problem. Alo, I would like o hank my econd upervior Profeor William Crawley-Boevey for helping me wih clearing concep and going hrough par of he hei poining ou miake. Boh very generouly len me heir ime and harp mind for my projec. For helpful dicuion and explaining ome argumen I am hankful o Dr. Greg Sevenon, Profeor Marku Reineke and Dr. Michael Bae. Alo, I hank Profeor Henning Kraue and he CRC 701 for nancial uppor during everal gue ay in Bielefeld. For accompanying me hrough hi ime I wan o hank my family, epecially my paren for heir nancial and moral uppor. My pecial hank go o my friend from he pg-aellie oce in Leed and o he algebra group for he nice amophere a he algebra dinner. 3

4 Abrac The hei coni of he following chaper: 1. Springer heory. For any projecive map E V, Chri and Ginzburg dened an algebra rucure on he (Borel-Moore homology Z := H (E V E, which we call Seinberg algebra. (Graded Projecive and imple Z-module are conrolled by he BBD-decompoiion aociaed o E V. We reric o collaping of union of homogeneou vecor bundle over homogeneou pace becaue we have he cellular braion echnique and for equivarian Borel-Moore homology we can ue localizaion o oru-xed poin. Example of Seinberg algebra include group ring of Weyl group, Khovanov-Lauda- Rouquier algebra, nil Hecke algebra. 2. Seinberg algebra. We chooe a cla of Seinberg algebra and give generaor and relaion for hem. Thi fail if he homogeneou pace are parial and no complee ag varieie, we call hi he parabolic cae. 3. The parabolic cae. In he parabolic cae, we realize he Seinberg algebra Z P a corner algebra in a Seinberg algebra Z B aociaed o Borel group (hi mean Z P = ez B e for an idempoen elemen e Z B. 4. Monoidal caegorie. We explain how o conruc monoidal caegorie from familie of collaping of homogeneou bundle. 5. Conruc collaping. We conruc collaping map over given loci which are reoluion of ingulariie or generic Galoi covering. For cloure of homogeneou decompoiion clae of he Kronecker quiver hee map are new. 6. Quiver ag varieie. Quiver ag varieie are he bre of cerain collaping of homogeneou bundle. We inveigae when quiver ag varieie have only niely many orbi and we decribe he caegory of ag of quiver repreenaion a a -lered ubcaegory for he quai-herediary algebra KQ KA n. 7. A n -equioriened. For he A n -equioriened quiver we nd a cell decompoiion of he quiver ag varieie, which are paramerized by cerain muli-ableaux. 4

5 Conen 1 A urvey on Springer heory Deniion of a Springer heory Convoluion module The Seinberg algebra The Seinberg algebra H[ ] A (Z a module over H A (p The Seinberg algebra H A (Z and HA (E Indecompoable projecive graded module over H[ ] A (Z and heir op for a dieren grading Indecompoable projecive in he caegory of graded lef H A [ ] (Z- module Simple in he caegory of graded niely generaed lef H A < >(Z- module Springer bre module in he caegory of graded H? A (Z-module The Springer funcor Orbial varieie, Springer bre and raa in he Seinberg variey Wha i Springer heory? Claical Springer heory Paramerizing imple module over Hecke algebra Quiver-graded Springer heory Monoidal caegoricaion of he negaive half of he quanum group Lieraure review Generalized quiver Hecke algebra Generalized quiver-graded Springer heory Relaionhip beween parabolic group in G and G The equivarian cohomology of ag varieie Compuaion of xed poin Noaion for he xed poin The bre over he xpoin Relaive poiion raicaion In he ag varieie In he Seinberg variey A hor lamenaion on he parabolic cae

6 2.7 Convoluion operaion on he equivarian Borel-Moore homology of he Seinberg variey Compuaion of ome Euler clae Localizaion o he oru xed poin The W-operaion on E G : Calculaion of ome equivarian mulipliciie Convoluion on he xed poin Generaor for Z G Relaion for Z G Parabolic Nil Hecke algebra and parabolic Seinberg algebra The parabolic (ane nil Hecke algebra On parabolic Seinberg algebra Reineke' Example (cp. end of [Rei03] Lieraure From Springer heory o monoidal caegorie (I-Graded Springer heory Monoidal caegoricaion of a muliplicaive equence of algebra Alernaive decripion of C a caegory of projecive graded module Luzig' pervere heave Example: Quiver-graded Springer heory Quiver-graded Springer heory - Borel cae Example: Symplecic quiver-graded Springer heory The Seinberg algebra and i horizonal produc Luzig' Pervere heave/projecive module correponding o he verice of he ymmeric quiver Monoidal caegoricaion A dicuion on he earch for Hall algebra for ymmeric quiver repreenaion Conrucing collaping of homogeneou bundle over quiver loci Explici equaion for he image of he Springer map The orbi lemma Quiver-graded Springer map The generic compoiion monoid When i he quiver-graded Springer map a reoluion of ingulariie of an orbi cloure? Reoluion pair for Dynkin quiver Reoluion pair for he oriened cycle Reoluion pair for exended Dynkin quiver Springer map for homogeneou decompoiion clae

7 5.3.1 Tube polynomial Springer map for he Jordan quiver Springer map for homogeneou decompoiion clae of he Kronecker quiver Quiver ag varieie of nie ype A locally rivial bre bundle Caegorie of ag Wha i a ag? On he enor produc KQ KA n Caegorie of monomorphim Decripion a -lered module over he quai-herediary algebra Λ = KQ KA ν When i X repreenaion-nie? Tangen mehod An example of a no generically reduced quiver ag variey Deecing irreducible componen Sraicaion Sraicaion in orbi Reineke' raicaion A conjecure on generic reducedne of Dynkin quiver ag varieie Scheme dened by rank condiion Quiver-relaed cheme dened by rank condiion Canonical decompoiion An example of a cloure of a Reineke raum which i no a union of Reineke raa A 2 -Gramannian Open problem A n -equioriened quiver ag varieie Noaion and baic properie for A n -equioriened repreenaion Reineke raa and roo ableaux Swapping numbered boxe in row roo ableaux Dimenion of roo ableau rb-raa and row roo ableaux Spli Module rb-raicaion a ane cell decompoiion Bei number for complee A n -equioriened quiver ag varieie Conjecural par Canonical decompoiion for A n -equioriened quiver ag varieie Submodule in erm of marix normal form Remark on parial A n -quiver ag

8 7.5 Roo ableau of hook ype Appendix on equivarian (cohomology A Lemma from Slodowy' book Equivarian cohomology Equivarian Borel-Moore homology Baic properie Se heoreic convoluion Convoluion in equivarian Borel-Moore homology Dualiy beween equivarian cohomology and equivarian Borel-Moore homology Equivarian derived caegory of heave afer Bernein and Lun The funcor formalim Dualiy Localizaion for equivarian Borel-Moore homology Cellular braion for equivarian Borel-Moore homology The Serre cohomology pecral equence wih arbirary coecien A lemma from he urvey on Springer heory The cohomology ring of ag varieie Forgeful map Equivarian pervere heave Pervere heave Equivarian pervere heave Bibliography 267 8

