CALOGERO-MOSER VERSUS KAZHDAN-LUSZTIG CELLS CE´DRIC BONNAFE´ AND RAPHAE¨L ROUQUIER 2 1. Introduction 1 0 In [KaLu], Kazhdan and Lusztig developed a combinatorial theory associated with Coxeter 2 groups. They defined in particular partitions of the group in left and two-sided cells. For Weyl n a groups, these have a representation theoretic interpretation in terms of primitive ideals, and J they play a key role in Lusztig’s description of unipotent characters for finite groups of Lie 3 type [Lu3]. Lusztig generalized this theory to Hecke algebras of Coxeter groups with unequal ] parameters [Lu2, Lu4]. T We propose a definition of left cells and two-sided cells for complex reflection groups, based R on ramification theory for Calogero-Moser spaces. These spaces have been defined via rational . h Cherednik algebras by Etingof and Ginzburg [EtGi]. We conjecture that these coincide with t a Kazhdan-Lusztig cells, for real reflection groups. Counterparts of families of irreducible char- m acters have been studied by Gordon and Martino [GoMa], and we provide here a version of left [ cell representations. The Calogero-Moser cells are studied in detail in [BoRou]. 1 v 5 8 2. Calogero-Moser spaces and cells 5 0 2.1. Rational Cherednik algebras at t = 0. Let us recall some constructions and results . 1 from [EtGi]. Let V be a finite-dimensional complex vector space and W a finite subgroup of 0 GL(V). Let betheset ofreflections of W, i.e., elements g such that ker(g 1)is ahyperplane. 2 S − 1 We assume that W is a reflection group, i.e., it is generated by . : S v We denote by / the quotient of by the conjugacy action of W and we let c be Xi a set of indetermSina∼tes. We put A = CS[CS/∼] = C[ c ]. Given s , let v{ s}sV∈S(/r∼esp. { s}s∈S/∼ ∈ S s ∈ r αs V∗) be an eigenvector for s associated to the non-trivial eigenvalue. a T∈he 0-rational Cherednik algebra H is the quotient of A T(V V∗)⋊W by the relations ⊗ ⊕ [x,x′] = [ξ,ξ′] = 0 x,α v ,ξ [ξ,x] = c h si·h s is for x,x′ V∗ and ξ,ξ′ V. X s v ,α ∈ ∈ s s s∈S h i We put Q = Z(H) and P = A S(V∗)W S(V)W Q. The ring Q is normal. It is a free ⊗ ⊗ ⊂ P-module of rank W . | | Date: January 4, 2012. 1 2 CE´DRICBONNAFE´ AND RAPHAE¨LROUQUIER 2.2. Galois closure. Let K = Frac(P) and L = Frac(Q). Let M be a Galois closure of the extension L/K and R the integral closure of Q in M. Let G = Gal(M/K) and H = Gal(M/L). Let = SpecP = AS/∼ V/W V∗/W, = SpecQ the Calogero-Moser space, and = C P × × Q R SpecR. We denote by π : the quotient by H, and by Υ : and φ : AS/∼ the C R → Q Q → P P → canonical maps. We put p = Υπ : the quotient by G. R → P 2.3. Ramification. Letr beaprimeidealofR. WedenotebyD(r) Gitsdecomposition ∈ R ⊂ group and by I(r) D(r) its inertia group. ⊂ We have a decomposition into irreducible components = , where = (x,π(g−1(x))) x R×P Q [ Og Og { | ∈ R} g∈G/H inducing a decomposition into irreducible components V(r) = (r), where (r) = (x,π(g−1g′(x))) x V(r), g′ I(r) . ×P Q a Og Og { | ∈ ∈ } g∈I(r)\G/H 2.4. Undeformed case. Let p = φ−1(0) = P c . We have P/p = C[V V∗]W×W, 0 Ps∈S/∼ s 0 ⊕ Q/p Q = C[V V∗]∆W, where ∆(W) = (w,w) w W W W. A Galois closure of the 0 ⊕ { | ∈ } ⊂ × extension of C(p Q) = C(V V∗)∆W over C(p ) = C(V V∗)W×W is C(V V∗)∆Z(W). 