Representation theory through the lens of categorical actions: part II

Transcription

1 Representaton theory through the lens of categorcal actons: part II Unversty of Vrgna June 10, 2015

2 Remnder from last tme Last tme, we dscussed what t meant to have sl 2 act on a category. Defnton An sl 2 -acton on an addtve category C conssts of: A drect sum decomposton C = n Z C n. Adjont functors E n : C n C n+2 and F n : C n C n 2 such that satsfy the relatons E n 2 F n = F n+2 E n d n (n 0) E n 2 F n d n = F n+1 E n (n 0) natural transformatons y: E n E n and ψ : E n+2 E n E n+2 E n that satsfy the relatons ( ) (and thus defne an acton of NH k on E k ).

3 Hgher rank The frst thng I want to dscuss s how to generalze ths: how a fnte dmensonal sem-smple (or more generally, Kac-Moody) Le algebra g wth smple roots I acts on a category. Defnton An g-acton on an addtve category C conssts of: A drect sum decomposton C = λ C λ over the ntegral weghts λ of g. Adjont functors E λ : C λ C λ+α and F λ : C λ C λ α for each I that satsfy the relatons E λ α F λ = F λ+α E λ d α (λ) (α (λ) 0)? E λ α F λ d α (λ) = F λ+α E λ (α (λ) 0)

4 Hgher rank We need some replacement of the nlhecke algebra for hgher rank. We consder an algebra R n generated by dempotents e for = ( 1...., n ) I n, polynomal generators y 1,..., y n, crossng generators ψ 1,..., ψ n 1, The algebra s graded wth deg(e y j ) = α j, α j, deg(e ψ j ) = α j, α j+1. We graphcally repesent these as 1 2 n 1 j n 1 j j+1 n e y j ψ j

5 Hgher rank KLR algebras Choose polynomals Q j (u, v) = t j u a j + t j u a j + l.o.t. Impose relatons: = unless = j = + j j = Q j (y 1, y 2 ) = unless = k j j j j k j k = 0 = + Q j (y 3, y 2 ) Q j (y 1, y 2 ) y 3 y 1 j j j

6 Hgher rank Defnton An g-acton on an addtve category C conssts of: A drect sum decomposton C = λ C λ over the ntegral weghts λ of g. Adjont functors E λ : C λ C λ+α and F λ : C λ C λ α for I that satsfy the relatons E λ α F λ = F λ+α E λ d α (λ) (α (λ) 0) E λ α F λ d α (λ) = F λ+α E λ (α (λ) 0) The functor (,λ E λ )n carres an acton of R n compatble wth horzontal composton.

7 Hgher rank Why s the KLR algebra the rght thng? Well, for one thng, t forces the Serre relatons to hold. If j by a sngle edge, then we can take Q j (u, v) = v u. The Serre relaton F F j F = F j F (2) + F (2) F j s categorfed by the decomposton of the dentty on F F j F nto dempotents: = j j j Ths s a consstent theme: relatons on the decategorfed level are encoded, sometmes n a subtle way, n the relatons between 2-morphsms whch force relatons.

8 Strctfcaton Ths defnton mght seem a lttle strange: shouldn t we have an actual object replacng the Le algebra (or more lkely ts unversal envelopng algebra)? Proposton ( Khovanov-Lauda, W.) There s a graded 2-category U whose graded Grothendeck group s the dempotented verson U q of the quantzed unversal enveloped algebra. Ths category s generated by 1-morphsms E and F, whch are badjont and carry the correct acton of the KLR algebra (by defnton). Any U-module has a g-module structure va the acton of these 1-morphsms. Proposton (Chuang-Rouquer, Brundan) Any g-module has a canoncal acton of U, nvertng ths operaton.

9 Categorfed smples The noton of the acton of a Le algebra s a powerful one. One can often understand a vector space much better once one has a Le algebra actng on t. What about a category? Theorem (Rouquer) For each smple hghest weght representaton V λ of g, there s a unversal categorfcaton ˇV λ of V λ ; any other abelan and Artnan categorfcaton V s obtaned from ths by base change along the functor ˇV λ λ V λ In any categorfcaton V of V λ wth each V µ abelan and Artnan, the hghest weght category V λ = S -mod for some local rng S, and every other weght space s naturally an S-lnear category. Thus, the relevant base change s va the rng map Š λ S, wth ˇV λ λ = Šλ -mod.