9 Chaper 1 A urvey on Springer heory Summary. A Springer map i for u a union of collaping of (complex homogeneou vecor bundle and a Seinberg variey i he careian produc of a Springer map wih ielf. Chri and Ginzberg conruced on he (equivarian Borel-Moore homology and on he (equivarian K-heory of a Seinberg variey a convoluion produc making i an aociaive algebra, we call hi a Seinberg algebra (cp. [CG97],2.7, 5.2 for he nonequivarian cae. The decompoiion heorem for pervere heave give he indecompoable, projecive graded module over he Seinberg algebra. Alo hi convoluion yield a module rucure on he repecive homology group of he bre under he Springer map, which we call Springer bre module. In hor, for u a Springer heory i he udy of a Seinberg algebra ogeher wih i graded module. We give wo example: Claical Springer heory and quiver-graded Springer heory. (1 Deniion and baic properie. (2 Example (a Claical Springer heory. (b Quiver-graded Springer heory. (3 We dicu lieraure on he wo example. 1.1 Deniion of a Springer heory Roughly, following he inroducion of Chri and Ginzburg' book ([CG97] 1, Springer heory i a uniform geomeric conrucion for a wide cla of (non-commuaive algebra ogeher wih familie of module over hee algebra. Example include (1 Group algebra of Weyl group ogeher wih heir irreducible repreenaion, (2 ane Hecke algebra ogeher wih heir andard module and irreducible repreenaion, 1 We ake a more general approach, wha uually i conidered a Springer heory you nd in he example claical Springer heory. Neverhele, our approach i ill only a pecial cae of [CG97], chaper 8. 9

10 (3 Hecke algebra wih unequal parameer, (4 Khovanov-Lauda-Rouquier-algebra (or horly KLR-algebra and alernaively called quiver Hecke algebra (5 Quiver Schur algebra For an algebraic group G and cloed ubgroup P (over C we call G G/P a principal homogeneou bundle. For a given P -variey F we have he aociaed bundle dened by he quoien G P F := G F/, (g, f (g, f : here i p P : (g, f = (g p, p 1 f and G P F G/P, (g, f gp. Given a repreenaion ρ: P Gl(F, i.e. a morphim of algebraic group, we call aociaed bundle of he form G P F G/P homogeneou vecor bundle (over a homogeneou pace. Deniion 1. The uniform geomeric conrucion in all cae i given by he following: Given (G, P i, V, F i i I wih I ome nie e, ( G a conneced reducive group wih parabolic ubgroup P i. We alo aume here exi a maximal oru T G which i conained in every P i. ( V a nie dimenional G-repreenaion, F i V a P i -ubrepreenaion of V, i I. We idenify V, F i wih he ane pace having he vecor pace a C-valued poin. Le E i := G P i F i, i I and conider he following morphim of algebraic varieie 2 : E := i I E i [(g, f i ] π µ V i I G/P i gf i gp i Then, E V i I G/P i, [(g, f i ] (gf i, gp i i a cloed embedding (ee [Slo80b], p.25,26, i follow ha π i projecive. We call he algebraic correpondence 3 (E, π, µ Springer riple, he map π Springer map, i bre Springer bre. Via rericion of E V i I G/P i o π 1 ( {x} i I G/P i one ee ha all Springer bre are via µ cloed ubcheme of i I G/P i. i a 2 algebraic variey = eparaed inegral cheme of nie ype over a eld 3 p wo cheme morpim X Z q Y are called algebraic correpondence, if p i proper and q 10

11 We alo have anoher induced roof-diagram Z := E V E p m V ( i I G/P i ( i I G/P i wih p: E V E pr E E π V projecive and m: E V E (pr E,pr E E E µ µ ( i I G/P i ( i I G/P i. Oberve, by deniion Z = i,j I Z i,j, Z i,j = E i V E j. We call he roof-diagram (Z, p, m Seinberg riple, he cheme Z Seinberg variey (even hough a a cheme Z migh be neiher reduced nor irreducible. Bu in view of our (co-homology choice below we only udy he underlying reduced cheme and look a i C-valued poin endowed wih he analyic opology. If all parabolic group P i are Borel group, he Seinberg variey Z i a cellular braion over i I G/P i via he map Z m i I G/P i i I pr 1 G/P i G/P i (ee deniion of cellular braion in [CG97], 5.5 or ubecion in he Appendix. We chooe a (co-homology heory which can be calculaed for pace wih cellular braion propery and which ha a localizaion o he T -xed poin heory. Le H A, A {p, T, G} be (A-equivarian Borel-Moore homology. We could alo chooe (equivarian K- heory, bu we ju give ome known reul abou i. There i a naural produc on H A (Z called convoluion produc conruced by Chri and Ginzburg in [CG97]. : H A (Z H A (Z H A (Z i I (c, c c c := (q 1,3 (p 1,2(c p 2,3(c where p a,b : E E E E E i he projecion on he a, b-h facor, q a,b i he rericion of p a,b o E V E V E, hen p a,b (c HA (p 1 a,b (E V E and : Hp A (X Hq A (Y (X Y i he inerecion pairing which i induced by he -produc in relaive H A p+q 2d ingular cohomology for X, Y M wo A-equivarian cloed ube of a d-dimenional complex manifold M (cp. [CG97], p.98, ( I hold H A p (Z i,j H A q (Z k,l δ j,k H A p+q 2e k (Z i,l, e k = dim C E k. We call (H A (Z, he (A-equivarian Seinberg algebra for (G, P i, V, F i i I. I i naurally an graded module over H A (p, ee Appendix ecion??, (6. We denoe by 11