0 0 ⊕ ⊕ ⊕ Let r above p . Since p Q is prime, we have G = D(r )H = HD(r ) and I(r ) = 1. Fix 0 0 0 0 0 0 an isomo∈rpRhism ι : C(r ) ∼ C(V V∗)∆Z(W) extending the canonical isomorphism of C(p Q) 0 0 → ⊕ with C(V V∗)∆W. ⊕ ∼ The application ι induces an isomorphism D(r ) (W W)/∆Z(W), that restricts to an 0 ∼ → × ∼ isomorphism D(r ) H ∆W/∆Z(W). This provides a bijection G/H (W W)/∆W. 0 ∩ → ∼ → × Composing with the inverse of the bijection W (W W)/∆W, w (w,1), we obtain a ∼ → × 7→ bijection G/H W. → From now on, we identify the sets G/H and W through this bijection. Note that this bijection depends on the choices of r and of ι. Since M is the Galois closure of L/K, we have 0 Hg = 1, hence the left action of G on W induces an injection G S(W). Tg∈G ⊂ 2.5. Calogero-Moser cells. Definition 2.1. Let r . The r-cells of W are the orbits of I(r) in its action on W. ∈ R Let c AS/∼. Choose r with p(r ) = c¯ 0 0. The r -cells are called the two-sided C c c c ∈ ∈ R × × Calogero-Moser c-cells of W. Choose now rleft contained in r with p(rleft) = c¯ V/W 0 c ∈ R c c × × ∈ . The rleft-cells are called the left Calogero-Moser c-cells of W. We have I(rleft) I(r ). P c c ⊂ c Consequently, every left cell is contained in a unique two-sided cell. The map sending w W to π(w−1(r )) induces a bijection from the set of two-sided cells to c ∈ Υ−1(c 0 0). × × CALOGERO-MOSER VERSUS KAZHDAN-LUSZTIG CELLS 3 2.6. Families and cell multiplicities. Let E be an irreducible representation of C[W]. We extend it to a representation of S(V)⋊W by letting V act by 0. Let 1 H ∆(E) = e·IndS(V)⋊W(A⊗C E), where e = W X w, | | w∈W be the spherical Verma module associated with E. It is a Q-module. Let c AS/∼ and let ∆left(E) = (R/rleft) ∆(E). ∈ C c ⊗P Definition 2.2. Given Γ a leftcell, we define the cell multiplicity m (E) of E as the multiplicity Γ of ∆left(E) at the component (rleft). OΓ c Note that m (E) [ (rleft)] is the support cycle of ∆left(E). PΓ Γ · OΓ c There is a unique two-sided cell Λ containing all left cells Γ such that m (E) = 0. Its image Γ 6 in is the unique q Υ−1(c 0 0) such that (Q/q) ∆(E) = 0. The corresponding map Q Q ∈ × × ⊗ 6 Irr(W) Υ−1(c 0 0) is surjective, and its fibers are the Calogero-Moser families of Irr(W), → × × as defined by Gordon [Go1]. 2.7. Dimension 1. Let V be a one-dimensional complex vector space, let d 2 and let W be ≥ the group of d-th roots of unity acting on V. Let ζ = exp(2iπ/d), let s = ζ W and c = c ∈ i si for 1 i d 1. We have A = C[c ,...,c ] and ≤ ≤ − 1 d−1 d−1 H = A x,ξ,s sxs−1 = ζ−1x, sξs−1 = ζξ and [ξ,x] = c si . h | X i i i=1 Let eu = ξx d−1(1 ζi)−1c si. We have P = A[xd,ξd] and Q = A[xd,ξd,eu]. Define − Pi=1 − i κ ,...,κ = κ by κ + +κ = 0 and d−1c si = d−1(κ κ )ε , where ε = 1 d−1ζijsj. 1 d 0 1 ··· d Pi=1 i Pi=0 i− i+1 i i d Pj=0 We have A = C[κ ,...,κ ]/(κ + +κ ). 1 d 1 ··· d The normalization of the Galois closure is described as follows. There is an isomorphism of A-algebras d A[X,Y,Z]/ XY (Z κ ) ∼ Q, X xd, Y ξd and Z eu. (cid:0) −Y − i (cid:1) → 7→ 7→ 7→ i=1 We have an isomorphism of A-algebras A[X,Y,λ ,...,λ ]/ e (λ) = e (κ),...,e (λ) = e (κ),e (λ) = e (κ)+( 1)d+1XY ∼ R 1 d 1 1 d−1 d−1 d d (cid:0) − (cid:1) → where Z = λ andwhere e denotes the i th elementary symmetric function. Wehave G = S , d i d − acting by permuting the λ ’s, and H = S . i d−1 Let p = (κ ,...,κ ) SpecP and r = (κ ,...,κ ,λ ζλ ,...,λ ζd−1λ ) SpecR. 