10 Categorfed smples Let s be a lttle more explct. What does ths categorfed smple look lke? Frst of all, there s a hghest weght object V wth End(V) = Šλ = Z[{γ,1,..., γ,α (λ)} I ]. The weght space V λ λ α s also 1-dmensonal, so the object F V should generate ˇV λ λ α. As a Š λ -module: End(F V) = Hom(V, E F V = V α (λ) ) = (Š λ ) α (λ). Key trck (Chuang-Rouquer) The endomorphsm y has to generate End(F V). Thus, we must have an somorphsm End(F V) = Z[y]/m(y) where m s the mnmal polynomal of y on F V).

11 Categorfed smples The pont of unversalty s that we take the unversal polynomal to be the mnmal one. On F V, we have the mnmal polynomal m (y) = y α (λ) + γ,1 y α (λ) γ,γ,α (λ) = 0. Note that ths s homogeneous f we let deg(γ,j ) = 2j. Theorem (Rouquer) For an arbtrary weght space ˇV λ µ, the generatng objects are F V for sequences wth = λ µ. Thus, ˇV λ µ s equvalent to representatons of the rng Ř λ µ = End ( F V ) = Rλ µ /m 1 (y 1 e ). =λ µ That s, after we mpose the relatons on F V, we need no more. The algebra Ř λ µ s called a deformed cyclotomc quotent, and m 1 (y 1 e ) = 0 the deformed cyclotomc relaton.

12 Categorfed smples In order to thnk about the Grothendeck group, t s better to base change wth a local rng (S, m) va a map Š λ S sendng γ,j to m. Theorem In ths case, K(R λ Šλ S -pmod) = V λ Z, the Stenberg ntegral form of Vλ. If S s graded local and the map s homogeneous, then we can mprove ths result. We can consder the graded projectve modules over R λ Šλ S, and ther graded Grothendeck group K q (Ř λ Šλ S -gpmod). Theorem K q (Ř λ Šλ S -gpmod) = V λ, the Lusztg ntegral form of the Z[q,q 1 ] correspondng representaton over the quantum group. For a general base change, we could end up wth a bgger Grothendeck group.

13 Categorfed smples The deformed part s because people more often thnk about the unque homogeneous quotent where S = Z, where we just set γ,j = 0. Ths base change R λ := Ř λ Šλ Z s the cyclotomc quotent. We can vsualze ths relaton as: α (λ) = 0

14 Categorfed smples Every approprately fnte categorfcaton has a Jordan-Hölder fltraton consstng of smple categorfcatons: Theorem If V s an (abelan and Artnan) categorcal g-module such that V µ = 0 for µ λ for some ν, then V has a unque fltraton by subcategores V λ such that V λ /V <λ s the base change of ˇV λ by an acton of Šλ λ on the subcategory {X V λ E X = 0 for all I} of hghest weght objects. As wth sl 2, one consequence of ths fact s that for any w W, we have that D b (V µ ) = D b (V wµ ). Actually, these equvalences gve an acton of the Artn brad group of g on these derved categores.

15 Geometrc examples The other explanaton of the KLR algebra s a geometrc one. Assume that the Cartan matrx of g s symmetrc, so t s assocated to a quver wth vertces I, and a j edges jonng and j, wth ether orentaton. For a gven dmenson vector d, we can consder the vector space E d := j Hom(C d, C d j ). For each, we have the Lusztg quver flag varety: Fl := {{V j } I,1 j n {0} = V 0 V n = C d wth dm V j /Vj 1 X := {(V, f ) Fl E d f e (V j t(e) ) Vj h(e) } = δ j,}

16 Geometrc examples Lusztg studed the sheaves Y = (π ) C X [dm X ] n hs consderaton of the canoncal bass. Under Rngel s homomorphsm to the Hall algebra, the canoncal bass vectors correspond to the smple consttuents of these sheaves. We can upgrade ths correspondence to the category generated by these sheaves: Theorem (Varagnolo-Vasserot, Rouquer) We have an somorphsm ( ( ) R ν = EndD b (E d /G d ) Y = H G d =ν = j =ν for the polynomals Q j (u, v) = (u v) #j (v u) # j. X Ed X j )