12 DA b (V he A-equivarian derived caegory of V dened by Bernein and Lun in [BL94]. There i a he following idenicaion. Theorem ([CG97], chaper 8 Le A {p, T, G} we wrie e i = dim C E i. There i an iomorphim of C-algebra H A (Z Ex D b A (V ( i I (π i C[e i ], i I (π i C[e i ], where C i he conan heaf aociaed o C on he appropriae pace. If we e H A [p] (Z := i,j I H A e i +e j p(z i,j hen H A [ ] (Z i a graded module over H A (p = HA (p. I i even a graded algebra over HA (p. The righ hand ide i naurally a graded algebra over H A (p = Ex DA b and he iomorphim i an iomorphim of graded HA (p-algebra. Furhermore, he (p(c, C Verdier dualiy on DA b (V induce an ani-involuion on he algebra on he righ hand ide. On he lef hand ide he ani-involuion i given by pullback along he wapping-hewo-facor map. The proof i only given for A = p, bu a Varagnolo and Vaero in [Var09] oberved, he ame proof can be rewrien for he A-equivarian cae. 1.2 Convoluion module Compare [CG97], ecion 2.7. Given wo ube S 1,2 M 1 M 2, S 2,3 M 2 M 3 he e-heoreic convoluion i dened a S 1,2 S 2,3 := {(m 1, m 3 m 2 M 2 : (m 1, m 2 S 1,2, (m 2, m 3 S 2,3 } M 1 M 3. Now, le S i,j M i M j be A-equivarian locally cloed ube of mooh complex A- varieie, le p i,j : M 1 M 2 M 3 M i M j be projecion on he (i, j-h facor and aume q 1,3 := p 1,3 p 1 12 (S 1,2 p 1 2,3 (S i proper. Then we ge a map 2,3 : H A p (S 1,2 H A q (S 2,3 H A p+q 2 dim C M 2 (S 1,2 S 2,3 c 1,2 c 2,3 := (q 1,3 (p 1,2c 1,2 p 2,3c 2,3. Thi way we dened he algebra rucure on he Seinberg algebra, bu i alo give a lef module ucure on H A (S for any A-variey S wih Z S = S and a righ module rucure when S Z = S. (a We chooe M 1 = M 2 = M 3 = E and embed Z = E V E E E, E = E p E E, hen i hold Z E = E. If we regrade he Borel-Moore homology (and he 12

13 Poincare dual A-equivarian cohomology of E a follow H A [p] (E := i I H A e i p(e i (= i I H e i+p A (E i =: H [p] A (E hen H[ ] A (E and H[ ] (E carry he rucure of a graded lef H[ ] A (Z-module. (b We chooe M 1 = M 2 = M 3 = E and embed E E E diagonally, hen E E = E, i hold H( A (E = H A (E a graded algebra where HA (p (E := i HA 2e i p (E i and he ring rucure on he cohomology i given by he cup produc. If we ake now Z = E V E E E hen E Z = Z and we ge a rucure a graded lef HA (E-module on HA [ ] (Z. (c We chooe M 1 = M 2 = M 3 = E, A = p and embed Z = E V E E E, π 1 ( = π 1 ( p E E, hen i hold Z π 1 ( = E. If we regrade he Borel-Moore homology and ingular cohomology of π 1 ( a follow H [p] (π 1 ( := i I H ei p(π 1 i (, H [p] (π 1 ( := i I H ei+p (πi 1 ( hen H [ ] (π 1 ( and H [ ] (π 1 ( are graded lef H [ ] (Z-module. We call hee he Springer bre module. Similarly in all example one can obain a righ module rucure (he eay wap are lef o he reader. Independenly, one can dene he ame graded module rucure on H (π 1 (, H (π 1 ( uing he decripion of he Seinberg algebra a Ex-algebra and a Yoneda operaion (for hi ee [CG97], , p.448. There i alo a reul of Johua (ee [Jo98] aying ha all hypercohomology group H A (Z, F, F DA b (Z carry he rucure of a lef (and righ HA (Z-module. 1.3 The Seinberg algebra The Seinberg algebra H[ ] A (Z a module over H (p. A We e W := i,j I W i,j wih W i,j := W i \ W/W j where W i he Weyl group for (G, T and W i W i he Weyl group for (L i, T wih L i P i he Levi ubgoup. We will x repreenaive w G for all elemen w W. Le C w = G (ep i, wp j be he G-orbi in G/P i G/P i correponding o w W i,j. pr Lemma 1. 1 (1 p: C w G/P i G/P j G/Pi i G-equivarian, locally rivial wih bre p 1 (ep i = P i wp j /P j. (2 P i wp j /P j admi a cell decompoiion ino ane pace via Schuber cell xb j x 1 vwp j /P j, v W i 13

14 where B j P j i a Borel ubgroup and x W uch ha x B j P i. In paricular, H odd (P i wp j /P j = 0 and H (P i wp j /P j = Cb i,j (v, b i,j (v := [xb j x 1 vwp j /P j ]. v W i I hold ha deg b i,j (v = 2l i,j (v where l i,j (v i he lengh of a minimal coe repreenaive in W for x 1 vww j W/W j. (3 For A {p, T, G} i hold Hodd A (C w = 0 and ince G/P i i imply conneced Hn A (C w = H p A (G/P i H q (P i wp j /P i, H A (C w = p+q=n u W/W i,v W i H A(pb i (u b i,j (v, where b i (u = [B i up i /P i ] i of degree 2 dim C G/P i 2l i (u wih l i (u i he lengh of a minimal coe repreenaive for u W/W i and b i,j (v a in (2. Proof: See lemma 80 in he Appendix. Thi implie uing degeneraion of Serre cohomology pecral equence (ee ecion in he Appendix he following properie for he homology of Z. Corollary (1 Z ha a lraion by cloed G-invarian ubvarieie uch ha he ucceive complemen are Z w := m 1 (C w, w W and he rericion of m o Z w i a vecor bundle over C w of rank d w (a complex vecor bundle. Furhermore, Hn A (Z = Hn A (Z w = Hn 2d A w (C w w W w W = HA(pb i (u b i,j (v i,j I u,v w W i,j where he index e of he la direc um i {u W/W i, v W i 2 dim G/P i 2l i (u + 2l i,j (v = n 2d w }. (2 We have H odd (Z = 0, H odd (Z = 0. (3 Z i equivarianly formal (for T and G, for Borel-Moore homology and cohomology. In paricular, for A {T, G} he forgeful map H A (Z H (Z and H A (Z H (Z are urjecive algebra homomorphim. I even hold he ronger iomorphim of C-algebra H (Z = H A (Z/H A <0(pH A (Z H (Z = H A(Z/H >0 A (ph A(Z 14

15 A a conequence we ge he following iomorphim. 1 H A (Z = H (Z C H A (p of H A (p-module 2 H A(Z = H (Z C H A(p of H A(p-module We can ee ha H[ ] A (Z ha nie dimenional graded piece and he graded piece are bounded from below in negaive degree The Seinberg algebra H A (Z and H A (E Recall from a previou ecion ha H A (E i a graded lef (and righ HA [ ] (Z-module and ha HA (E ha a H A (p-algebra rucure wih repec o he cup produc, he H A [ ] (Z-operaion i H A (p-linear. Remark. Le q i : E i p, i I, here i an iomorphim of algebra End H A (p(h A(E = H A (E E = Ex D A (p ( i I (q i C[e i ], i I (q i C[e i ], he r equaliy follow from [CG97], Ex , p.123, for he econd: Ue he Thom iomorphim o replace E E by a union of ag varieie, hen ue heorem for he Springer map given by he projecion o a poin. Furhermore, under he idenicaion, he following hree graded H A (p-algebra homomorphim are equal. (1 The map H A (Z End H A (p(ha (E, c (e c e. (2 i : H A (Z H A (E E where i: Z E E i he naural embedding. (3 Se A π := i I (π i C[e i ]. Ex D A (V (A π, A π Ex D A (p (a (A π, a (A π, f a (f where a: V p. We do no prove hi here. Lemma 2. ([VV11], remark afer Prop.3.1, p.12 Aume ha T i P i i a maximal oru and Z T = E T E T, E T = i I (G/P i T. Le A {T, G}. The map from (1 H A (Z End H A (p(h A(E, c (e c e i an injecive homomorphim of HA (p-algebra. Le be he Lie algebra of T, hen i hold HG (E = C[] I, where C[] i he ring of regular funcion on he ane pace. 15