0 1 d ∈ 0 1 d 1 − d d−1 − d ∈ We have D(r ) = (1,2,...,d) S and C(r ) = C(X,Y,λ = √d XY) = C(X,Y,Z = 0 d 0 d √d XY). The compohsite bijectioni ⊂D(r ) ∼ G/H ∼ W is an isomorphism of groups given by 0 → → (1,...,d) s. 7→ Fix c Cd−1 and let κ ,...,κ C corresponding to c. Consider r = r or rleft as in 2.5. ∈ 1 d ∈ c c § Then I(r) is the subgroup of S stabilizing (κ ,...,κ ). The left c-cells coincide with the two- d 1 d sided c-cells and two elements si and sj are in the same cell if and only if κ = κ . Finally, the i j multiplicity m (detj) is 1 if sj Γ and 0 otherwise. Γ ∈ 4 CE´DRICBONNAFE´ AND RAPHAE¨LROUQUIER 3. Coxeter groups 3.1. Kazhdan-Lusztig cells. Following Kazhdan-Lusztig [KaLu] and Lusztig [Lu2, Lu4], let us recall the construction of cells. We assume here V is the complexification of a real vector space VR acted on by W. We choose a connected component C of VR ker(s 1) and we denote by S the set of s such that ker(s 1) C¯ has codimension−1Sins∈SC¯. This−makes (W,S) into a Coxeter group, ∈anSd − ∩ we denote by l the length function. Let Γ be a totally ordered free abelian group and let L : W Γ be a weight function, i.e., → a function such that L(ww′) = L(w) + L(w′) if l(ww′) = l(w) + l(w′). We denote by vγ the element of the group algebra Z[Γ] corresponding to γ Γ. ∈ We denote by H the Hecke algebra of W: this is the Z[Γ]-algebra generated by elements T s with s S subject to the relations ∈ (T vL(s))(T +v−L(s)) = 0 and T T T = T T T for s,t S with m = s s s t s t s t st − ··· ··· ∈ 6 ∞ mst terms mst terms | {z } | {z } where m is the order of st. Given w W, we put T = T T , where w = s s is a st ∈ w s1··· sn 1··· n reduced decomposition. Let i be the ring involution of H given by i(vγ) = v−γ for γ Γ and i(T ) = T−1. We denote ∈ s s by C the Kazhdan-Lusztig basis of H. It is uniquely defined by the properties that w w∈W { } i(C ) = C and C T Z[Γ ]T . w w w − w ∈ Lw′∈W <0 w′ We introduce the partial order on W. It is the transitive closure of the relation given L ≺ by w′ w if there is s S such that the coefficient of C in the decomposition of C C in L w′ s w ≺ ∈ the Kazhdan-Lusztig basis is non-zero. We define w w′ to be the corresponding equivalence L ∼ relation: w w′ if and only if w w′ and w′ w. The equivalence classes are the left L L L ∼ ≺ ≺ cells. We define as the partial order generated by w w′ if w w′ or w−1 w′−1. LR LR L L ≺ ≺ ≺ ≺ As above, we define an associated equivalence relation . Its equivalence classes are the LR ∼ two-sided cells. When Γ = Z, L = l, and W is a Weyl group, a definition of left cells based on primitive ideals in enveloping algebras was proposed by Joseph [Jo]: let g be a complex semi-simple Lie algebra with Weyl group W. Let ρ be the half-sum of the positive roots. Given w W, let I w ∈ be the annihilator in U(g) of the simple module with highest weight w(ρ) ρ. Then, w and − − w′ are in the same left cell if and only if I = I . w w′ 3.2. Representations and families. Let Γ be a left cell. Let W (resp. W ) be the set ≤Γ <Γ of w W such that there is w′ Γ with w w′ (resp. w w′ and w Γ). The left cell L L ∈ ∈ ≺ ≺ 6∈ representation of W over C associated with Γ [KaLu, Lu4] is the unique representation, up to isomorphism, that deforms into the left H-module Z[Γ]C / Z[Γ]C . (cid:16) M w(cid:17) (cid:16) M w(cid:17) w∈W≤Γ w∈W<Γ Lusztig [Lu1, Lu4] has defined the set of constructible characters of W inductively as the smallest set of characters with the following properties: it contains the trivial character, it is stable under tensoring by the sign representation and it is stable under J-induction from a CALOGERO-MOSER VERSUS KAZHDAN-LUSZTIG CELLS 5 parabolic subgroup. Lusztig’s families are the equivalences classes of irreducible characters of W for the relation generated by χ χ′ if χ and χ′ occur in the same constructible character. ∼ Lusztig has determined constructible characters and families for all W and all parameters. Lusztig has shown for equal parameters, and conjectured in general, that the set of left cell characters coincides with the set of constructible characters. 