17 Geometrc examples Ths means that we can readly cook up actons of the KLR algebra on geometrc categores, and thus g-actons. The most mportant source of these s the geometry of Nakajma quver varetes M λ µ: Theorem (Cauts-Kamntzer-Lcata, W.) The category C µ = D b Coh(M λ µ) of coherent sheaves on M λ µ carres a categorcal g acton. The acton s by Fourer-Muka transform wth sheaves E and F supported on Nakajma s Hecke correspondences. Ths the classcal lmt of a smlar acton on quantum coherent sheaves D b Coh (M λ µ). Ths should extend further to the Fukaya category of M λ µ wth Hecke correspondences themselves thought of as Lagrangan correspondences.

18 Canoncal bases By applyng the unqueness theorem, we can actually understand a pece of these categores: Proposton The subcategory C generated under the categorfcaton functors by sheaves on M λ λ = n D b Coh(M λ µ) or D b Coh (M λ µ) s equvalent to the category of perfect dg-modules over R λ µ Z C. Ths geometrc realzaton allows us to connect these categorfcatons and Lusztg s canoncal bases: Theorem (Varagnolo-Vasserot, W.) Lusztg s canoncal bass of V λ corresponds to smple objects n the heart of C, and (part of) the canoncal bass of U to the smples n the subcategory of Coh (M λ µ M λ µ ) generated by E and F under convoluton.

19 Canoncal bases Thus, we also have an algebrac realzaton of these canoncal bases. Theorem (Varagnolo-Vasserot, W.) Wth the parameters we have fxed, the classes of ndecomposable self-dual projectve modules over R λ Z Q correspond to Lusztg s canoncal bass. In fnte type, the canoncal bass of U q corresponds to the ndecomposable 1-morphsms of U g Z Q. If we take R λ Z F p, then for small prmes we obtan a dfferent p-canoncal bass. If we choose dfferent parameters Q j, then we can also change ths bass. For those of you who worry about these thngs: Q and F p aren t mportant; passng to an overfeld doesn t change the classes (all ndec projectves are absolutely ndec).

20 Connectons to Hecke algebras There s an alternate way of understandng categorcal actons for (affne) type A, usng the (degenerate) affne Hecke algebra. Defnton The degenerate affne Hecke algebra h m of rank m s generated by t 1,..., t m 1, x 1,..., x m wth relatons: t 2 = 1 t t ±1 t = t ±1 t t ±1 t t j = t j t ( j ± 1) x x j = x j x t x t = x +1 t x t j = t x j ( j, j + 1)

21 Connectons to Hecke algebras There s an alternate way of understandng categorcal actons for (affne) type A, usng the (degenerate) affne Hecke algebra. Defnton The affne Hecke algebra H m (v) of rank m s generated by T 1,..., T m 1, X ±1 1,..., X±1 m wth relatons: (T + 1)(T v) = 0 T T ±1 T = T ±1 T T ±1 T T j = T j T ( j ± 1) X X j = X j X T X T = vx +1 X T j = T X j ( j, j + 1)

22 Connectons to Hecke algebras Fx a feld k and let U k be a fnte subset. We gve U a graph structure by addng an edge u 1 u 2 f u 2 = vu 1. Defnton Fx an nteger l 0 (the level). Consder an k-lnear addtve category C equpped wth adjont functors E, F : C C for each I that satsfy the relatons EF = FE d l natural transformatons X : E E and T : EE EE that generate an acton of the affne Hecke algebra wth v 1, wth X satsfyng a mnmal polynomal wth roots n U k. The generalzed egenspaces F = z U F z, E = z U E z for X defne a categorcal acton of g U.

23 Connectons to Hecke algebras Fx a feld k and let u k be a fnte subset. We gve u a graph structure by addng an edge u 1 u 2 f u 2 = u Defnton Fx an nteger l 0 (the level). Consder an k-lnear addtve category C equpped wth adjont functors E, F : C C for each I that satsfy the relatons EF = FE d l natural transformatons x: E E and t : EE EE that generate an acton of the degenerate affne Hecke algebra, wth x satsfyng a mnmal polynomal wth roots n u k. The generalzed egenspaces F = z u F z, E = z u E z for x defne a categorcal acton of g u.