16 Proof: For G-equivarian Borel-Moore homology we claim ha he following diagram i commuaive H T (Z T C K H T (E T E T C K H T (Z H T (E E H G (Z H G (E E where K = Quo(HT (p The commuaiviy of he lowe quare ue funcorialiy of he forgeful map. By aumpion Z T = (E E T, he highe horizonal map i an iomorphim. Now, ince H T (Z, H T (E E are free HT (p-module, we ge ha he map H T (Z H T (Z K, H T (E E H T (E E K are injecive. By he localizaion heorem ee Appendix, heorem or [Bri00], lemma 1, we ge he iomorphim H T (Z K = H T (Z T K, H T (E E K = H T (E T E T K. Tha implie ha he middle horizonal map ha o be injecive, ogeher wih (2 from he previou remark i implie he claim for T -equivarian Borel-Moore homology. Bu by he pliing principle, i.e. he idenicaion of he G-equivarian Borel-Moore homology wih he W -invarian ubpace in he T -equivarian Borel-Moore homology, he forgeful map become he incluion of he W -invarian ubpace. Thi mean he wo verical map in he lower quare are injecive. Thi implie ha he lowe horizonal map i injecive. Togeher, wih (2 of he previou remark he claim follow for A = G. The main ingredien o he previou lemma i a weak verion of Gorezky', Kowiz' and MacPheron' localizaion heorem (ee [GKM98]. developed by Gonzale for K-heory in [Gon]. Similar mehod are currenly The previou lemma i fale for no equivarian Borel-Moore homology a he following example how. Example. Le G be a reducive group wih a Borel ubgroup B and u be he Lie algebra of i unipoen radical. Z := (G B u g (G B u, hen i hold ha he algebra H (Z can under he iomorphim in Kwon (ee [Kwo09] be idenied wih C[]/I W #C[W ] where I W C[] i he ideal generaed by he kernel of he map C[] W C, f f(0. The kew ring C[]/I W #C[W ] i dened a he C-vecor pace C[]/I W C C[W ] wih he muliplicaion (f w (g v := fw(g wv. Furhermore, we can idenify End C (H (E via he Thom-iomorphim and he Borel map wih End C lin (C[]/I W. The canonical map idenie wih C[]/I W #C[W ] End C lin (C[]/I W f w (p fw(p Thi map i neiher injecive nor urjecive. For example w W 1 w 0 in C[]/I W #C[W ] bu i image (p w W w(p i zero becaue w W w(p I W. Becaue boh pace have he ame C-vecor pace dimenion, i i clear ha i i alo no urjecive. 16

17 Furhermore, H A (Z i naurally a HA (E-module. In fac le e i,j := e W i,j be he double coe of he neural elemen, hen HA (E = H A ( i,j I Ze i,j i even a ubalgebra of H A (Z. Corollary In he iuaion of he previou lemma, i.e. T i P i i a maximal oru and Z T = E T E T, E T = i I (G/P i T and le A {T, G}. There are injecive homomorphim of HG (p-algebra H A (p H A (E H A (Z End H A (p(h A(E, where he r incluion i given by he pullback along he map E p. In paricular, HA (p i conained in he cenre of HA (Z (we only know example where i i equal o he cenre. Le w W. Oberve, ha HA (E already operae on HA (Z w and he compoiion H A (Z = w HA (Z w i a direc um compoiion of HA (E-module. Uing he Thom-iomorphim (ee Appendix, ubecion??, (5, up o a degree hif we can alo udy H A (C w a module over HA ( i I G/P i. Now, le e i be he idempoen in HA ( i I G/P i = i I H A (G/P i which correpond o he projecion on he i-h direc ummand. Since for w W i,j i hold H A (C w = HA (G/P i C H (P i wp j /P j alo a HA (G/P i-module, we conclude ha H A (C w i alway a projecive module over HA ( i I G/P i. In oher word hi dicuion yield. Lemma 3. form (1 Le w W i,j. Each H A (Z w i a projecive graded HA (E-module of he v W i (H A(Ee i [2d w + deg b i,j (v], where [d] denoe he degree hif by d. In paricular, H A (Z i a projecive graded H A (E-module. (2 If all P i = B i are Borel ubgroup of G, hen H A (Z = w,j W I( i I (H A(Ee i [d w,i,j ] a graded H A (E-module for cerain d w,i,j Z. In paricular, if we forge he grading H A (Z i a free HA (E-module of rank #W #I. 1.4 Indecompoable projecive graded module over H A [ ] (Z and heir op for a dieren grading Le X be an irreducible algebraic variey, we call a decompoiion X = a A S a ino niely many irreducible mooh locally cloed ube a weak raicaion. Since π : E = i I E i V i a G-equivarian projecive map, here exi (and we x i a weak rai- 17

18 caion ino G-invarian ube V = a A S a uch ha π 1 (S a π S a i a locally rivial 4 braion wih conan bre F a := π 1 ( a where a S a one xed poin, for every a A. (For projecive map of complex algebraic varieie one can alway nd uch a weak raicaion, ee [Ara01], Recall ha for any G-equivarian projecive map of complex varieie, he decompoiion heorem hold (compare [BBD82] for he no equivarian verion and [BL94] for he equivarian verion. Le run over all imple 6 G-equivarian local yem L on ome raum S = S a, a A, we wrie IC A := (i S (IC A (S, L [d S ] wih d S = dim C S for he imple pervere heaf in he caegory of A-equivarian pervere heave P erv A (V DA b (V, ee again [BL94], p. 41. Le e i = dim C E i, i I, hen C Ei [e i ] i a imple pervere heaf in D b A (E. For a graded vecor pace L = d Z L d we dene L(n o be he graded vecor pace wih L(n d := L n+d, n Z. We ee C a he graded vecor pace concenraed in degree zero. For an elemen F D b A (X for an A-variey X we wrie F [n] for he (cla of he complex (F [n] d := F d+n, n Z. Now given F DA b (X and a nie dimenional graded vecor pace L := r i=1 C(d i we dene L gr F := r F [d i ] DA(X b i=1 The A-equivarian decompoiion heorem applied o π give (π i C Ei [e i ] = i I L gr IC A D b A(V where he L := d Z L,d are complex nie dimenional graded vecor pace. Le D be he Verdier-dualiy on V, i hold D(π (C[d] = π (C[d], D(IC A = IC A where we dene for = (S, L he aociaed dual local yem a = (S, L, L := Hom(L, C. Thi implie L = L for all Indecompoable projecive in he caegory of graded lef H A [ ] (Z- module We e P A := Ex D b A (V (ICA, i I (π i C[e i ]. 4 wih repec o he analyic opology 5 If he image of π i irreducible, by [Ara01], heorem we can rene hi raicaion o a (nie Whiney raicaion, bu i i no clear if we can nd a Whiney raicaion ino G-invarian ube. 6 a local yem i imple if he by monodromy aociaed repreenaion of he fundamenal group ha no nonrivial ubrepreenaion. Uually hi i called irreducible. 18