3.3. A conjecture. Let c RS/∼. Let Γ be the subgroup of R generated by Z and c . s s∈S ∈ { } We endow it with the natural order on R. Let L : W Γ be the weight function determined → by L(s) = c if s S. s ∈ The following conjecture is due to Gordon and Martino [GoMa]. A similar conjecture has been proposed independently by the second author1. It is known to hold for types A , B , D n n n and I (n) [Go2, GoMa, Be, Ma1, Ma2]. 2 Conjecture 3.1. The Calogero-Moser families of irreducible characters of W coincide with the Lusztig families. We propose now a conjecture involving partitions of elements of W, via ramification. The part dealing with left cell characters could be stated in a weaker way, using Q and not R, and thus not needing the choice of prime ideals, by involving constructible characters. Conjecture 3.2. There is a choice of rleft r such that c ⊂ c the Calogero-Moser two-sided cells (resp. left cells) coincide with the Kazhdan-Lusztig • two-sided cells (resp. left cells) the representations m (E)E, where Γ is a Calogero-Moser left cell, coincide • PE∈Irr(W) Γ with the left cell representations of Kazhdan-Lusztig. VariousparticularcasesandgeneralresultssupportingConjecture3.2areprovidedin[BoRou]. In particular, the conjecture holds for W = B , for all choices of parameters. 2 References [Be] G. Bellamy, The Calogero-Moser partition for G(m,d,n), preprint arXiv:0911.0066, to appear in Nagoya Math. J. [BoRou] C. Bonnaf´e and R. Rouquier, Calogero-Moser cells, in preparation. [EtGi] P.EtingofandV.Ginzburg,Symplecticreflectionalgebras,Calogero-Moser spaceanddeformedHarish- Chandra homomorphism, Inv. Math. 147 (2002), 243–348. [Go1] I. Gordon, Baby Verma modules for rational Cherednik algebras, Bull. London Math. Soc. 35 (2003), 321–336. [Go2] I.Gordon,Quiver varieties, category for rational Cherednik algebras, and Heckealgebras,Int.Math. O Res. Papers (2008), Article ID rpn006, 69 pages. [GoMa] I.GordonandM.Martino,Calogero-Moser space, restrictedrationalCherednik algebras, andtwo-sided cells, Math. Res. Lett. 16 (2009), 255–262. [Jo] A. Joseph, Goldie rank in the enveloping algebra of a semisimple Lie algebra. I, II, J. Algebra 65 (1980), 269–283,284–306. [KaLu] D. Kazhdan and G. Lusztig, Representations of Coxeter groups and Hecke algebras, Inv. Math. 53 (1979), 165–184. [Lu1] G.Lusztig,Aclass ofirreduciblerepresentations ofaWeylgroup.II,Indag.Math.44(1982),219–226. [Lu2] G.Lusztig,Leftcells in Weylgroups,in“Liegrouprepresentations,I”,99–111,LectureNotesinMath. 1024, Springer, Berlin, 1983 1Talk at the Enveloping algebra seminar, Paris, December 2004. 6 CE´DRICBONNAFE´ AND RAPHAE¨LROUQUIER [Lu3] G. Lusztig, “Characters of reductive groups over a finite field”, Ann. of Math. Studies, vol. 107, Princeton Univ. Press, 1984. [Lu4] G. Lusztig, “Hecke algebras with unequal parameters”, American Mathematical Society, 2003. [Ma1] M. Martino, The Calogero-Moser partition and Rouquier families for complex reflection groups, J. Algebra 323 (2010), 193–205. [Ma2] M. Martino, Blocks of restricted rational Cherednik algebras for G(m,d,n), preprint arXiv:1009.3200. C´edric Bonnaf´e : Universit´e Montpellier 2, Institut de Math´ematiques et de Mod´elisation deMontpellier,CaseCourrier051,PlaceEug`eneBataillon,34095MontpellierCedex,FRANCE E-mail address: [email protected] Rapha¨el Rouquier : Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford, OX13LB,UKandDepartmentofMathematics, UCLA,Box951555,LosAngeles,CA90095-1555, USA E-mail address: [email protected]