24 Connectons to Hecke algebras The proof s an explct somorphsm between completons of the (degenerate) affne Hecke algebra fxng the spectrum of x /X and the KLR algebra of the correspondng graph completed wth respect to ts gradng. Ths sn t precsely easy, but t s a straghtforward computaton wth the polynomal representatons of these algebras. Note that ths somorphsm badly breaks the gradng: the gradng s extremely non-obvous from the Hecke perspectve. One partcularly nterestng consequence s an somorphsm between the cyclotomc KLR algebra R ω 0 Z F p for ŝl p and the group algebra F p [S n ], puttng a surprsng gradng on the latter.

25 Connectons to Hecke algebras Ths theorem has a number of nterestng applcatons: There s a level 1 acton on the category n k[s n ] -mod wth F, E the nducton and restrcton functors wth u = Im(Z k). There s a level l = deg(p) acton on modules over degenerate cyclotomc Hecke algebras m h m /p(x 1 ) -mod wth u = Z + {roots of p}. For any category of U(gl N )-modules wth sutable fnteness propertes (e.g. category O), the functors F = V, E = V for V the vector representaton defnes a level 0 acton wth u the spectrum of the Casmr on modules n ths category. Ths approach also works for ratonal representons of the algebrac group.

26 Connectons to Hecke algebras Ths theorem has a number of nterestng applcatons: There s a level 1 acton on the category n H fn n (v) -mod wth F, E the nducton and restrcton functors wth U = v Z. There s a level l = deg(p) acton on modules over cyclotomc Hecke algebras m H m (v)/p(x 1 ) -mod wth U = v Z {roots of p}. For any category of U v (gl N )-modules wth sutable fnteness propertes (e.g. category O), the functors F = V, E = V for V the vector representaton defnes a level 0 acton wth U the spectrum of the quantum Casmr on modules n ths category. Ths approach also works for ratonal representons of the algebrac group.

27 Connectons to Hecke algebras For a fnte feld F, nducton and restrcton functors nduce a level 1 acton on k[gl(n, F)] -mod. The set U s the subgroup of k generated by #F. Recent work of Dudas-Varagnolo-Vasserot extends ths to level 2 acton on modules over the fnte untary group k[u(n, F)] -mod, wth U the subgroup of k generated by #F. There s a level l acton on the categores O over the ratonal Cherednk algebra for the complex reflecton group Z/lZ S n for n 0. The functors F, E are the Bezrukavnkov-Etngof nducton and restrcton functors; the set U s the same as for the assocated cyclotomc Hecke algebras assocated under the Knzhnk-Zamolodchkov functor: U = l k=1 e2πk(z+s ).

28 Connectons to Hecke algebras The categorfcaton m h m /p(x 1 ) -mod s rreducble: the underlyng Grothendeck groups gve all the hghest weght rreps of ŝl e f k s characterstc e and all roots of p are n F p and of sl f k s characterstc 0 and all roots of p are n Z. Thus, any categorfcaton of a smple module over these Le algebras and felds can be realzed usng cyclotomc degenerate Hecke algebras. On the other hand, the other cases we consdered categorfy Fock spaces, whch are not rreducble. Thus, we do not have a unqueness theorem that allows us to characterze them.

29 Connectons to Hecke algebras The categorfcaton m H m (v)/p(x 1 ) -mod s rreducble: the underlyng Grothendeck groups gve all the hghest weght rreps of ŝl e f v s a prmtve eth root of unty and the roots of p are v-connected and of sl f v has nfnte multplcatve order and the roots of p are v-connected. Thus, any categorfcaton of a smple module over these Le algebras and felds can be realzed usng cyclotomc Hecke algebras. On the other hand, the other cases we consdered categorfy Fock spaces, whch are not rreducble. Thus, we do not have a unqueness theorem that allows us to characterze them.

30 Connectons to Hecke algebras Actually, there are dfferent categores whch correspond to the same underlyng representaton (for example, dfferent parameters for Cherednk algebras). We genunely need more data n order to pn down our category. Luckly, ths data exsts! Next tme, we ll dscuss how to fx unqueness.