19 I i a graded (lef H A [ ] (Z-module. I i indecompoable becaue ICA i hold a lef graded H A [ ] (Z-module i imple. Clearly H A [ ] (Z = d Z, L,d [ n Z = d Z, L,d C P A [d] Ex n+d DA b (V (ICA, (π i C[e i ]] i I = L gr P A ha implie ha P A i a projecive module and ha (P A i a complee e of iomorphim clae up o hif of indecompoable projecive graded H[ ] A (Z-module. Lemma 4. Aume ha H A (p i a graded ubalgebra of he cenre of HA [ ] (Z. elemen H A >0(p operae on any graded imple HA [ ](Z-module S by zero. In paricular, by lemma we ee ha S i a graded imple module over H [ ] (Z. Any graded imple module i nie-dimenional and here exi up o iomorphim and hif only niely many graded imple module. For any graded imple module S here i no nonzero degree zero homomorphim S S(a, a 0. Proof: By aumpion ha HA (p i cenral, we obain ha H>0 A (p S i a graded lef (Z-module, clearly i i a ubmodule of S. Since S i imple i hold H>0(p S i H A [ ] zero or S. Aume i i S, hen here exi x HA d (p for a d > 0 uch ha x S 0. Since x i cenral, hi i a ubmodule of S and we have x S = S. Le y S, y 0, homogeneou. Then, i hold S = H[ ] A (Z y = HA [ ](Z xy conradicing he fac ha here i a uniquely deermined minimal nonzero degree for S. Therefore H A >0 (p S = 0. By [NO82], II.6, p.106, we know ha he graded imple module conidered a module over he ungraded ring H A (Z, H (Z are ill imple module. Since he nie-dimenional algebra H (Z ha up o iomorphim only niely many imple, he claim follow. Any nonzero degree 0 homomorphim φ: S S(a ha o be an iomorphim. Le S = H[ ] A (Z y a before, e deg y = m. Then S(a = HA [ ](Z φ(y, deg φ(y = m which give a conradicion when conidering he minimal nonzero degree of S and S(a. Corollary There i a bijecion beween iomorphim clae up o hif of (1 indecompoable projecive graded H A [ ] (Z-module (2 indecompoable projecive graded H [ ] (Z-module (3 imple graded H [ ] (Z-module. The bijecion beween (1 and (2 i clear from he decompoiion heorem, i map P P/H A >0 (pp. We pa from (3 o (2 by aking he projecive cover and we pa from (2 o (3 by aking he op (which i graded becaue for a nie dimenional graded algebra he radical i given by a graded ideal. A The 19

20 Example. (due o Khovanov and Lauda, [KL09] Le G B T be a reducive group conaining a Borel ubgroup conaining a maximal oru, Z = G/B G/B. Then, i i known ha H G (Z = End C[] W (C[] =: NH where W i he Weyl group aociaed o (G, T and = Lie(T. The G-equivarian puhforward (o he poin of he hif of he conan heaf i a direc um of hif of copie of he conan heave on he poin, herefore here exi preciely one indecompoable projecive graded H[ ] G (Z-module up o iomorphim and hif. I i eay o ee ha P := C[] i an indecompoable projecive module and P/H G >0(pP = C[]/I W i he only graded imple NH-module which i he op of P. Alo, one check ha H [ ] (Z = End C (C[]/I W i a emi-imple algebra which ha up o iomorphim and hif only he one graded imple module C[]/I W. In he following ubecionwe equip he Seinberg algebra wih a grading by poiive ineger which lead o a decripion of graded imple module in erm of he mulipliciy vecor pace L in he BBD-decompoiion heorem Simple in he caegory of graded niely generaed lef H A < >(Z- module Given a graded vecor pace L, we wrie L := d Z L d for he underlying (ungraded vecor pace. If we regrade H A (Z a follow H A <n>(z :=, Hom C ( L, L C Ex n D b A (V (ICA, IC A ], in oher word H A < >(Z = Ex ( L C IC A, L C IC A a graded algebra. Thi i a an ungraded algebra iomorphic o H A (Z. Wih he ame argumen a in he previou ecion one ee ha P A := Ex DA b (V (ICA, π C are a complee repreenaive yem for he iomorphim clae of he indecompoable projecive graded H< >(Z-module. A We claim ha here i a graded H< >(Z-module A rucure on he (mulipliciy-vecor pace L uch ha he family { L } i a complee e of he iomorphim clae up o hif of graded imple module. Uing Hom(IC A, IC A = Cδ,, Ex n (IC A, IC A = 0 for n < 0 we ge H< >(Z A = End( L, }{{} deg=0 Hom( L, L C Ex >0 (IC A, IC A. Now, he econd ummand i he graded radical, i.e. he elemen of degree > 0 (wih 20

21 repec o he new grading. I follow H A < >(Z H A < >(Z/(H A < >(Z >0 = End C ( L. Thi give L a naural graded H< >(Z-module A rucure concenraed in degree zero (he poiive degree elemen in H< >(Z A operae by zero. Oberve, ha L doe no depend on A, i.e. in fac hey are module over H< >(Z A via he forgeful morphim H< >(Z A H < > (Z. Tha mean we can inead look for he imple graded module of H < > (Z. Remark. Le H be a nie dimenional poiively graded algebra uch ha H 0 = H /H >0 = End(L i a emi-imple algebra. Then H >0 i he e of nilpoen elemen, i.e. Jacobon radical of H. Furhermore all imple and projecive H -module are graded module. * (L i he uple of (pairwie diinc iomorphim clae of all imple module. * For each pick an e End(L H 0 which correpond o projecion and hen incluion of a one dimenional ubpace of L. (P := H e i he uple of (pairwie diinc iomorphim clae of all indecompoable projecive module. We can apply hi remark o H = H < > (Z. A a conequence we ee ha up o hif ( L i he uple of (pairwie diinc iomorphim clae of all imple graded H A (Z-module. From now on, he cae where he wo grading coincide will play a pecial role. Remark. The following condiion are equivalen (1 H A [ ] (Z = HA < >(Z a graded algebra for every A {p, T, G}. (1 H A [ ] (Z = HA < >(Z a graded algebra for a lea one A {p, T, G}. (2 (π i C[e i ] i A-equivarian pervere for every i I for every A {p, T, G}. (2 (π i C[e i ] i A-equivarian pervere for every i I for a lea one A {p, T, G}. (3 π i : E i V i emi-mall for every i I, hi mean by deniion dim Z i,i = e i for every i I. In hi cae, we ay he Springer map i emi-mall. Alo, π emi-mall i equivalen o H op (Z i,i = H [0] (Z i,i, i I. Oberve, ha H [0] (Z i alway a ubalgebra of H [ ] (Z and in he emi-mall cae iomorphic o he quoien algebra H [ ] (Z/(H [ ] (Z >0. Aume π emi-mall, hen i hold 2 dim πi 1 ( e i d S, i I where x S belong o he raicaion and H op (π 1 ( := i: 2 dim π 1 i (=e i d S H 2 dim π 1 i ( (π 1 i ( i a lef H [0] (Z-module via he rericion of he convoluion. If I coni of a ingle elemen, H op (Z = H [0] (Z and H 2 dim π 1 ((π 1 ( i a H [0] (Z-module. 21

22 Remark. If one applie he decompoiion heorem o π i, i I one ge ha L = i I L(i (a graded vecor pace where L (i i he mulipliciy vecor pace for IC in (π i C[e i ]. I hold {L (i H (Z i,i -module. L (i 0} i he complee e of iomorphim clae of imple Remark. In fac, Syu Kao poined ou ha he caegorie of niely generaed graded module over H A [ ] (Z and HA < >(Z are equivalen. Thi ha been ued in [Ka13]. Remark. Now, we know ha he forgeful (=forgeing he grading funcor from nie dimenional graded H [ ] (Z-module o nie dimenional H (Z-module map graded imple module o imple module. We can ue he fac ha we know ha imple and graded imple are paramerized by he ame e, o ee: Every imple H (Z-module L ha a grading uch ha i become a graded imple H [ ] (Z-module and every graded imple i of hi form. For he decompoiion marix for he nie dimenional algebra H (Z, here i he following reul of Chri and Ginzburg. Theorem ([CG97], hm Aume H odd (π 1 ( = 0 for all x V. Then, he following marix muliplicaion hold [P : L] = IC D IC where all are marice indexed by = (S, L, = (S, L uch ha L 0, L 0 and ( denoe he ranpoed marix. [P : L], := [P : L ] = k dim Ex k (IC, IC IC, := [H k (i S(IC : L] k D, := δ S,S ( 1 k dim H k (S, (L L k According o Kao in [Ka13], he whole heory of hee algebra i reminicen of quaiherediary algebra (bu we have innie dimenional algebra. He inroduce andard and coandard module for H< >(Z G in [Ka13], hm 1.3, under ome aumpion 7.He how ha under hee aumpion, H< >(Z G ha nie global dimenion (ee [Ka13], hm Springer bre module in he caegory of graded H A? (Z-module Recall, ha Springer bre module H [ ] (π 1 (, H [ ] (π 1 (, V are naurally graded module over H [ ] (Z, bu if we forge abou he grading and we can how ha hey are acually emi-imple in H (Z-mod, hen, we can ee hem a emi-imple graded H< >(Z-module A for A {G, T, p} by he previou ecion. 7 = niely many orbi wih conneced abilizer group in he image of he Springer map, H< >(Z G and he in he decompoiion heorem occurring IC are pure of weigh zero 22

23 Le A = p. Since he map π i locally rivial over S := S a we nd ha i S( i I R k (π i C[e i ], i! S( i I R k (π i C[e i ] are local yem on S, via monodromy hey correpond o he π 1 (S, -repreenaion H [k] (π 1 ( = i I H ei+k (πi 1 (, i I H ei k(πi 1 ( = H [k] (π 1 ( wih e i := dim C E i repecively (for a xed poin = a S, cp. [CG97], Lemma Now, le u make he exra aumpion ha he image of he Springer map i irreducible and he raicaion {S a } a A i a Whiney raicaion (every algebraic raicaion of an irreducible variey can be rened o a Whiney raicaion ee [Ara01], hm , p.30, which i oally ordered by incluion ino he cloure. Le S S be an incuion for wo raa S, S, we wrie Ind S S (L := i S H (IC (S,L, i.e. we conider he funcor for k [ d S, d S ] Ind S S ( k : LocSy(S LocSy(S L Ind S S (L k := i S H k (IC (S,L where LocSy(S i he caegory of local yem on S, i.e. locally conan heave on S of nie dimenional vecor pace (for oher k Z hi i he zero funcor. If we apply he funcor i S Hk on he righ hand ide of he decompoiion heorem we noice he following (for he cohomology group of IC-heave, ee [Ara01], ecion 4.1, p.41, le = (S, L. L, if d S = d S, k = d S i SH k (IC = Ind S S (L k if d S < d S, k [ d S, d S 1] 0 ele. and i! SH k (IC [d] = H k+d (D S i SIC = i SH k d 2d S (IC implie L, if d S = d S, k + d = d S i! SH k (IC [d] = Ind S S (L k d 2dS if d S < d S, k d 2d S [ d S, d S 1] 0 ele. 23

24 where d S = dim C S. Thi implie and H [k] (π 1 ( = H [k] (π 1 ( = = = L,d C i SH k+d (IC d Z =(S,L L, ds k C L L,d C i! SH i+d (IC d Z =(S,L a π 1 (S, -repreenaion. L, ds k C L =(S,L,d S <d S d S 1 r= d S L,r k C Ind S S (L r } {{ } =:H [k] (π 1 ( >S =(S,L,d S <d S d S 1 r= d S L, r 2dS k C Ind S S (L r } {{ } =:H [k] (π 1 ( >S We call he direc ummand iomorphic o Ind S S (L r, r [ d S, d S 1] he unwaned ummand. Now we can explain how you can recover from he π 1 (S, -repreenaion H [k] (π 1 (, k Z he daa for he decompoiion heorem (i.e. he local yem and he graded mulipliciy pace. If d S i he maximal one, i hold H [ ] (π 1 ( = k Z =(S,L L, ds k C L and we can recover he graded mulipliciy pace L wih = (S,? for he dene raum ocurring in he decompoiion heorem. For arbirary S we conider H [ ] (π 1 (/H [ ] (π 1 ( >S = k Z =(S,L L, ds k C L and by inducion hypohei we know he π 1 (S, -repreenaion H [ ] (π 1 ( >S, herefore we can recover he L wih = (S,? from he above repreenaion. Now aume ha π i emi-mall. Then, we know ha L,d = 0 for all = (S, L whenever d 0. We can alo reric our aenion on a direc ummand (π i C[e i ] for one i I and nd he decompoiion ino imple pervere heave. Tha mean we only need H e i d S (πi 1 ( o recover he daa for he decompoiion heorem. I alo hold 2 dim πi 1 ( e i d S, i I and ince H e i d S (πi 1 ( = 0 whenever 2 dim πi 1 ( < e i d S, we only need o conider he raa S wih 2 dim πi 1 ( = e i d S, hen H e i d S (πi 1 ( = H op (πi 1 ( 0 and we call S a relevan raum for i ( I. We call a raum relevan if i i relevan for a mo one i I. Analogouly, one can replace H [k] (π 1 ( by H [ k] (π 1 ( and alk by coalk. 24

25 Le V be arbirary. By a previou ecion we know ha H [ ] (π 1 ( and H [ ] (π 1 ( are lef (and righ graded H [ ] (Z-module. The following lemma explain heir pecial role. Unforunaely, he following aemen i only known if all raa S conain a G-orbi G := O S uch ha π 1 (O, = π 1 (S,. For local yem on he raa hi i by monodromy he ame a he aumpion ha all raa are G-orbi. Le C be a nie group, we wrie Simp(C for he e of iomorphim clae of imple CC-module and denoe by 1 Simp(C he rivial repreenaion 8. Lemma 5. ([CG97], Lemma , p.436, Lemma 3.5.3, p.170 Aume ha he image of he Springer map conain only niely many G-orbi. (a Le O = G V be a G-orbi. There i an equivalence of caegorie beween {G-equivarian local yem on O} C( mod where C( = Sab G (/(Sab G ( o i he componen group of he abilizer of. In paricular, via monodromy alo he π 1 (O, -repreenaion which correpond o G-equivarian local yem on O are equivalen o C( mod. (b The C(-operaion and he H [ ] (Z-operaion on H [ ] (π 1 ( (and on H [ ] (π 1 ( commue. The emi-impliciy of C( mod implie ha H [ ] (π 1 ( = (H [k] (π 1 ( χ C χ k Z χ Simp(C( where Simp(C( i he e of iomorphim clae of imple C(-module and for any C(-module M we call M χ := Hom C( mod (χ, M an ioypic componen. Since he wo operaion commue i hold (H [ ] (π 1 ( χ naurally ha he rucure of a graded H [ ] (Z-module. Bu we will from now ju ee i a a module over H (Z. A H (Z C(- bimodule decompoiion we can wrie he previou decompoiion a H [ ] (π 1 ( = H (π 1 ( χ χ χ Simp(C( where H (π 1 ( χ χ i he obviou bimodule H (π 1 ( χ χ. A an immediae conequence of hi we ge, if G i a dene orbi in he image of he Springer map, hen L, ( d G = H [ ] (π 1 ( χ, for = (, χ, χ Simp(C(, in paricular, H [ ] (π 1 ( i a emiimple H (Z-module (graded and no graded, even a emiimple H (Z C(-bimodule. For more general orbi, we do no know if i i emiimple. In he cae of a emi-mall Springer map we have he following reul. 8 In he lieraure hi i called Irr(C, we ue he word irreducible only for a propery of opological pace 25

26 Theorem Aume he image of he Springer map π ha only niely many orbi and π i emi-mall. There i a bijecion beween he following e (1 {(, χ O = G, χ Simp(C(, H [do ](π 1 ( χ 0} where he in V are in a nie e of poin repreening he G-orbi in he image of he Springer map. (2 Simp(H <0> (Z mod := imple H <0> (Z-module up o iomorphim (3 Simp(H A < >(Z mod Z := imple graded H A < >(Z-module up o iomorphim and hif for any A {p, T, G}. Beween (1 and (2, i i given by (, χ H [do ](π 1 ( χ. We call hi bijecion he Springer correpondence. For a relevan orbi O (for a lea one i I i hold H [do ](π 1 ( 1 = i: 2 dim π 1 i 1 H op(π (=e i d O i ( C( 0 and C( operae on he op-dimenional irreducible componen of πi 1 ( by permuaion. Thi implie we ge an injecion kech of proof: {relevan G-orbi in Im(π} Simp(H <0> (Z mod O = G H [do ](π 1 ( C( For k = d O look a he decompoiion for H [k] (π 1 ( and ue ha L,d = 0 whenever d 0 o ee ha he unwaned ummand vanih. Then how ha he decompoiion coincide wih he econd decompoiion (wih repec o he irreducible characer of C( of H [k] (π 1 ( which give he idenicaion of he L wih he H [do ](π 1 ( χ. I i an open queion o underand Springer bre module more generally. Alo, Springer correpondence hin a a hidden equivalence of caegorie. Thi funcorial poin of view we inveigae in he nex ubecion. 1.5 The Springer funcor We conider H A [ ] (Z again wih he grading from he heorem Le projz H A (Z be he caegory of niely generaed projecive Z-graded lef H[ ] A (Z-module, morphim are he module homomorphim which are homogeneou of degree 0. Le P A DA b (X be he full ubcaegory cloed under direc um and hif generaed by IC A, = (S, L be he uple of raa wih imple local yem on i which occur in he decompoiion heorem (wih nonzero mulipliciy pace L. The following lemma i in a pecial cae due o Sroppel and Weber, ee [SW11]. 26

27 Lemma 6. The funcor proj Z H A [ ] (Z P A M i I (π i C Ei [e i ] H A [ ] (Z M i an equivalence of emiimple caegorie mapping P A funcor 9. IC A. We call hi he Springer Proof: By heorem we know H A [ ] (Z = Ex D b A (V ( i I (π i C Ei [e i ], i I (π i C Ei [e i ] i an iomorphim of graded algebra. Thi make he funcor well-dened. The direc um decompoiion of i I (π i C Ei [e i ] by he decompoiion heorem in P A correpond o idempoen elemen in H[0] A (Z, which correpond (up o iomorphim and hif o he indecompoable projecive graded module, le for example P = H[ ] A (Ze. Shif of graded module are mapped o hif in P A, herefore he funcor i eenially urjecive. I i fully faihful becaue of he menioned equaliy Hom proj Z H A [ ] (Z(P, P (n = e H A [n] (Ze = Hom D b A (V (IC, IC [n] Le P A (V DA b (V be he caegory of A-equivarian pervere heave on V. Aume for a momen ha he map π i emi-mall. Then, we know ha i I (π i C Ei [e i ] i an objec of P A (V. In hi iuaion he wo grading of he Seinberg algebra coincide. The op-dimenional Borel-Moore homology H op (Z i,i coincide wih he degree zero ubalgebra H [0] (Z i,i. We wan he Springer funcor o go o a caegory of pervere heave, i.e. we do no wan o allow hif of he grading for module. Therefore, we pa o H [0] (Z = H < > (Z/(H < > (Z >0 = H A < >(Z/(H A < >(Z >0, A {p, T, G} and replace projecive graded module over H[ ] A (Z by he addiive caegory of imple module over H [0] (Z, hi equal he caegory H [0] (Z mod of nie dimenional (ungraded module over H [0] (Z becaue he algebra i emi-imple. In paricular, i hold H [0] (Z = Ex 0 D b A (V ( i I (π i C Ei [d i ], i I (π i C Ei [d i ] = End P A (V ( i I (π i C Ei [d i ], A {p, T, G}. The following lemma i for claical Springer heory due o Duin Clauen, cp. Thm 1.2 in [Cla08]. Lemma 7. If he Springer map π i emi-mall, we have he following verion of he 9 Thi name i due o Duin Clauen in hi hei. 27

28 Springer funcor S : H [0] (Z mod P G (V M i I (π i C Ei [e i ] H[0] (Z M. I hold ha S i an exac funcor (beween abelian caegorie and i i fully faihful. If e i + e j i even for all i, j I hen S idenie H [0] (Z mod wih a emi-imple Serre ubcaegory of P G (V (i.e. i i an exac ubcaegory which i alo exenion cloed and cloed under direc ummand. Furhermore i i invarian under Verdier dualiy on P G (V. Remark. Aume ha he Springer map i emi-mall, he image of he Springer map conain only niely many G-orbi and each G-orbi i relavan and imply conneced, hen he Springer funcor from above induce an equivalence of caegorie S : H [0] (Z mod P G (Im(π. (The only known example for hi i he claical Springer map for G = Gl n, ee laer. Proof: A imilar proof a in he lemma above how ha he Springer funcor induce an equivalence on he full ubcaegory of P G (V generaed by nie direc um of direc ummand of i I (π i C Ei [e i ]. Thi i a emi-imple caegory. Aume ha e i + e j i even for all i, j I, we have o ee ha i i exenion cloed. By compoiion wih he forgeful funcor we ge a funcor H [0] (Z mod S P G (V F P p (V =: P(V, by [Cla08] he forgeful funcor F i fully faihful. Now, by [Ara01], he caegory P(V of D b (V i cloed under exenion and admiible becaue i i he hear of a - rucure. By he Riemann Hilber correpondence here exi an abelian caegory A (= regular holonomic D-module on V and an equivalence of riangulaed caegorie (= he de Rham funcor DR V : D b (A D b (V uch ha he andard -rucure on D b (A i mapped o he pervere -rucure and i reric o an equivalence of caegorie A P(V. Thi implie ha for X = DR V (X, Y = DR V (Y in P(V and n N 0 Ex n P(V (X, Y = Exn A(X, Y = Hom D b (A(X, Y [n] = Hom D b (V (X, Y [n] where he r and he hird equaliy follow from he de Rham funcor and he econd equaliy hold becaue i i he andard -rucure, cp. for example [GM03], p

29 Now, ince we know Hom D b (V ( i I (π i C Ei [e i ], ( i I (π i C Ei [e i ][1] = H <1> (Z = i,j I H ei +e j 1(Z = 0 becaue H odd (Z = 0 by lemma and he aumpion ha e i + e j i even for every i, j I. We obain ha Ex 1 P(V ( i I (π i C Ei [e i ], i I (π i C Ei [e i ] = 0, i.e. he emi-imple caegory generaed by he direc image of he Springer map i exenion cloed. The nex ecion coni of concep of claical Springer heory in he conex of a more general collaping of a homogeneou bundle. 1.6 Orbial varieie, Springer bre and raa in he Seinberg variey In hi ecion we work over an arbirary algebraically cloed eld K. In he example of claical Springer heory (ee laer orbial varieie have been inroduced by Spalenein in [Spa77]. He proved hem o be in bijecion o irreducible componen of Springer bre. Thi idea ha been furher applied by everal auhor (for example Reineke [Rei03], Varga [Var79], Melnikov and Pagnon [MP06]. We give he analogue here o an arbirary collaping of a homogeneou bundle under he (reaonable aumpion on he orbi O v V o be iomorphic o he quoien G/Sab(v, where Sab(v = {g G gv = v} i he abilizer of v. Thi propery can be characerized a follow. We hink hi lemma i well-known (bu we do no have a ource for i. Lemma 8. Le G be an algebraic group over an algebraically cloed eld K. Le V be a G-cheme of nie ype over K. Le v V (K and denoe by O v V he orbi endowed wih he reduced ubcheme rucure. Le m: G O v, g gv be he muliplicaion map. Then, he following are equivalen. 1 m induce an iomorphim O v = G/Sab(v. 2 m i eparable. 3 T e m: T e G T v O v i urjecive where e G(K i he neural elemen. Furhermore, if one of he condiion i fullled he map m i open and cloed. Example. In characeriic zero, he map m i alway eparaed. Alo in he example of quiver-graded Springer heory (ee a laer ecion, for Gl d -orbi in R Q (d he propery 3 i rue over any algebraically cloed eld becaue of Voig' lemma [Gab75], Prop

30 Deniion 2. Le (G, P, V, F be he conrucion daa for a Springer heory (i.e. we aume he nie e I coni of a ingle elemen. Then he irreducible componen of O v F are called orbial varieie (for v. Lemma 9 (Reineke, [Rei03], Lemma 3.1. There i an iomorphim G Sab(v π 1 (v = π 1 (O v = G P (O v F. Proof: The r iomophim follow from lemma 77. Looking a π π : G P F V, (g, f gf give π 1 (O v = {(g, f G P F gf O v } = {(g, f G P F f O v } = G P (O v F We ge he immediae corollary. Corollary There i an iomorphim of equivarian Chow group enored wih Q A Sab(v j+dim Sab(v (G π 1 (v Z Q = A P j+dim P (G (O v F Z Q, where equivarian Chow group are mean in he ene of Edidin and Graham (ee [EG98a]. Secondly, here i a more inimae relaion beween he opology of aociaed bre bundle and heir bre, we cie from Bongarz he following Lemma 10. ([Bon98], Lemma 5.16 Le G be a conneced algebraic group wih a cloed ubgroup P. Le F be a quai-projecive P -variey. Then, he map U G P U induce a bijecion beween P -invarian ubvarieie of F and G-invarian ubvarieie of G P F. The bijecion repec incluion, cloure and geomeric properie like irreducibiliy, moohne and normaliy. Thi induce he bijecion beween he Sab(v-invarian ubvarieie of π 1 (v and P -invarian ubvarieie of O v F. Deniion 3. Le (G, P, F, V be he conrucion daa for a Springer heory. A dene P -orbi in F will be called a Richardon orbi. Corollary Le (G, P, F be a in he previou lemma. Then he following are equivalen. (1 G P F ha a dene G-orbi. 30